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Polynomial families of quantum semisimple coajoint orbits via deformed quantum enveloping algebras

Mao Hoshino Department of Mathematical Sciences, The University of Tokyo
Komaba 3-8-1, Tokyo 153-8914, Japan mhoshino@ms.u-tokyo.ac.jp

Abstract.

We construct a polynomial family of semisimple left module categories over the representation category of the Drinfeld-Jimbo deformation, with the fusion rule of the representation category of each Levi subalgebra. In this construction we perform a kind of generalized parabolic induction using a deformed quantum enveloping algebra, whose definition depends on an arbitrary choice of a positive system and corresponds to De Commer’s definition for the standard positive system. These algebras define a sheaf of algebras on the toric variety associated to the root system, which contains the moduli of equivariant Poisson brackets. This fact finally produces the family of $2$ -cocycle. We also obtain a comparison theorem between our module categories and module categories induced from our construction for intermediate Levi subalgebras. The construction of deformed quantum enveloping algebras and the comparison theorem are discussed in the integral setting of Lusztig’s sense.

Key words and phrases:

quantum group, deformation quantization, representation theory, toric variety

2020 Mathematics Subject Classification:

Primary 17B37, Secondary 17B10

This work was supported by JSPS KAKENHI Grant Number JP23KJ0695 and WINGS-FoPM Program at the University of Tokyo.

1. Introduction

Deformation quantization is one of rigorous formulations of quantization, which naturally arose in quantum physics. This physical background is a reason why we typically consider an algebra of suitable functions on a space with a Poisson structure as an algebra to be quantized, though it is possible to introduce the notion of deformation quantization for general Poisson algebras. Actually finding and classifying deformation quantizations of such function algebras is a natural problem to consider and has been attracting many physicists and mathematicians. One of most successful results was accomplished by Kontsevich ([MR2062626]), who proved the existence of deformation quantizations of each Poisson manifold and their classification theorem. See [MR1321655] for developments before Kontsevich’s work, including Fedosov’s construction for deformation quantizations of symplectic manifolds ([MR1293654]).

Theory of quantum groups is also a relevant and important branch of mathematical physics around quantization. Not surprisingly, deformation quantization deeply relates to it. For instance a Drinfeld-Jimbo deformation $U_{q}(\mathfrak{g})$ of a complex semisimple Lie algebra $\mathfrak{g}$ is a kind of dual of a quantization $\mathcal{O}_{q}(G)$ of the function algebra on the complex semisimple Lie group $G$ with respect to the so-called standard Poisson bracket. This fact invokes a further topic: equivariant quantization of actions of complex semisimple Lie groups as actons of $U_{q}(\mathfrak{g})$ , or equivalently coactions of $\mathcal{O}_{q}(G)$ , on quantized spaces. One of important researches on this direction is Podleś’s study [MR0919322] on equivariant quantizations of $2$ -sphere. Nowadays his work can be understood as an example of quantized compact symmetric spaces, initiated by Letzter ([MR1913438]).

Nontheless, in this paper, we would like to explore an alternative approach to generalizing the Podleś quantum spheres, focusing on quantum semisimple coadjoint orbits, or equivalently quantum flag manifolds. The latter terminology is used in papers considering the algebra of smooth or continuous functions on partial flag manifolds with the action of the compact real form $K$ of $G$ , for instance ([MR3376147, MR1697598]). In this case a $\ast$ -structure also seems to be taken in consideration. On the other hand, the former is used by researchers considering $L\backslash G$ with a Levi subgroup $L$ and the algebra $\mathcal{O}(L\backslash G)$ of algebraic functions on $L\backslash G$ , usually which is not equipped with any $\ast$ -structure.

Quantization of such actions has already been studied by several researchers. In [MR1817512] Donin gives an explicit description of the moduli of all Poisson structures compatible with the action and also shows that such deformation quantizaions can be classified by the set of formal paths on the moduli space. We explain this fact in more detail. Fix a Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ and consider the associated root system $R$ with a positive system $R^{+}$ . The set of simple roots is denoted by $\Delta$ . We also choose a subset $S$ of $\Delta$ , from which we can generate a closed subsystem $R_{S}$ and a Levi subgroup $L_{S}$ . In this setting, Donin shows that the set of Poisson brackets on $L_{S}\backslash G$ compatible with the action of the Poisson-Lie group $G$ can be identified with the set $X_{L_{S}\backslash G}$ of $\varphi=(\varphi_{\alpha})_{\alpha\in R\setminus R_{S}}\in\mathbb{C}^{R\setminus R_{S}}$ satisfying the following relations:

(i)

$\varphi_{-\alpha}=-\varphi_{\alpha}$ for $\alpha\in R\setminus R_{S}$ ,
(ii)

$\varphi_{\alpha}\varphi_{\beta}+1=\varphi_{\alpha+\beta}(\varphi_{\alpha}+\varphi_{\beta})$ when $\alpha,\beta,\alpha+\beta\in R\setminus R_{S}$ ,
(iii)

$\varphi_{\alpha}=\varphi_{\beta}$ when $\alpha,\beta\in R\setminus R_{S}$ and $\alpha-\beta\in R_{S}$ .

He also shows the existence of a holomorphic family $\{\mathcal{O}_{h,\varphi}(L_{S}\backslash G)\}_{\varphi\in X_{L_{S}\backslash G}}$ of equivariant deformation quantizations and that any equivariant deformation quantization is equivalent to $\mathcal{O}_{h,\varphi(h)}(L_{S}\backslash G)$ , where $\varphi(h)\in\mathbb{C}\llbracket h\rrbracket^{R\setminus R_{S}}$ satisfies the same relations (i), (ii), (iii). We would like to emphasize here that his result was establised as a consequence of some vanishing theorems of relevant cohomological obstructions. In particular any explicit construction of those deformations was not presented, at least in the paper.

If $\varphi\in X_{L_{S}\backslash G}$ is generic in the sense that $\varphi_{\alpha}\neq-1$ for all $\alpha\in R\setminus R_{S}$ , Enriquez-Etingof-Marshall ([MR2126485, MR2349621]) and Mudrov ([MR2304470]) give an explicit construction of corresponding equivariant deformation quantizations. Namely they construct left $\operatorname{\mathrm{Rep}}^{\mathrm{f}}_{h}\mathfrak{g}$ -module categories using the parabolic induction twisted by $\mathbb{C}\llbracket h\rrbracket_{\lambda/h}$ , where $\lambda\in\mathfrak{h}^{*}$ is an appropriate weight. It should be mentioned that this approach also appears in the investigation of equivariant quantization with respect to the trivial Poisson-Lie structure ([MR1952112, MR2182701, MR2141466]).

De Commer ([MR3208147]) also followed these pieces of research, with a slight and significant modification, which enables us to discuss quantizations with respect to $\varphi$ with $\varphi_{\alpha}\neq-1$ for $\alpha\in R^{+}\setminus R_{S}^{+}$ . There he uses a deformed quantum enveloping algebra instead of the parabolic induction twisted by $\mathbb{C}\llbracket h\rrbracket_{\lambda/h}$ , which cannot be formulated in this case as the generic case. He also discusses some operator algebraic aspects, motivated by construction of real semisimple quantum groups as locally compact quantum groups.

Contents of this paper

The main purpose is to give an explicit way to construct an algebraic family of $2$ -cocyle twists of left $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{g}$ -module categories $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{l}_{S}$ , parametrized by $X_{L_{S}\backslash G}$ . To achieve this we define a deformed quantum enveloping algebra (Definition 3.6), which is inspired by De Commer’s work. Namely such a deformed quantum enveloping algebra is defined whenever we fix a positive system $R_{0}^{+}$ and a character on $2Q_{0}^{-}$ , the cone generated by $-2R_{0}^{+}$ . If $R_{0}^{+}=R^{+}$ , this definition corresponds to De Commer’s definition. It is worth to remark here that the PBW theorem still holds for such algebras, which involves some observations on quantum PBW vectors in the usual quantum enveloping algebras. This enables us to investigate the twisted parabolic induction defined in Subsection 3.3. Any construction in Section 3 is conducted in the integral setting of Lusztig’s sense.

In Section 4, we discuss a kind of semisimplicity of the twisted parabolic induction in the formal setting. Using this fact we construct left $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{g}$ -module categories which are $2$ -cocycle twists of $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{l}_{S}$ . We also construct quantum semisimple coadjoint orbits using these $2$ -cocycles and determine their semi-classical limits.

In Section 5, we discuss the compatibility of the $2$ -cocycles constructed from different choices of $R_{0}^{+}$ . At the beginning we discuss an immersion of the moduli space $X_{L_{S}\backslash G}$ to the toric scheme $X_{R}$ associated to the root system $R^{+}$ . Then we construct a sheaf $\mathscr{U}_{q,X_{R}}(\mathfrak{g}{}^{\sim})$ of deformed quantum enveloping algebras on $X_{R}$ , which also induces a sheaf $\mathscr{U}_{q,X_{L_{S}\backslash G}}(\mathfrak{g}{}^{\sim})$ . This fact immediately implies that the $2$ -cocycles constructed in Section $4$ can be glued up to a $2$ -cocycle with coefficients in $\mathcal{O}(X_{L_{S}\backslash G})$ .

Finally, in Section 6, we compare quantum semisimple coadjoint orbits constructed in Section 4 and those induced from quantum semisimple coadjoint orbits of intermediate Levi subgroups. Namely we show that an induction functor becomes a left $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{g}$ -module functor in the specific situation.

2. Preliminaries

In this paper $\mathfrak{g}$ and $\mathfrak{h}$ denote a complex semisimple Lie algebra and its Cartan subalgebra respectively. The associated set of roots is denoted by $R$ , which forms a root system with a bilinear form $(\textendash,\textendash)$ on $\mathfrak{h}^{*}$ induced by a normalized Killing form so that $(\alpha,\alpha)=2$ for a short root $\alpha$ . The reflection with respect to $\alpha\in R$ is denoted by $s_{\alpha}$ . The associated Weyl group is denoted by $W$ .

We fix a positive system $R^{+}$ , which induces a triangular decomposition $\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$ and defines a set $\Delta=\{\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{r}\}$ of simple roots. Here $r$ , the number of simple roots, is the rank of $\mathfrak{g}$ . We also set $N=\lvert R^{+}\rvert$ . The set of reflections with respect to simple roots generates $W$ , and defines the length function $\ell\colon W\longrightarrow\mathbb{Z}_{\geq 0}$ . The unique longest element is denoted by $w_{0}$ , whose length is $N$ .

We set $d_{\alpha},\alpha^{\vee},a_{ij}$ as follows:

d_{\alpha}=\frac{(\alpha,\alpha)}{2},\quad\alpha^{\vee}=d_{\alpha}^{-1}\alpha,\quad a_{ij}=(\delta_{i}^{\vee},\delta_{j}).

The fundamental weights, which are dual to $(\varepsilon_{i}^{\vee})_{i}$ with respect to $(\textendash,\textendash)$ , are denoted by $\varpi_{i}$ . The root lattice $Q$ (resp. $P$ ) is $\mathbb{Z}$ -linear span of $\Delta$ (resp. $(\varpi_{i})_{i}$ ). We also use the positive cone $Q^{+}$ and $P^{+}$ :

	$\displaystyle Q^{+}$	$\displaystyle=\mathbb{Z}_{\geq 0}\varepsilon_{1}+\mathbb{Z}_{\geq 0}\varepsilon_{2}+\cdots+\mathbb{Z}_{\geq 0}\varepsilon_{r},$
	$\displaystyle P^{+}$	$\displaystyle=\mathbb{Z}_{\geq 0}\varpi_{1}+\mathbb{Z}_{\geq 0}\varpi_{2}+\cdots+\mathbb{Z}_{\geq 0}\varpi_{r}.$

We usually replace $\varepsilon_{i}$ by the symbol $i$ when $\varepsilon_{i}$ appears as a subscript. For instance we use $s_{i},d_{i},K_{i}$ instead of using $s_{\varepsilon_{i}},d_{\varepsilon_{i}},K_{\varepsilon_{i}}$ .

For $S\subset\Delta$ , we define $R_{S}$ as $R\cap\operatorname{\mathrm{span}}_{\mathbb{Z}}S$ and $R_{S}^{+}$ as $R^{+}\cap R_{S}$ . The number of positive roots in $R_{S}$ is denote by $N_{S}$ . Then we have Lie subalgebras $\mathfrak{n}_{S}^{\pm}=\bigoplus_{\alpha\in R_{S}^{+}}\mathfrak{g}_{\pm\alpha}$ , $\mathfrak{b}_{S}^{\pm}=\mathfrak{h}\oplus\mathfrak{n}_{S}^{\pm}$ , $\mathfrak{l}_{S}=\mathfrak{n}_{S}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{S}^{+}$ , $\mathfrak{u}_{S}^{\pm}=\bigoplus_{\alpha\in R^{+}\setminus R_{S}^{+}}\mathfrak{g}_{\pm\alpha}$ , $\mathfrak{p}_{S}=\mathfrak{l}_{S}\oplus\mathfrak{u}_{S}^{+}$ . We also define $Q_{S},P_{S},Q_{S}^{+},P_{S}^{+}$ for each subset $S\subset\Delta$ .

For $q$ -integers, we use the following symbols:

q_{\alpha}=q^{d_{\alpha}},\quad[n]_{q}=\frac{q^{n}-q^{-n}}{q-q^{-1}},\quad[n]_{q}!=[1]_{q}[2]_{q}\cdots[n]_{q},\quad\begin{bmatrix}n\\ k\end{bmatrix}_{q}=\frac{[n]_{q}!}{[k]_{q}![n-k]_{q}!}

A quantum commutator is defined for vectors in suitable algebras which admits a weight space decomposition as stated in the next section:

\displaystyle[x,y]_{q}=xy-q^{-(\operatorname{\mathrm{wt}}{x},\operatorname{\mathrm{wt}}{y})}yx.

Finally we prepare some notations on a multi-index $\Lambda=(\lambda_{i})_{i}\in\mathbb{Z}_{\geq 0}^{n}$ .

•

$\lvert\Lambda\rvert=\sum_{i}\lambda_{i}$ .
•

$\operatorname{\mathrm{supp}}\Lambda=\{i\mid\lambda_{i}\neq 0\}$ .
•

$\Lambda\subset(k,l)\overset{\text{def}}{\Longleftrightarrow}\operatorname{\mathrm{supp}}\Lambda\subset\{k+1,k+2,\cdots,l-1\}$ . For an interval $I$ , like $[k,l]$ , $\Lambda\subset I$ is defined in a similar way.
•

$\Lambda<k\overset{\text{def}}{\Longleftrightarrow}\Lambda\subset(0,k)$ . Similarly $\Lambda\leq k,\Lambda>k,\Lambda\geq k$ are defined.
•

$\Lambda\cdot\alpha=\sum_{i}\lambda_{i}\alpha_{i}$ for a sequence $(\alpha_{i})_{i}$ of vectors.
•

$x^{\Lambda}=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{n}^{\lambda_{n}}$ for a sequence $(x_{i})_{i}$ in a (possibly non-commutative) ring.

2.1. Quantum enveloping algebras of semisimple Lie algebras

Basically we refer the convension in [MR4162277] and [MR1492989]. Though our main purpose is to construct deformation quantizations, we will present a definition of the deformed quantum enveloping algebra in the integral form for generality. Therefore begin with the definition of the Drinfeld-Jimbo deformation over $\mathbb{Q}(s)$ , the function field of one variables over $\mathbb{Q}$ with a deformation parameter $q=s^{2L}$ where $L$ is the smallest positive integer such that $(\lambda,\mu)\in L^{-1}\mathbb{Z}$ for any $\lambda,\mu\in P$ . We set $q^{r}:=s^{2Lr}$ for $r\in(2L)^{-1}\mathbb{Z}$ and $q_{i}=q^{d_{i}}$ . The Drinfeld-Jimbo deformation of $\mathfrak{g}$ is a Hopf algebra $U_{q}(\mathfrak{g})$ generated by $E_{i},F_{i},K_{\lambda}$ for $1\leq i\leq r$ and $\lambda\in P$ , with relations

	$\displaystyle K_{0}=1,$	$\displaystyle K_{\lambda}E_{i}K_{\lambda}^{-1}=q^{(\lambda,\varepsilon_{i})}E_{i},$	$\displaystyle[E_{i},F_{j}]=\delta_{ij}\frac{K_{i}-K_{i}^{-1}}{q_{i}-q_{i}^{-1}},$
	$\displaystyle K_{\lambda}K_{\mu}=K_{\lambda+\mu},$	$\displaystyle K_{\lambda}F_{i}K_{\lambda}^{-1}=q^{-(\lambda,\varepsilon_{i})}F_{i},$

and the quantum Serre relations:

	$\displaystyle\sum_{k=0}^{1-a_{ij}}(-1)^{k}\begin{bmatrix}1-a_{ij}\\ k\end{bmatrix}_{q_{i}}E_{i}^{1-a_{ij}-k}E_{j}E_{i}^{k}=0,$
	$\displaystyle\sum_{k=0}^{1-a_{ij}}(-1)^{k}\begin{bmatrix}1-a_{ij}\\ k\end{bmatrix}_{q_{i}}F_{i}^{1-a_{ij}-k}F_{j}F_{i}^{k}=0.$

The coproduct $\Delta$ , the antipode $S$ and the counit $\varepsilon$ are given as follows on the generators:

$\displaystyle\Delta(K_{\lambda})=K_{\lambda}\otimes K_{\lambda},$	$\displaystyle S(K_{\lambda})=K_{\lambda}^{-1},$	$\displaystyle\varepsilon(K_{\lambda})=1,$
$\displaystyle\Delta(E_{i})=E_{i}\otimes K_{i}+1\otimes E_{i},$	$\displaystyle S(E_{i})=-E_{i}K_{i}^{-1},$	$\displaystyle\varepsilon(E_{i})=0,$
$\displaystyle\Delta(F_{i})=F_{i}\otimes 1+K_{i}^{-1}\otimes F_{i},$	$\displaystyle S(F_{i})=-K_{i}F_{i},$	$\displaystyle\varepsilon(F_{i})=0.$

Next we introduce some subalgebras of $U_{q}(\mathfrak{g})$ . The most fundamental ones are $U_{q}(\mathfrak{n}^{+}),U_{q}(\mathfrak{n}^{-})$ and $U_{q}(\mathfrak{h})$ , which are genereted by $E_{i},F_{i},K_{\lambda}$ respectively. These allow us to decompose $U_{q}(\mathfrak{g})$ into the tensor products $U_{q}(\mathfrak{n}^{\pm})\otimes U_{q}(\mathfrak{h})\otimes U_{q}(\mathfrak{n}^{\mp})$ via the multiplication maps. We also use $U_{q}(\mathfrak{b}^{\pm})$ for the subalgebras generated by $U_{q}(\mathfrak{h})$ and $U_{q}(\mathfrak{n}^{\pm})$ respectively. Note that $U_{q}(\mathfrak{b}^{\pm})$ are Hopf subalgebras of $U_{q}(\mathfrak{g})$ .

For a $U_{q}(\mathfrak{h})$ -module $M$ and $v\in M$ , we say that $v$ is a weight vector of weight $\lambda\in P$ when $K_{\mu}v=q^{(\mu,\lambda)}v$ for all $\mu\in P$ . In this case $\lambda$ is denoted by $\operatorname{\mathrm{wt}}v$ . The submodule of elements of weight $\lambda$ is denoted by $M_{\lambda}$ . To consider the weight of an element of $U_{q}(\mathfrak{g})$ , we regard $U_{q}(\mathfrak{g})$ as a $U_{q}(\mathfrak{h})$ -module by the restriction of the left adjoint action $x\triangleright y=x_{(1)}yS(x_{(2)})$ .

Next we describe the braid group action on $U_{q}(\mathfrak{g})$ and the quantum PBW bases. At first we have an algebra automorphism $\mathcal{T}_{i}$ on $U_{q}(\mathfrak{g})$ for each $\delta_{i}\in\Delta$ , which satisfies

\displaystyle\mathcal{T}_{i}(K_{\lambda})=K_{s_{i}(\lambda)},\quad\mathcal{T}_{i}(E_{i})=-K_{i}F_{i},\quad\mathcal{T}_{i}(F_{i})=-E_{i}K_{i}^{-1}

and other formulae in [MR4162277, Theorem 3.58] which determine $\mathcal{T}_{i}$ uniquely.

Then the family $(\mathcal{T}_{i})_{i}$ satisfies the Coxeter relations and defines a braid group action on $U_{q}(\mathfrak{g})$ . Especially we have $\mathcal{T}_{w}$ for each $w\in W$ , which is given by $\mathcal{T}_{w}=\mathcal{T}_{i_{1}}\mathcal{T}_{i_{2}}\cdots\mathcal{T}_{i_{\ell(w)}}$ where $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell(w)}}$ is a reduced expression.

This action produces PBW bases of $U_{q}(\mathfrak{g})$ . Let $w_{0}$ be the longest element in $W$ and fix its reduced expression $w_{0}=s_{\boldsymbol{i}}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{N}}$ , where $\boldsymbol{i}=(i_{1},i_{2},\dots,i_{N})$ . Then each $\alpha\in R^{+}$ has a unique positive integer $k\leq N$ with $\alpha=\alpha^{\boldsymbol{i}}_{k}:=s_{i_{1}}s_{i_{2}}\cdots s_{i_{k-1}}(\varepsilon_{i_{k}})$ . Finally we set $E_{\boldsymbol{i},\alpha}$ and $F_{\boldsymbol{i},\alpha}$ , the quantum root vectors, as follows:

	$\displaystyle E_{\boldsymbol{i},\alpha}$	$\displaystyle=E_{\boldsymbol{i},k}:=\mathcal{T}_{s_{i_{1}}s_{i_{2}}\cdots s_{i_{k-1}}}(E_{i_{k}})=\mathcal{T}_{s_{i_{1}}}\mathcal{T}_{s_{i_{2}}}\cdots\mathcal{T}_{s_{i_{k-1}}}(E_{i_{k}}),$
	$\displaystyle F_{\boldsymbol{i},\alpha}$	$\displaystyle=F_{\boldsymbol{i},k}:=\mathcal{T}_{s_{i_{1}}s_{i_{2}}\cdots s_{i_{k-1}}}(F_{i_{k}})=\mathcal{T}_{s_{i_{1}}}\mathcal{T}_{s_{i_{2}}}\cdots\mathcal{T}_{s_{i_{k-1}}}(F_{i_{k}}).$

Though these elements depend on $\boldsymbol{i}$ , we still have an analogue of the Poincaré-Birkhoff-Witt theorem in $U_{q}(\mathfrak{g})$ i.e. $\{F_{\boldsymbol{i}}^{\Lambda^{-}}K_{\mu}E_{\boldsymbol{i}}^{\Lambda^{+}}\}_{\Lambda^{\pm},\mu}$ forms a basis of $U_{q}(\mathfrak{g})$ . Each element of this family is called a quantum PBW vector.

2.2. The Drinfeld pairing and the quantum Killing form

Next we recall the Drinfeld pairing and the quantum Killing form. The Drinfeld pairing is a unique bilinear form $\tau\colon U_{q}(\mathfrak{b}^{+})\times U_{q}(\mathfrak{b}^{-})\longrightarrow k$ with

\displaystyle\tau(K_{\lambda},K_{\mu})=q^{-(\lambda,\mu)},\quad\tau(E_{i},K_{\lambda})=0=\tau(F_{i},K_{\mu}),\quad\tau(E_{i},F_{j})=-\frac{\delta_{ij}}{q_{i}-q_{i}^{-1}}

and the following conditions:

	$\displaystyle\tau(xx^{\prime},y)=\tau(x,y_{(1)})\tau(x^{\prime},y_{(2)}),\quad$	$\displaystyle\tau(x,1)=\varepsilon(x),\quad$
	$\displaystyle\tau(x,yy^{\prime})=\tau(x_{(1)},y^{\prime})\tau(x_{(2)},y),\quad$	$\displaystyle\tau(1,y)=\varepsilon(y).$

Here $\Delta(x)=x_{(1)}\otimes x_{(2)}$ is Sweedler’s notation.

The PBW bases are orthogonal with respect to this form:

\displaystyle\tau(E_{\boldsymbol{i}}^{\Lambda^{+}},F_{\boldsymbol{i}}^{\Lambda^{-}})=\delta_{\Lambda^{+}\Lambda^{-}}\prod_{k=1}^{N}(-1)^{\lambda_{k}^{+}}q_{\alpha^{\boldsymbol{i}}_{k}}^{\lambda^{+}_{k}(\lambda^{+}_{k}-1)/2}\frac{[\lambda^{+}_{k}]_{q_{\alpha^{\boldsymbol{i}}_{k}}!}}{(q_{\alpha^{\boldsymbol{i}}_{k}}-q^{-1}_{\alpha^{\boldsymbol{i}}_{k}})^{\lambda^{+}_{k}}}.

The following formulae will be used frequently:

(1)			$\displaystyle\tau(xK_{\mu},yK_{\nu})=q^{-(\mu,\nu)}\tau(x,y),$	$\displaystyle x\in U_{q}(\mathfrak{n}^{+}),y\in U_{q}(\mathfrak{n}^{-}),$
(2)			$\displaystyle xy=\tau(S(x_{(1)}),y_{(1)})y_{(2)}x_{(2)}\tau(x_{(3)},y_{(3)}),$	$\displaystyle x\in U_{q}(\mathfrak{b}^{+}),y\in U_{q}(\mathfrak{b}^{-}).$

The quantum Killing form $\kappa\colon U_{q}(\mathfrak{g})\times U_{q}(\mathfrak{g})\longrightarrow k$ is given by the following formula:

\displaystyle\kappa(X^{-}K_{\mu}S^{-1}(X^{+}),Y^{-}K_{\nu}S(Y^{+}))=q^{(\mu,\nu)/2}\tau(X^{+},Y^{-})\tau(Y^{+},X^{-}).

Ad-invariance, an important property of the classical Killing form, still holds in this case:

\kappa(Z\triangleright X,Y)=\kappa(X,S(Z)\triangleright Y)

2.3. The restricted integral form

At the end of this section we recall the restricted integral form $U_{q}^{\mathcal{A}}(\mathfrak{g})$ . Set $\mathcal{A}:=\mathbb{Z}[s^{2},s^{-2}]\subset\mathbb{Q}(s)$ . Then $U_{q}(\mathfrak{g})$ has a Hopf subalgebra $U_{q}^{\mathcal{A}}(\mathfrak{g})$ over $\mathcal{A}$ , called the restricted integral form, which is generated by the following elements:

\displaystyle E_{i}^{(r)}=\frac{1}{[r]_{q_{i}}!}E_{i}^{r},\quad F_{i}^{(r)}=\frac{1}{[r]_{q_{i}}!}F_{i}^{r},\quad K_{\lambda},\quad[K_{i};0]_{q_{i}}=\frac{K_{i}-K_{i}^{-1}}{q_{i}-q_{i}^{-1}}.

The upper (resp. lower) triangular subalgebra $U_{q}^{\mathcal{A}}(\mathfrak{n}^{\pm})$ are defined similarly to the non-integral case. They are free over $\mathcal{A}$ since the following elements form a basis:

\displaystyle E_{\boldsymbol{i}}^{(\Lambda)}=E_{\boldsymbol{i},1}^{(\lambda_{1})}E_{\boldsymbol{i},2}^{(\lambda_{2})}\cdots E_{\boldsymbol{i},N}^{(\lambda_{N})},\quad F_{\boldsymbol{i}}^{(\Lambda)}=F_{\boldsymbol{i},1}^{(\lambda_{1})}F_{\boldsymbol{i},2}^{(\lambda_{2})}\cdots F_{\boldsymbol{i},N}^{(\lambda_{N})}

We also have the Cartan subalgebra and the Borel subalgebras, defined as $U_{q}^{\mathcal{A}}(\mathfrak{h})=U_{q}(\mathfrak{h})\cap U_{q}^{\mathcal{A}}(\mathfrak{g})$ and $U_{q}^{\mathcal{A}}(\mathfrak{b}^{\pm})=U_{q}^{\mathcal{A}}(\mathfrak{n}^{\pm})U_{q}^{\mathcal{A}}(\mathfrak{g})$ respectively.

Finally, for a commutative $\mathcal{A}$ -algebra $k$ , we define $U_{q}^{k}(\mathfrak{g})$ as $k\otimes_{\mathcal{A}}U_{q}^{\mathcal{A}}(\mathfrak{g})$ . we also define $U_{q}^{k}(\mathfrak{n}^{\pm}),U_{q}^{k}(\mathfrak{h})$ and $U_{q}^{k}(\mathfrak{b}^{\pm})$ in a similar way.

3. Deformed quantum enveloping algebras

3.1. More on quantum PBW bases

Before defining deformed quantum enveloping algebras, we collect some technical facts on quantum PBW vectors in the usual quantum enveloping algebra. For a reduced expression $s_{\boldsymbol{i}}$ of $w_{0}$ , we set $\acute{E}_{\boldsymbol{i},\alpha}=E_{\boldsymbol{i},\alpha}K_{\alpha}^{-1},\acute{K}_{\mu}=K_{\mu}$ and $\acute{F}_{\boldsymbol{i},\alpha}=F_{\boldsymbol{i},\alpha}$ .

Remark 3.1.

In the following, various expansion formulae with respect to the modified quantum PBW vectors $\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-})}K_{\mu}\acute{E}_{\boldsymbol{i}}^{(\Lambda^{+})}$ are stated. We sometimes apply them to other generators, like $\hat{E}_{\boldsymbol{i},\alpha},\hat{K}_{\mu},\hat{F}_{\boldsymbol{i},\alpha}$ . Moreover we use their variations: for instance, using $\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}$ instead of $\hat{E}_{\boldsymbol{i}}^{(\Lambda^{+})}$ . These application can be validated by considering the “easiest” case, in which these generators are scalar multiple of the original generators. Then the other cases are reduced to this case by changing the base ring.

The following lemma is fundamental in this subsection.

Lemma 3.2.

Let $s_{\boldsymbol{i}}$ be a reduced expression of $w_{0}$ and set $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$ for a fixed $k$ with $1\leq k\leq N$ . Then there is another reduced expression $s_{\boldsymbol{j}}$ which satisfies the following:

	$\displaystyle\mathcal{T}_{w}(\acute{E}_{\boldsymbol{j},l})$	$\displaystyle=\begin{cases}\acute{E}_{\boldsymbol{i},k+l}&(1\leq l\leq N-k)\\ -q_{\alpha_{l-(N-k)}^{\boldsymbol{i}}}^{-2}\acute{F}_{\boldsymbol{i},l-(N-k)}\acute{K}_{2\alpha^{\boldsymbol{i}}_{l-(N-k)}}&(N-k<l\leq N),\end{cases}$
	$\displaystyle\mathcal{T}_{w}(\acute{F}_{\boldsymbol{j},l})$	$\displaystyle=\begin{cases}\acute{F}_{\boldsymbol{i},k+l}&(1\leq l\leq N-k)\\ -\acute{E}_{\boldsymbol{i},l-(N-k)}&(N-k<l\leq N),\end{cases}$

Proof.

At first we assume $k=1$ . Set $\varepsilon=s_{i_{N}}s_{i_{N-1}}\cdots s_{i_{2}}(\varepsilon_{i_{1}})$ . Then this is a simple root and $s_{\boldsymbol{j}}=s_{i_{2}}s_{i_{3}}\cdots s_{i_{N}}s_{\varepsilon}$ is a reduced expression of $w_{0}$ , which has the required property. Actually we can see from the construction that $\mathcal{T}_{i_{1}}(\acute{E}_{\boldsymbol{j},l})=\acute{E}_{\boldsymbol{i},l+1}$ and $\mathcal{T}_{i_{1}}(\acute{F}_{\boldsymbol{j},l})=\acute{F}_{\boldsymbol{i},l+1}$ for $1\leq l\leq N-1$ .

For the case $l=N$ , $\acute{E}_{\boldsymbol{j},N}=\acute{E}_{i_{1}}$ and $\acute{F}_{\boldsymbol{j},N}=\acute{F}_{i_{1}}$ hold since $\alpha_{N}^{\boldsymbol{j}}=\varepsilon_{i_{1}}$ is a simple root. Hence we have

	$\displaystyle\mathcal{T}_{i_{1}}(\acute{E}_{\boldsymbol{j},N})$	$\displaystyle=\mathcal{T}_{i_{1}}(\acute{E}_{i_{1}})=-K_{i_{1}}\acute{F}_{i_{1}}K_{i_{1}}=-q_{\alpha^{\boldsymbol{i}}_{1}}^{-2}\acute{F}_{\boldsymbol{i},1}\acute{K}_{2\alpha^{\boldsymbol{i}}_{1}},$
	$\displaystyle\mathcal{T}_{i_{1}}(\acute{F}_{\boldsymbol{j},N})$	$\displaystyle=\mathcal{T}_{i_{1}}(\acute{F}_{i_{1}})=-\acute{E}_{i_{1}}=-\acute{E}_{\boldsymbol{i},1}.$

Now we show the statement by induction on $k$ . Assume the statement with $k=k_{0}$ holds for all reduced expressions. Then we can take $\boldsymbol{i^{\prime}}$ for $\boldsymbol{i}$ and $k=k_{0}$ , which automatically satisfies $i^{\prime}_{1}=i_{k+1}$ . Then we can take $\boldsymbol{j}$ for $\boldsymbol{i^{\prime}}$ and $k=1$ , which is the required one for $\boldsymbol{i}$ and $k=k_{0}+1$ . ∎

Next we investigate quantum commutators, whose definition in this paper is $[x,y]_{q}:=xy-q^{-(\operatorname{\mathrm{wt}}x,\operatorname{\mathrm{wt}}y)}yx$ for weight vectors $x,y$ .

The following is an easy consequence of a well-known result, called the Levendörskii-Soibelman relation in literature.

Proposition 3.3 (c.f. [MR1116413, Proposition 5.5.2]).

For $1\leq k<l\leq N$ , we have the following expansion with $C_{\Lambda}^{\pm}\in\mathcal{A}$ :

\displaystyle[\acute{E}_{\boldsymbol{i},l},\acute{E}_{\boldsymbol{i},k}]_{q}=\sum_{\begin{subarray}{c}\Lambda\subset(k,l)\\ \Lambda\cdot\alpha^{\boldsymbol{i}}=\alpha^{\boldsymbol{i}}_{k}+\alpha^{\boldsymbol{i}}_{l}\end{subarray}}C_{\Lambda}^{+}\acute{E}_{\boldsymbol{i}}^{(\Lambda)},\quad[\acute{F}_{\boldsymbol{i},l},\acute{F}_{\boldsymbol{i},k}]_{q}=\sum_{\begin{subarray}{c}\Lambda\subset(k,l)\\ \Lambda\cdot\alpha^{\boldsymbol{i}}=\alpha^{\boldsymbol{i}}_{k}+\alpha^{\boldsymbol{i}}_{l}\end{subarray}}C_{\Lambda}^{-}\acute{F}_{\boldsymbol{i}}^{(\Lambda)},

Combining this with Lemma 3.2, we can obtain a “mixed” version of these formulae.

Proposition 3.4.

Fix a reduced expression $s_{\boldsymbol{i}}$ of the longest element $w_{0}$ . and positive integers $1\leq k<l\leq N$ .

(i)

The following $\mathcal{A}$ -submodules are closed under multiplication:

	$\displaystyle\operatorname{\mathrm{span}}_{\mathcal{A}}\{\acute{F}_{\boldsymbol{i}}^{\Lambda^{-}}\acute{E}_{\boldsymbol{i}}^{\Lambda^{+}}\mid\Lambda^{+}\leq k<l\leq\Lambda^{-}\},$
	$\displaystyle\operatorname{\mathrm{span}}_{\mathcal{A}}\{\acute{E}_{\boldsymbol{i}}^{\Lambda^{+}}\acute{K}_{\mu}\acute{F}_{\boldsymbol{i}}^{\Lambda^{-}}\mid\mu\in P,\,\Lambda^{-}\leq l<k\leq\Lambda^{+}\}.$

(ii)

The latter subalgebra coincides with the following:

\displaystyle\operatorname{\mathrm{span}}_{\mathcal{A}}\{\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\acute{K}_{\mu}\acute{E}_{\boldsymbol{i}}^{(\Lambda^{+})}\mid\mu\in P,\Lambda^{-}\leq l<k\leq\Lambda^{+}\}.

(iii)

We have the following expansion with $C_{k,l}(\Lambda^{-},\Lambda^{+}),C_{l,k}(\Lambda^{-},\Lambda^{+})\in\mathcal{A}$ :

(3)		$\displaystyle[\acute{E}_{\boldsymbol{i},k},\acute{F}_{\boldsymbol{i},l}]_{q}$	$\displaystyle=\sum_{\begin{subarray}{c}\Lambda^{+}<k<l<\Lambda^{-},\\ \mu:=\alpha_{k}^{\boldsymbol{i}}-\Lambda^{+}\cdot\alpha^{\boldsymbol{i}}=\alpha_{l}^{\boldsymbol{i}}-\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}\in Q^{+}\end{subarray}}C_{k,l}(\Lambda^{-},\Lambda^{+})\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\acute{E}_{\boldsymbol{i}}^{(\Lambda^{+})},$
(4)		$\displaystyle[\acute{E}_{\boldsymbol{i},l},\acute{F}_{\boldsymbol{i},k}]_{q}$	$\displaystyle=\sum_{\begin{subarray}{c}\Lambda^{-}<k<l<\Lambda^{+}\\ \mu:=\alpha_{k}^{\boldsymbol{i}}-\Lambda^{+}\cdot\alpha^{\boldsymbol{i}}=\alpha_{l}^{\boldsymbol{i}}-\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}\in Q^{+}\end{subarray}}C_{l,k}(\Lambda^{-},\Lambda^{+})\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\acute{K}_{2\mu}^{-1}\acute{E}_{\boldsymbol{i}}^{(\Lambda^{+})}.$

Proof.

(i) This follows from Lemma 3.2 immediately.

(ii) Note that the latter subalgebra in (i) is generated by $(\acute{E}_{\boldsymbol{i},n})_{n\leq l}$ , $(\acute{F}_{\boldsymbol{i},m})_{m\geq k}$ and $(\acute{K}_{\mu})_{\mu\in P}$ . Hence it suffices to show that the following $\mathcal{A}$ -submodules are closed under multiplication for all $0\leq m\leq l$ :

\displaystyle\mathcal{I}_{m}:=\operatorname{\mathrm{span}}_{\mathcal{A}}\{\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-}_{1})}\acute{E}_{\boldsymbol{i}}^{(\Lambda^{+})}\acute{K}_{\mu}\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-}_{2})}\mid\mu\in P,\Lambda^{-}_{2}\leq m<\Lambda^{-}_{1}\leq l<k\leq\Lambda^{+}\}.

If $m=l$ , there is nothing to prove. For the downward induction step, assume that the statement holds for a fixed $m$ . It suffices to show $\mathcal{I}_{m-1}$ is invariant under the right multiplication by $\acute{E}_{\boldsymbol{i},m}^{(r)}$ for all $r\geq 1$ . There is nothing to prove when $r=0$ . For a general $r$ , we can see $\mathcal{I}_{m-1}\acute{E}_{\boldsymbol{i},m}^{(r)}\subset\mathcal{I}_{m-1}\acute{E}_{\boldsymbol{i},m}^{(r-1)}$ using Lemma 3.2 and Proposition 3.3. Hence an induction argument works.

(iii) Take $\boldsymbol{j}$ using Lemma 3.2 for $\boldsymbol{i}$ and $k$ . Then we have

\displaystyle\acute{E}_{\boldsymbol{i},k}\acute{F}_{\boldsymbol{i},l}-q^{-(\alpha^{\boldsymbol{i}}_{k},\alpha^{\boldsymbol{i}}_{l})}\acute{F}_{\boldsymbol{i},l}\acute{E}_{\boldsymbol{i},k}=q^{(\alpha_{k}^{\boldsymbol{i}},\alpha_{l}^{\boldsymbol{i}})}\mathcal{T}_{w}(\acute{F}_{\boldsymbol{j},l-k}\acute{F}_{\boldsymbol{j},n}-q^{(\alpha_{l-k}^{\boldsymbol{j}},\alpha_{n}^{\boldsymbol{j}})}\acute{F}_{\boldsymbol{j},n}\acute{F}_{\boldsymbol{j},l-k}).

Now Proposition 3.3 leads us to the desired expansion without the constraint on $\mu$ . A similar argument works for the other case.

Finally we show the positivity of $\mu$ . Since $\Delta(F_{i})=F_{i}\otimes 1+K_{i}^{-1}\otimes F_{i}$ holds, $\Delta(F_{\boldsymbol{i},\alpha})$ is an $\mathcal{A}$ -linear combination of $F_{\boldsymbol{i}}^{(\Lambda_{1})}K_{\Lambda_{2}\cdot\alpha^{\boldsymbol{i}}}^{-1}\otimes F_{\boldsymbol{i}}^{(\Lambda_{2})}$ with $\Lambda_{1}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{2}\cdot\alpha^{\boldsymbol{i}}=\alpha$ . Using this fact twice we can see that $(\Delta\otimes\operatorname{\mathrm{id}})\Delta(\acute{F}_{\boldsymbol{i},\alpha})$ is an $\mathcal{A}$ -linear combination of $F_{\boldsymbol{i}}^{(\Lambda_{1})}K_{\Lambda_{2}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{3}\cdot\alpha^{\boldsymbol{i}}}^{-1}\otimes\acute{F}_{\boldsymbol{i}}^{(\Lambda_{2})}\acute{K}_{\Lambda_{3}\cdot\alpha^{\boldsymbol{i}}}^{-1}\otimes F_{\boldsymbol{i}}^{(\Lambda_{3})}$ with $\Lambda_{1}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{2}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{3}\cdot\alpha^{\boldsymbol{i}}=\alpha$ . A similar fact holds for $(\Delta\otimes\operatorname{\mathrm{id}})\Delta(\acute{E}_{\boldsymbol{i},\beta})$ . Then the formula (2) implies that $\acute{E}_{\boldsymbol{i},\alpha^{+}}\acute{F}_{\boldsymbol{i},\alpha^{-}}$ is an $\mathcal{A}$ -linear combination of elements of the following form:

\displaystyle\tau(S(E_{\boldsymbol{i}}^{(\Lambda_{1}^{+})}K_{\alpha^{+}}^{-1}),F_{\boldsymbol{i}}^{(\Lambda_{1}^{-})}K_{\Lambda_{2}^{-}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{3}^{-}\cdot\alpha^{\boldsymbol{i}}}^{-1})\tau(E_{\boldsymbol{i}}^{(\Lambda_{3}^{+})}K_{\Lambda_{3}^{+}\cdot\alpha^{\boldsymbol{i}}}^{-1},F_{\boldsymbol{i}}^{(\Lambda_{3}^{-})})\acute{F}_{\boldsymbol{i}}^{(\Lambda_{2}^{-})}\acute{K}_{\Lambda_{3}^{-}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{3}^{+}\cdot\alpha^{\boldsymbol{i}}}^{-1}\acute{E}_{\boldsymbol{i}}^{(\Lambda_{2}^{+})}

with $\Lambda_{1}^{\pm}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{2}^{\pm}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{3}^{\pm}\cdot\alpha^{\boldsymbol{i}}=\alpha^{\pm}$ . Since this term does not vanish only when $\Lambda_{3}^{+}=\Lambda_{3}^{-}$ , we have $\mu=\Lambda_{3}^{\pm}\cdot\alpha^{\boldsymbol{i}}\in Q^{+}$ . ∎

As an application we can obtain formulae on the coproduct.

Proposition 3.5.

Fix a reduced expression $s_{\boldsymbol{i}}$ of $w_{0}$ . Then we have the following expansion with $C_{k}^{\pm}(\Lambda^{l},\Lambda^{r})\in\mathcal{A}$ :

(5)			$\displaystyle\begin{aligned} \Delta(\acute{E}_{\boldsymbol{i},k})=E_{\boldsymbol{i},k}&K_{\alpha^{\boldsymbol{i}}_{k}}^{-1}\otimes 1+K_{\alpha^{\boldsymbol{i}}_{k}}^{-1}\otimes\acute{E}_{\boldsymbol{i},k}\\ &+\sum_{\Lambda_{r}<k<\Lambda_{l},\Lambda_{l}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{r}\cdot\alpha^{\boldsymbol{i}}=\alpha^{\boldsymbol{i}}_{k}}C_{k}^{+}(\Lambda_{l},\Lambda_{r})K_{\alpha^{\boldsymbol{i}}_{k}}^{-1}E_{\boldsymbol{i}}^{(\Lambda_{l})}\otimes\acute{E}_{\boldsymbol{i}}^{(\Lambda_{r})},\end{aligned}$
(6)			$\displaystyle\begin{aligned} \Delta(\acute{F}_{\boldsymbol{i},k})=F_{\boldsymbol{i},k}&\otimes 1+K_{\alpha^{\boldsymbol{i}}_{k}}^{-1}\otimes\acute{F}_{\boldsymbol{i},k}\\ &+\sum_{\Lambda_{l}<k<\Lambda_{r},\Lambda_{l}\cdot\alpha^{\boldsymbol{i}}+\Lambda_{r}\cdot\alpha^{\boldsymbol{i}}=\alpha^{\boldsymbol{i}}_{k}}C_{k}^{-}(\Lambda_{l},\Lambda^{r})F_{\boldsymbol{i}}^{(\Lambda^{l})}K_{\Lambda^{r}\cdot\alpha^{\boldsymbol{i}}}^{-1}\otimes\acute{F}_{\boldsymbol{i}}^{(\Lambda^{r})}.\end{aligned}$

We need some preparations to prove this proposition. Recall the quantum exponential function:

\displaystyle\exp_{q}x=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}}{[k]_{q}!}x^{k}

Let $s_{\boldsymbol{i}}$ be a reduced expression of an element of $W$ .

\displaystyle\Delta(\mathcal{T}_{i}(z))=A_{i}(\mathcal{T}_{i}\otimes\mathcal{T}_{i})(\Delta(z))A_{i}^{-1},

where $A_{i}:=\exp_{q_{i}}((q_{i}-q_{i}^{-1})K_{i}F_{i}\otimes\acute{E}_{i})$ , whose inverse is given by $A_{i}^{-1}=\exp_{q_{i}^{-1}}(-(q_{i}-q_{i}^{-1})K_{i}F_{i}\otimes\acute{E}_{i})$ . Applying this formula iteratively to a reduced expression $s_{\boldsymbol{i}}$ of an element $w\in W$ , we also have

\displaystyle\Delta(\mathcal{T}_{w}(z))=A_{\boldsymbol{i},1}A_{\boldsymbol{i},2}\cdots A_{\boldsymbol{i},\ell(w)}(\mathcal{T}_{w}\otimes\mathcal{T}_{w})(\Delta(z))A_{\boldsymbol{i},\ell(w)}^{-1}A_{\boldsymbol{i},\ell(w)-1}^{-1}\cdots A_{\boldsymbol{i},1}^{-1}

where

	$\displaystyle A_{\boldsymbol{i},l}$	$\displaystyle=\exp_{q_{\alpha^{\boldsymbol{i}}_{l}}}((q_{\alpha^{\boldsymbol{i}}_{l}}-q_{\alpha^{\boldsymbol{i}}_{l}}^{-1})K_{\alpha^{\boldsymbol{i}}_{l}}F_{\boldsymbol{i},l}\otimes\acute{E}_{\boldsymbol{i},l}),$
	$\displaystyle A_{\boldsymbol{i},l}^{-1}$	$\displaystyle=\exp_{q_{\alpha^{\boldsymbol{i}}_{l}}^{-1}}(-(q_{\alpha^{\boldsymbol{i}}_{l}}-q_{\alpha^{\boldsymbol{i}}_{l}}^{-1})K_{\alpha^{\boldsymbol{i}}_{l}}F_{\boldsymbol{i},l}\otimes\acute{E}_{\boldsymbol{i},l}).$

Proof.

Let $s_{\boldsymbol{i}}$ be a reduced expression of the longest element compatible with $R_{0}^{+}$ and $w_{k}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$ . Then, for any $1\leq k\leq n$ , we have

\displaystyle\Delta(\acute{E}_{\boldsymbol{i},k})=A_{\boldsymbol{i},1}A_{\boldsymbol{i},2}\cdots A_{\boldsymbol{i},k-1}(\mathcal{T}_{w_{k-1}}\otimes\mathcal{T}_{w_{k-1}})(\Delta(\acute{E}_{i_{k}}))A_{\boldsymbol{i},k-1}^{-1}A_{\boldsymbol{i},k-2}^{-1}\cdots A_{\boldsymbol{i},1}^{-1}.

On the other hand we have

\displaystyle(\mathcal{T}_{w_{k-1}}\otimes\mathcal{T}_{w_{k-1}})(\Delta(\acute{E}_{i_{k}}))=E_{\boldsymbol{i},k}K_{\alpha^{\boldsymbol{i}}_{k}}^{-1}\otimes 1+K_{\alpha^{\boldsymbol{i}}_{k}}^{-1}\otimes\acute{E}_{\boldsymbol{i},k}

Combining with Proposition 3.3, Proposition 3.4 (i) and $\Delta(\acute{E}_{\boldsymbol{i},k})\in U_{q}^{\mathcal{A}}(\mathfrak{b}^{+})\otimes U_{q}^{\mathcal{A}}(\mathfrak{b}^{+})$ , we can see (5).

The same argument works for (6) since the following formula holds:

\displaystyle(\mathcal{T}_{w_{k-1}}\otimes\mathcal{T}_{w_{k-1}})(\Delta(\acute{F}_{i_{k}}^{(r)}))=\sum_{j=0}^{r}q_{\alpha^{\boldsymbol{i}}_{k}}^{j(r-j)}F_{\boldsymbol{i},k}^{(r-j)}K_{\alpha^{\boldsymbol{i}}_{k}}^{-j}\otimes\acute{F}_{\boldsymbol{i},k}^{(j)}.

For a reference, see [MR4162277, Proof of Lemma 3.18]. ∎

3.2. Deformed quantum enveloping algebra

We fix a positive system $R_{0}^{+}$ , possibly different from $R^{+}$ . Then there is a unique element $w\in W$ such that $w(R^{+})=R_{0}^{+}$ . Note $R^{+}\setminus R_{0}^{+}=\{\alpha\in R^{+}\mid w^{-1}(\alpha)\in-R^{+}\}$ for such $w$ .

We say that a reduced expression $s_{\boldsymbol{i}}$ of the longest element is compatible with $R_{0}^{+}$ if it begins with a reduced expression of $w$ i.e. $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell(w)}}$ . Such an expression always exists since $\ell(w_{0})=\ell(w)+\ell(w^{-1}w_{0})$ . Note that $s_{\boldsymbol{i}}$ is compatible with $R_{0}^{+}$ if and only if $R^{+}\setminus R_{0}^{+}=\{\alpha^{\boldsymbol{i}}_{n}\}_{n=1}^{\ell(w)}$ .

For a monoid $M$ , the monoid algebra with coefficients in a commutative ring $k$ is denoted by $k[M]$ . It has a canonical $k$ -basis $\{e_{m}\}_{m\in M}$ , for which we have $e_{m}e_{m^{\prime}}=e_{mm^{\prime}}$ .

Let $Q_{0}^{-}$ be the additive submonoid of $Q$ generated by $-R_{0}^{+}$ . By the universal property of monoid algebras, we can identify an $\mathcal{A}[2Q_{0}^{-}]$ -algebra with a pair of an $\mathcal{A}$ -algebra $k$ and a monoid homomorphism $\chi\colon 2Q_{0}^{-}\longrightarrow k;\lambda\longmapsto\chi_{\lambda}$ with respect to the multiplication on $k$ . In light of this fact, we refer such a pair as a commutative $\mathcal{A}[2Q_{0}^{-}]$ -algebra.

Definition 3.6.

Let $(k,\chi)$ be a commutative $\mathcal{A}[2Q_{0}^{-}]$ -algebra and fix a reduced expression $s_{\boldsymbol{i}}$ compatible with $R_{0}^{+}$ . We define $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ as $k\otimes_{\mathcal{A}[2Q_{0}^{-}]}U_{q,e}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})$ , where $U_{q,e}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})$ is an $\mathcal{A}[2Q_{0}^{-}]$ -subalgebra of $U_{q}^{\mathcal{A}[P]}(\mathfrak{g})$ generated by the following elements:

•

${\displaystyle\hat{F}_{\boldsymbol{i},\alpha}^{(r)}:=\begin{cases}e_{2\alpha}^{r}\acute{F}_{\boldsymbol{i},\alpha}^{(r)}&\text{for }\alpha\in R^{+}\setminus R_{0}^{+},r\geq 0\\ \acute{F}_{\boldsymbol{i},\alpha}^{(r)}&\text{for }\alpha\in R^{+}\cap R_{0}^{+},r\geq 0,\end{cases}}$
•

$\hat{K}_{\lambda}:=e_{-\lambda}\acute{K}_{\lambda}$ for $\lambda\in P$ .
•

$\hat{E}_{\boldsymbol{i},\alpha}:=(q_{\alpha}-q_{\alpha}^{-1})\acute{E}_{\boldsymbol{i},\alpha}$ for $\alpha\in R^{+}$ .

We also define $U_{q,\chi}^{k}({}^{\sim}\mathfrak{g})$ by using $\check{F}_{\boldsymbol{i},\alpha}=(q_{\alpha}-q_{\alpha}^{-1})\hat{F}_{\boldsymbol{i},\alpha},\check{K}_{\lambda}=\hat{K}_{\lambda},\check{E}_{\boldsymbol{i},\alpha}^{(r)}=\acute{E}_{\boldsymbol{i},\alpha}^{(r)}$ .

Remark 3.7.

These definitions actually do not depend on the choice of $s_{\boldsymbol{i}}$ since $\mathrm{span}_{\mathcal{A}}\{F_{\boldsymbol{i}}^{(\Lambda)}\}_{\Lambda\leq\lvert R^{+}\setminus R_{0}^{+}\rvert}=U_{q}^{\mathcal{A}}(\mathfrak{n}^{-})\cap\mathcal{T}_{w}^{-1}(U_{q}^{\mathcal{A}}(\mathfrak{b}^{+}))$ and $\mathrm{span}_{\mathcal{A}}\{F_{\boldsymbol{i}}^{(\Lambda)}\}_{\Lambda>\lvert R^{+}\setminus R_{0}^{+}\rvert}=U_{q}^{\mathcal{A}}(\mathfrak{n}^{-})\cap\mathcal{T}_{w}(U_{q}^{\mathcal{A}}(\mathfrak{n}^{-}))$ are independent of $s_{\boldsymbol{i}}$ , where $w$ is the unique element with $w(R^{+})=R_{0}^{+}$ .

On the other hand, though we can consider a subalgebra of $U_{q}^{\mathcal{A}[P]}(\mathfrak{g})$ generated by the elements above constructed from a non-compatible reduced expression $s_{\boldsymbol{i}}$ , it can be different from $U_{q,e}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})$ . As an example we consider $\mathfrak{g}=\mathfrak{sl}_{3}$ . The simple roots is denoted by $\alpha$ and $\beta$ . Then the reduced expressions of the longest element are $s_{\boldsymbol{i}}=s_{\alpha}s_{\beta}s_{\alpha}$ and $s_{\boldsymbol{j}}:=s_{\beta}s_{\alpha}s_{\beta}$ . Consider $R_{0}^{+}:=\{\beta,-\alpha,-\alpha-\beta\}$ . Then $s_{\boldsymbol{i}}$ is compatible with $R_{0}^{+}$ and $s_{\boldsymbol{j}}$ is not. In this case we have

\displaystyle e_{2(\alpha+\beta)}F_{\boldsymbol{j},\alpha+\beta}=e_{2(\alpha+\beta)}(F_{\beta}F_{\alpha}-q^{-1}F_{\alpha}F_{\beta})=-q^{-1}\hat{F}_{\boldsymbol{i},\alpha+\beta}-e_{2\beta}(q-q^{-1})\hat{F}_{\beta}\hat{F}_{\alpha}.

Since $e_{2\beta}$ is not invertible in $\mathcal{A}[2Q_{0}^{-}]$ and the PBW theorem holds for $U_{q,e}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})$ , as shown in Proposition 3.11, $e_{2(\alpha+\beta)}F_{\boldsymbol{i},\alpha+\beta}$ is not in $U_{q,e}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})$ .

Remark 3.8.

We cannot remove $\hat{F}_{\boldsymbol{i},\alpha}$ with non-simple root $\alpha$ from the set of generators. As an example, we consider $\mathfrak{sl}_{3}$ and a positive system $R_{0}^{+}:=\{\alpha+\beta,\beta,-\alpha\}$ . Then $s_{\boldsymbol{i}}$ is compatible with $R_{0}^{+}$ . Then we can calculate some commutation relations of the generators as follows:

\displaystyle[\hat{F}_{\alpha},\hat{F}_{\beta}]_{q}=e_{2\alpha}\hat{F}_{\alpha+\beta},\,[\hat{E}_{\alpha+\beta},\hat{F}_{\alpha}]_{q}=\hat{K}_{2\alpha}^{-1}\hat{E}_{\beta},\,[\hat{E}_{\alpha+\beta},\hat{F}_{\beta}]_{q}=-\hat{E}_{\alpha},

here we omit the subscript $\boldsymbol{i}$ . Since $e_{2\alpha}$ is not invertible in $\mathcal{A}[2Q_{0}^{-}]$ , the subalgebra generated by $\hat{F}_{\alpha}^{(r)},\hat{F}_{\beta}^{(r)},\hat{K}_{\lambda},\hat{E}_{\alpha}$ , $\hat{E}_{\alpha+\beta},\hat{E}_{\beta}$ does not contain $\hat{F}_{\alpha+\beta}$ .

Proposition 3.9.

The canonical left coaction of $U_{q}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g})$ on itself restricts to a left coaction on $U_{q,e}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})$ . In particular $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ has a canonical left $U_{q}^{k}(\mathfrak{g})$ -coaction. Similarly $U_{q,\chi}^{k}({}^{\sim}\mathfrak{g})$ has a canonical left $U_{q}^{k}(\mathfrak{g})$ -coaction.

Proof.

Proof of Proposition 3.5 shows the statement since $\Delta(\hat{F}_{\boldsymbol{i},\alpha}^{(r)})$ , $\Delta(\hat{K}_{\mu}),\Delta(\hat{E}_{\boldsymbol{i},\alpha})$ and $\pm K_{\alpha}F_{\boldsymbol{i},\alpha}\otimes\hat{E}_{\boldsymbol{i},\alpha}$ , whose quantum exponential is $A_{\boldsymbol{i},n}^{\pm 1}$ , are contained in $U_{q}^{\mathcal{A}}(\mathfrak{g})\otimes U_{q,e}^{\mathcal{A}[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})$ . A similar argument works for $U_{q,\chi}^{k}({}^{\sim}\mathfrak{g})$ . ∎

Next we prove a PBW-type result for $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ . Let us define $\widetilde{E}_{\boldsymbol{i},\alpha},\widetilde{K}_{\mu},\widetilde{F}_{\boldsymbol{i},\alpha}\in U_{q}^{\mathcal{A}[P]}(\mathfrak{g})$ as $(q_{\alpha}-q_{\alpha}^{-1})\acute{E}_{\boldsymbol{i},\alpha},e_{-\mu}K_{\mu},(q_{\alpha}-q_{\alpha}^{-1})\acute{F}_{\boldsymbol{i},\alpha}$ respectively.

Lemma 3.10.

Set $\mathcal{U}:=\mathrm{span}_{\mathcal{A}[2Q^{-}]}\{\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}\widetilde{K}_{\mu}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}\}_{(\Lambda^{\pm},\mu)}\subset U_{q}^{\mathcal{A}[P]}(\mathfrak{g})$ . Then this is closed under multiplication. Moreover, for all $1\leq l\leq N$ , it coincides with the $\mathcal{A}[2Q^{-}]$ -linear span of elements of the form $\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{2}^{+}}\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}\widetilde{K}_{\mu}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}}$ where $\Lambda_{1}^{+}\leq l<\Lambda_{2}^{+}$ .

Proof.

It is not difficult to see that $\mathcal{U}^{-}:=\mathrm{span}_{\mathcal{A}[2Q^{-}]}\{\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}\widetilde{K}_{\mu}\}$ and $\mathcal{U}^{+}:=\mathrm{span}_{\mathcal{A}[2Q^{-}]}\{\widetilde{K}_{\mu}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}\}$ are closed under multiplication. To see $xy\in\mathcal{U}$ for $x\in\mathcal{U}^{+}$ and $y\in\mathcal{U}^{-}$ , one should note that the argument in the proof of Proposition 3.4 (iii) works for computing the expansion of $\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}$ . Hence this product can be expressed as an $\mathcal{A}$ -linear combination of elements of the form $\widetilde{F}_{\boldsymbol{i}}^{\Gamma^{-}}K_{2\mu}^{-1}\widetilde{E}_{\boldsymbol{i}}^{\Gamma^{+}}$ with $\mu\in Q^{+}$ , which shows the required property since $K_{2\mu}^{-1}=e_{-2\mu}\widetilde{K}_{2\mu}^{-1}$ .

Next we show the latter half of the statement by downward induction on $l$ . If $l=N$ , there is nothing to prove. For the induction step, assume the statement holds for a fixed $l$ . It suffices to show that $\mathcal{U}^{\prime}:=\mathrm{span}_{\mathcal{A}[2Q^{-}]}\{\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{2}^{+}}\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}\widetilde{K}_{\mu}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}}\mid\Lambda_{1}^{+}<l\leq\Lambda_{2}^{+}\}$ is closed under multiplication, which, combining with the induction hypothesis, implies that the linear span is the $\mathcal{A}[2Q^{-}]$ -subalgebra generated by $(\widetilde{F}_{\boldsymbol{i},\alpha})_{\alpha\in R^{+}},(\widetilde{K}_{\mu})_{\mu\in P}$ and $(\widetilde{E}_{\boldsymbol{i},\alpha})_{\alpha\in R^{+}}$ , which is nothing but $\mathcal{U}$ .

We only prove that $\mathcal{U}^{\prime}$ is invariant under the right multiplication by $\widetilde{E}_{\boldsymbol{i},l}$ since this fact and the induction hypothesis imply the invariance for other elements.

Set $\mathcal{U}_{l}^{\pm}$ as follows:

	$\displaystyle\mathcal{U}_{l}^{-}$	$\displaystyle=\operatorname{\mathrm{span}}_{\mathcal{A}[2Q^{-}]}\{\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}\widetilde{K}_{\mu}\widetilde{E}^{\Lambda^{+}}\mid\mu\in P,\Lambda^{+}<l<\Lambda^{-}\}$
	$\displaystyle\mathcal{U}_{l}^{+}$	$\displaystyle=\operatorname{\mathrm{span}}_{\mathcal{A}[2Q^{-}]}\{\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}\widetilde{K}_{\mu}\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}\mid\mu\in P,\Lambda^{-}<l<\Lambda^{+}\}.$

These are closed under multiplication by Proposition 3.4 (i). Also note that $\mathcal{U}^{\prime}$ is spanned by elements of the form $\widetilde{E}_{\boldsymbol{i},l}^{p}x^{+}\widetilde{F}_{\boldsymbol{i},l}^{q}x^{-}$ with $x^{\pm}\in\mathcal{U}_{l}^{\pm}$ and we have

\displaystyle\widetilde{F}_{\boldsymbol{i},l}\widetilde{E}_{\boldsymbol{i},l}=q_{\alpha^{\boldsymbol{i}}_{l}}^{-2}\widetilde{E}_{\boldsymbol{i},l}\widetilde{F}_{\boldsymbol{i},l}-(q_{\alpha^{\boldsymbol{i}}_{l}}-q_{\alpha^{\boldsymbol{i}}_{l}}^{-1})(1-e_{-2\alpha^{\boldsymbol{i}}_{l}}\widetilde{K}_{-2\alpha^{\boldsymbol{i}}_{l}}).

Therefore it suffices to show that $\mathcal{U}_{l}^{\pm}\widetilde{E}_{\boldsymbol{i},l}\subset\widetilde{E}_{\boldsymbol{i},l}\mathcal{U}_{l}^{\pm}+\mathcal{U}_{l}^{\pm}$ . At first take $x^{-}\in\mathcal{U}_{l}^{-}$ . By using the triangular decomposition and Proposition 3.3, we may assume $x^{-}=\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}$ with $\Lambda^{-}>l$ . If $\Lambda^{-}=\delta_{m}$ for some $m>l$ , the mixed Levendörskii-Soibelman relation (3) proves $\widetilde{F}_{\boldsymbol{i},m}\widetilde{E}_{\boldsymbol{i},l}-q^{-(\alpha^{\boldsymbol{i}}_{l},\alpha^{\boldsymbol{i}}_{m})}\widetilde{E}_{\boldsymbol{i},l}\widetilde{F}_{\boldsymbol{i},m}\in\mathcal{U}_{l}^{-}$ . Then we have

\displaystyle\widetilde{F}_{\boldsymbol{i},m_{1}}\widetilde{F}_{\boldsymbol{i},m_{2}}\cdots\widetilde{F}_{\boldsymbol{i},m_{n}}\widetilde{E}_{\boldsymbol{i},l}-q^{-(\alpha^{\boldsymbol{i}}_{l},\alpha^{\boldsymbol{i}}_{m_{1}}+\alpha^{\boldsymbol{i}}_{m_{2}}+\cdots+\alpha^{\boldsymbol{i}}_{m_{l}})}\widetilde{E}_{\boldsymbol{i},l}\widetilde{F}_{\boldsymbol{i},m_{1}}\widetilde{F}_{\boldsymbol{i},m_{2}}\cdots\widetilde{F}_{\boldsymbol{i},m_{n}}\in\mathcal{U}_{l}^{-}

for any sequence $(m_{1},m_{2},\cdots,m_{n})$ with $m_{j}>l$ , which is nothing but the required property. A similar argument works for $x^{+}\in\mathcal{U}_{l}^{+}$ . ∎

Proposition 3.11.

Fix a reduced expression $s_{\boldsymbol{i}}$ of the longest element compatible with $R_{0}^{+}$ . Then $\{\hat{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\hat{K}_{\mu}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}\}_{(\Lambda^{\pm},\mu)}$ is a basis of $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ .

Proof.

We may assume $k=\mathcal{A}[2Q_{0}^{-}]$ . Set $l:=\lvert R^{+}\cap R_{0}^{+}\rvert$ . Then there is $w\in W$ and a reduced expression $s_{\boldsymbol{j}}$ of $w_{0}$ as Lemma 3.2 for $s_{\boldsymbol{i}}$ and $N-l$ . Let $t_{w}$ be an automorphism of $\mathcal{A}[P]$ which sends $t_{w}(e_{\lambda})=e_{w(\lambda)}$ and $\varphi$ be an automorphism on $U_{q}^{\mathcal{A}[P]}$ defined as $t_{w}\otimes\mathcal{T}_{w}$ . Then $\varphi(\mathcal{U})$ is closed under multiplication and spanned by elements of the form $\mathcal{T}_{w}(\widetilde{E}_{\boldsymbol{j}})^{\Lambda_{2}^{+}}\mathcal{T}_{w}(\widetilde{F}_{\boldsymbol{j}})^{\Lambda^{-}}\widetilde{K}_{\mu}\mathcal{T}_{w}(\widetilde{E}_{\boldsymbol{j}})^{\Lambda_{1}^{+}}$ with $\Lambda_{1}^{+}\leq l<\Lambda_{2}^{+}$ . Hence Lemma 3.10 implies that $\mathrm{span}_{k}\{\check{F}_{\boldsymbol{i}}^{\Lambda^{-}}\hat{K}_{\mu}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}\}_{(\Lambda^{\pm},\mu)}$ is closed under multiplication.

Now fix $(\Lambda_{1}^{\pm},\mu_{1})$ and $(\Lambda_{2}^{\pm},\mu_{2})$ and consider the following two expansions with $C_{\Lambda^{\pm},\mu}\in\mathcal{A}$ :

\displaystyle(\acute{F}_{\boldsymbol{i}}^{(\Lambda_{1}^{-})}K_{\mu_{1}}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}})(\acute{F}_{\boldsymbol{i}}^{(\Lambda_{2}^{-})}K_{\mu_{2}}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{2}^{+}})

\displaystyle=\sum_{\Lambda^{\pm},\mu}C_{\Lambda^{\pm},\mu}\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-})}K_{\mu}\widetilde{E}^{\Lambda^{+}}.

Then there is $\nu(\Lambda^{\pm},\mu)\in P$ for each $(\Lambda^{\pm},\mu)$ with which we have

\displaystyle(\hat{F}_{\boldsymbol{i}}^{(\Lambda_{1}^{-})}\hat{K}_{\mu_{1}}\hat{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}})(\hat{F}_{\boldsymbol{i}}^{(\Lambda_{2}^{-})}\hat{K}_{\mu_{2}}\hat{E}_{\boldsymbol{i}}^{\Lambda_{2}^{+}})=\sum_{\Lambda^{\pm},\mu}e_{\nu(\Lambda^{\pm},\mu)}C_{\Lambda^{\pm},\mu}\hat{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\hat{K}_{\mu}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}.

On the other hand, this weight appears also in the following comparison:

	$\displaystyle(\widetilde{F}_{\boldsymbol{i}}^{\Lambda_{1}^{-}}K_{\mu_{1}}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}})(\widetilde{F}_{\boldsymbol{i}}^{\Lambda_{2}^{-}}K_{\mu_{2}}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{2}^{+}})$	$\displaystyle=\sum_{\Lambda^{\pm},\mu}c_{\Lambda^{\pm},\mu}\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}K_{\mu}\widetilde{E}^{\Lambda^{+}},$
	$\displaystyle(\check{F}_{\boldsymbol{i}}^{\Lambda_{1}^{-}}\hat{K}_{\mu_{1}}\hat{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}})(\check{F}_{\boldsymbol{i}}^{\Lambda_{2}^{-}}\hat{K}_{\mu_{2}}\hat{E}_{\boldsymbol{i}}^{\Lambda_{2}^{+}})$	$\displaystyle=\sum_{\Lambda^{\pm},\mu}e_{(\Lambda^{\pm},\mu)}c_{\Lambda^{\pm},\mu}\check{F}_{\boldsymbol{i}}^{\Lambda^{-}}\hat{K}_{\mu}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}.$

Then the discussion above shows that each $\nu(\Lambda^{\pm},\mu)$ is contained in the image of $t_{w}(2Q^{-})=2Q_{0}^{-}$ , which completes the proof. ∎

A similar argument works to prove the following.

Proposition 3.12.

Fix a reduced expression $s_{\boldsymbol{i}}$ of the longest element compatible with $R_{0}^{+}$ . Then $\{\check{F}_{\boldsymbol{i}}^{\Lambda^{-}}\check{K}_{\mu}\check{E}_{\boldsymbol{i}}^{(\Lambda^{+})}\}_{(\Lambda^{\pm},\mu)}$ is a basis of $U_{q}^{k}({}^{\sim}\mathfrak{g})$ .

3.3. Twisted parabolic induction via deformed QEA

At the end of this section, we define twisted parabolic induction, which play an important role to construct quantum semisimple coadjoint orbits.

Let $S$ be a set of simple roots and $R_{0}^{+}$ be a positive system of $R$ containing $R_{S}^{+}$ . Fix a reduced expression $s_{\boldsymbol{i}}$ compatible with $R_{0}^{+}$ . We also assume that $R_{S}^{+}=\{\alpha^{\boldsymbol{i}}_{k}\}_{k=N-N_{S}+1}^{N}$ holds:

\displaystyle\underbrace{\alpha^{\boldsymbol{i}}_{1},\,\alpha^{\boldsymbol{i}}_{2},\,\dots,\,\alpha^{\boldsymbol{i}}_{l}}_{R^{+}\setminus R_{0}^{+}},\,\underbrace{\alpha^{\boldsymbol{i}}_{l+1},\,\alpha^{\boldsymbol{i}}_{l+2},\,\dots,\alpha^{\boldsymbol{i}}_{N-N_{S}}}_{R_{0}^{+}\setminus R_{S}^{+}},\,\underbrace{\alpha^{\boldsymbol{i}}_{N-N_{S}+1},\,\alpha^{\boldsymbol{i}}_{N-N_{S}+2}\dots,\,\alpha^{\boldsymbol{i}}_{N}}_{R_{S}^{+}}.

where $l=\lvert R^{+}\setminus R_{0}^{+}\rvert$ . For an $\mathcal{A}[2Q_{0}^{-}]$ -algebra $(k,\chi)$ with $\chi_{-2\alpha}=1$ for $\alpha\in R_{S}^{+}$ , we introduce the following $k$ -subalgebras:

•

$U_{q,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim})$ , generated by $(\hat{K}_{\mu})_{\mu\in P},\,(\hat{E}_{\boldsymbol{i},\alpha})_{\alpha\in R^{+}}$ and $(\hat{F}_{\boldsymbol{i},\alpha}^{(r)})_{\alpha\in R_{S}^{+},r\geq 0}$ .
•

$U_{q,\chi}^{k}(\mathfrak{l}_{S}{}^{\sim})$ , generated by $(\hat{K}_{\mu})_{\mu\in P},\,(\hat{E}_{\boldsymbol{i},\alpha})_{\alpha\in R_{S}^{+}}$ and $(\hat{F}_{\boldsymbol{i},\alpha}^{(r)})_{\alpha\in R_{S}^{+},r\geq 0}$ .
•

$U_{q,\chi}^{k}(\mathfrak{u}_{S}^{+}{}^{\sim})$ , generated by $(\hat{E}_{\boldsymbol{i},\alpha})_{\alpha\in R^{+}\setminus R_{S}^{+}}$ .
•

$U_{q,\chi}^{k}(\mathfrak{u}_{S}^{-}{}^{\sim})$ , generated by $(\hat{F}_{\boldsymbol{i},\alpha}^{(r)})_{\alpha\in R^{+}\setminus R_{S}^{+},r\geq 0}$ .

Due to the additional requirement on $s_{\boldsymbol{i}}$ , these do not depend on the choice of such a reduced expression. Moreover these are spanned by the PBW basis introduced in Proposition 3.11. Hence the multiplication map induces the following tensor product decompositions:

(7)		$\displaystyle U_{q,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim})$	$\displaystyle\cong U_{q,\chi}^{k}(\mathfrak{l}_{S}{}^{\sim})\otimes U_{q,\chi}^{k}(\mathfrak{u}_{S}^{+}{}^{\sim}),$
(8)		$\displaystyle U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$	$\displaystyle\cong U_{q,\chi}^{k}(\mathfrak{u}_{S}^{-}{}^{\sim})\otimes U_{q,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim}).$

For $\chi\equiv 1$ , these algebras are simply denoted by $U_{q}^{k}(\mathfrak{g}{}^{\sim}),U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim}),U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})$ and $U_{q}^{k}(\mathfrak{u}_{S}^{\pm}{}^{\sim})$ respectively. These can be thought as subalgebras of $U_{q}^{k}(\mathfrak{g})$ by the identifications $\hat{F}_{\boldsymbol{i},\alpha}^{(r)}=F_{\boldsymbol{i},\alpha}^{(r)},\hat{K}_{\mu}=K_{\mu},\hat{E}_{\boldsymbol{i},\alpha}=\widetilde{E}_{\boldsymbol{i},\alpha}$ .

Lemma 3.13.

There is a canonical isomorphism from $U_{q,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim})$ to $U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})$ , which is defined as follows on the generators:

\displaystyle\hat{F}_{\boldsymbol{i},\alpha}^{(r)}\longmapsto F_{\boldsymbol{i},\alpha}^{(r)},\quad\hat{K}_{\mu}\longmapsto K_{\mu},\quad\hat{E}_{\boldsymbol{i},\alpha}\longmapsto\widetilde{E}_{\boldsymbol{i},\alpha}.

Remark 3.14.

In light of this fact, we identify $U_{q,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim})$ with $U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})$ .

Proof.

Using Proposition 3.11 we can define a $k$ -linear map from $U_{q,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim})$ to $U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})$ sending $\hat{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\hat{K}_{\mu}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}$ to $F_{\boldsymbol{i}}^{(\Lambda^{-})}K_{\mu}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}$ . What we have to show is multiplicativity of this map. By Proposition 3.4, $\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{0}^{+}}F_{\boldsymbol{i}}^{(\Lambda_{0}^{-})}$ with $\Lambda^{+}\leq N-N_{S}<\Lambda^{-}$ is a linear combination of $F_{\boldsymbol{i}}^{(\Lambda^{-})}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}$ with $\Lambda_{0}^{+}\leq N-N_{S}<\Lambda_{0}^{-}$ . Hence we have

\displaystyle F_{\boldsymbol{i}}^{(\Lambda_{1}^{-})}K_{\mu_{1}}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}}F_{\boldsymbol{i}}^{(\Lambda_{1}^{-})}K_{\mu_{1}}\widetilde{E}_{\boldsymbol{i}}^{\Lambda_{1}^{+}}=\sum_{\begin{subarray}{c}\Lambda^{\pm},\mu\in Q_{S}\\ \Lambda^{-}>N-N_{S}\end{subarray}}C_{\Lambda^{\pm},\mu}F_{\boldsymbol{i}}^{(\Lambda^{-})}K_{\mu_{1}+\mu_{2}+\mu}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}

when $\Lambda_{i}^{-}>N-N_{S}$ . Now the multiplicativity follows from this expansion and the definition of the generators $\hat{F}_{\boldsymbol{i},\alpha}^{(r)},\hat{K}_{\mu},\hat{E}_{\boldsymbol{i},\alpha}$ . ∎

Note that there is a canonical homomorphism $\pi_{S}\colon U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})\longrightarrow U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})$ which is identical on $U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})$ and equal to $\varepsilon$ on $U_{q}^{k}(\mathfrak{u}_{S}^{+}{}^{\sim})$ . All $U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})$ -modules are regarded as $U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})$ -modules via this homomorphism.

Definition 3.15.

We define a functor $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}\colon U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})\text{-}\mathrm{Mod}\longrightarrow U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})\text{-}\mathrm{Mod}$ as $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})\otimes_{U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})}\textendash$ . For a $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ -module $V$ , $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ is called the $(R_{0}^{+},\chi)$ -twisted parabolic induction module of $V$ .

Remark 3.16.

By the tensor product decomposition (8), $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ can be naturally identified with $U_{q,\chi}^{k}(\mathfrak{u}_{S}^{-}{}^{\sim})\otimes V$ as $U_{q,\chi}^{k}(\mathfrak{u}_{S}^{-}{}^{\sim})$ -modules.

Let $M$ be a $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ -module. We say that $m\in M$ is $S$ -maximal when $xm=\pi_{S}(x)m$ for $x\in U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})$ , which is the same thing as saying that $\hat{E}_{\boldsymbol{i},\beta}m=0$ for $\beta\in R^{+}\setminus R_{S}^{+}$ . The following universal property of $(R_{0}^{+},\chi)$ -twisted parabolic induction is proven in an analogous way to the usual setting.

Lemma 3.17.

Let $M$ be a $U_{q}^{k}(\mathfrak{g}{}^{\sim})$ -module. Then $M_{S\text{-}\max}$ , the set of $S$ -maximal vectors in $M$ , forms a $U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})$ -module. Moreover we have a canonical isomorphism:

\displaystyle\operatorname{\mathrm{Hom}}_{U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})}(V,M_{S\text{-}\max})\cong\operatorname{\mathrm{Hom}}_{U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})}(\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V,M).

Let $\mathrm{ind}_{\mathfrak{p}_{S},q}^{\mathfrak{g}}\colon U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})\text{-}\mathrm{Mod}\longrightarrow U_{q}^{k}(\mathfrak{g}{}^{\sim})\text{-}\mathrm{Mod}$ be the usual parabolic induction $U_{q}^{k}(\mathfrak{g}{}^{\sim})\otimes_{U_{q}^{k}(\mathfrak{p}_{S}{}^{\sim})}\textendash$ . In order to compare the $(R_{0}^{+},\chi)$ -twisted parabolic induction with the usual parabolic induction, we consider subalgebras $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})_{2Q}$ defined by restricting the Cartan part $\operatorname{\mathrm{span}}_{k}\{\hat{K}_{\mu}\}_{\mu\in P}$ to $\operatorname{\mathrm{span}}_{k}\{\hat{K}_{2\mu}\}_{\mu\in Q}$ and similar variants for the other subalgebras. Note that $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ is naturally isomorphic to $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})_{2Q}\otimes_{U_{q,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim})_{2Q}}V$ as $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})_{2Q}$ -module and the same thing can be said for $\mathrm{ind}_{\mathfrak{p}_{S},q}^{\mathfrak{g}}V$ .

Lemma 3.18.

Assume that $\chi\colon 2Q^{-}\longrightarrow k$ extends to a group homomorphism $\chi\colon 2Q\longrightarrow k^{\times}$ with $\chi|_{2Q_{S}}\equiv 1$ . Let $k_{\chi}$ be a $U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})$ -module which is $k$ as a $k$ -module and $\hat{F}_{\boldsymbol{i},\alpha}^{(r)},\hat{K}_{2\mu},\hat{E}_{\boldsymbol{i},\beta}$ act on it as $0,\chi_{2\mu},0$ respectively.

(i)

There is an isomorphism $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})_{2Q}\cong U_{q}^{k}(\mathfrak{g}{}^{\sim})_{2Q}$ with:

\displaystyle\hat{F}_{\boldsymbol{i},\alpha}^{(r)}\longmapsto\begin{cases}\chi_{2\alpha}^{r}F_{\boldsymbol{i},\alpha}^{(r)}&(\alpha\in R^{+}\setminus R_{0}^{+})\\ F_{\boldsymbol{i},\alpha}^{(r)}&(\alpha\in R^{+}\cap R_{0}^{+})\end{cases},\quad\hat{K}_{2\mu}\longmapsto\chi_{-2\mu}K_{2\mu},\quad\hat{E}_{\boldsymbol{i},\alpha}\longmapsto\widetilde{E}_{\boldsymbol{i},\alpha}.

(ii)

Let $V$ be a $U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})$ -module. Then $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ is canonically isomorphic to $\mathrm{ind}_{\mathfrak{p}_{S},q}^{\mathfrak{g}}(V\otimes k_{\chi})$ as a $U_{q}^{k}(\mathfrak{g}{}^{\sim})_{2Q}$ -module.

Proof.

(i) This can be immediately verified by the definition of $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ .

(ii) This can be seen from the universal property of the $(R_{0}^{+},\chi)$ -twisted parabolic induction and that of the usual parabolic induction. ∎

4. $S$ -maximal vectors in the formal setting

4.1. Notation

In Section 4 and 5, we work on $(k\llbracket h\rrbracket,s=e^{h/2L})$ where $k$ is a $\mathbb{Q}$ -algebra. In this case it is convenient to consider $h$ -adically completed tensor products, which is denoted by $\textendash\hat{\otimes}\textendash$ in this paper.

In the formal setting we consider the $h$ -adic Drinfeld-Jimbo deformation $U_{h}^{k}(\mathfrak{g}):=k\llbracket h\rrbracket\hat{\otimes}U_{h}(\mathfrak{g})$ , where the definition of $U_{h}(\mathfrak{g})$ is described in [MR1492989, Section 6.1.3, Definition 2] for instance. This is isomorphic to $(k\otimes U(\mathfrak{g}))\llbracket h\rrbracket$ as topological $k\llbracket h\rrbracket$ -algebra. Moreover we can see that a topological $U_{h}^{k}(\mathfrak{g})$ -module admitting a finite $k\llbracket h\rrbracket$ -basis is isomorphic to $(k\otimes V)\llbracket h\rrbracket$ where $V$ is a finite dimensional representation of $\mathfrak{g}$ ([MR1492989, Section 7.1.3, Proposition 10]). Such a module is called a finite integrable module in this paper and the category of finite integrable modules is denoted by $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{g}$ .

Fix $S,R_{0}^{+},s_{\boldsymbol{i}}$ and $\chi$ as those in Subsection 3.3. In the formal setting it is rather appropriate to add $\hat{H}_{\alpha}$ to $U_{q,\chi}^{k\llbracket h\rrbracket}(\mathfrak{g}{}^{\sim})$ so that $\hat{K}_{\alpha}=\exp{hd_{\alpha}\hat{H}_{\alpha}}$ . The precise definition is the following.

Definition 4.1.

Set $U^{k}(\mathfrak{h}):=k\otimes U(\mathfrak{h})$ and $U_{h}^{k}(\mathfrak{h})$ be the $h$ -adic completion of $U^{k}(\mathfrak{h})\otimes k\llbracket h\rrbracket$ . This is equipped with a canonical coproduct such that $\Delta(H)=H\otimes 1+1\otimes H$ for $H\in\mathfrak{h}$ . Consider $U_{q,\chi}^{k\llbracket h\rrbracket}(\mathfrak{g}{}^{\sim})\otimes_{k\llbracket h\rrbracket}U_{h}^{k}(\mathfrak{h})$ with the following multiplication:

(x\otimes f)(y\otimes g)=(\operatorname{\mathrm{wt}}y)(f_{(1)})(xy\otimes f_{(2)}g),

where $\operatorname{\mathrm{wt}}y$ , which is an element of $\mathfrak{h}^{*}$ , is considered as a character on $U_{h}^{k}(\mathfrak{h})$ . Then We define $U_{h,\chi}^{k}(\mathfrak{g}{}^{\sim})$ as a quotient of $U_{q,\chi}^{k\llbracket h\rrbracket}(\mathfrak{g}{}^{\sim})\otimes_{k\llbracket h\rrbracket}U_{h}^{k}(\mathfrak{h})$ by the ideal generated by $\hat{K}_{\lambda}\otimes 1-1\otimes\exp(h\lambda)$ , where $\lambda$ is considered as an element of $\mathfrak{h}$ uniquely determined by $\alpha(\lambda)=(\alpha,\lambda)$ for $\alpha\in\mathfrak{h}^{*}$ .

We refer definitions and properties developed in Section 3 to apply them to $U_{h,\chi}^{k}(\mathfrak{g}{}^{\sim})$ in a suitably modified form. For instance $U_{h,\chi}^{k}(\mathfrak{g}{}^{\sim})$ has a canonical left coaction of $U_{h}^{k}(\mathfrak{g})$ , the $h$ -adic Drinfeld-Jimbo deformation of $\mathfrak{g}$ .

4.2. Generalized maximal vector

Though what we would like to show in this section is a kind of semisimplicity of the tensor product of finite integrable $U_{h}^{k}(\mathfrak{g})$ -modules and $(R_{0}^{+},\chi)$ -twisted parabolically induced modules, we show a stronger result: there is an operator in $U_{h}^{k}(\mathfrak{g}\times\mathfrak{l}_{S})$ which produces $S$ -maximal vectors in the tensor product modules. To discuss such an operator, we introduce the following generalization of $S$ -maximal vectors. Like the Sweedler notation, $x_{(\mathfrak{u}_{S}^{-})}\otimes x_{(\mathfrak{p}_{S})}$ denotes the image of $x\in U_{h,\chi}^{k}(\mathfrak{g}{}^{\sim})$ under the isomorphism $U_{h,\chi}^{k}(\mathfrak{g}{}^{\sim})\cong U_{h,\chi}^{k}(\mathfrak{u}_{S}^{-}{}^{\sim})\otimes U_{h,\chi}^{k}(\mathfrak{p}_{S}{}^{\sim})$ . Similarly $x\in W\hat{\otimes}V$ is denoted by $x_{(W)}\otimes x_{(V)}$ . For instance, the image of $x$ under a linear mapping $w\otimes v\longmapsto w\otimes a\otimes v$ is denoted by $x_{(W)}\otimes a\otimes x_{(V)}$ .

Definition 4.2.

Let $V$ be a complete $U_{h}^{k}(\mathfrak{l}_{S})$ -module and $W$ be a complete $U_{h}^{k}(\mathfrak{g})$ -module. We say that $(m_{\Lambda})_{\Lambda\leq N-N_{S}}\subset W\hat{\otimes}V$ is a generalized $S$ -maximal vector if it satisfies the following equation for all $\beta\in R^{+}\setminus R_{S}^{+}$ :

(9)

\displaystyle\sum_{\Lambda}\hat{E}_{\boldsymbol{i},\beta,(1)}m_{\Lambda,(W)}\otimes(\hat{E}_{\boldsymbol{i},\beta,(2)}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{u}_{S}^{-})}\otimes\pi_{S}((\hat{E}_{\boldsymbol{i},\beta,(2)}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{p}_{S})})m_{\Lambda,(V)}=0.

Remark 4.3.

Strictly speaking, these equalities should be interpreted after expanding $(\hat{E}_{\boldsymbol{i},\beta,(2)}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{u}_{S}^{-})}$ along the PBW basis. Then we obtain well-defined equations since, for any fixed $\Gamma$ , only finitely many terms produce $\hat{F}_{\boldsymbol{i}}^{(\Gamma)}$ .

In the rest of this subsection, we assume $\chi_{-2\alpha}=1$ for $\alpha\in R_{S}^{+}$ and $\chi_{-2\alpha}-1\in k\llbracket h\rrbracket^{\times}$ for $\alpha\in R_{0}^{+}\setminus R_{S}^{+}$ .

Proposition 4.4.

Let $V$ (resp. W) be a complete $U_{h}^{k}(\mathfrak{l}_{S})$ -module (resp. $U_{h}^{k}(\mathfrak{g})$ -module). Then, for any $m\in W\hat{\otimes}V$ , a generalized $S$ -maximal vector $(m_{\Lambda})_{\Lambda}$ with $m_{0}=m$ exists and is unique.

To prove this proposition, we consider a generalized version. Let $(p_{\Gamma})_{\Gamma\leq N-N_{S}}$ be the dual basis of the PBW basis $(\hat{F}_{\boldsymbol{i}}^{(\Gamma)})_{\Gamma\leq N-N_{S}}$ of $U_{h,\chi}^{k}(\mathfrak{u}_{S}^{-}{}^{\sim})$ . Then, for $x\in W\hat{\otimes}U_{h,\chi}^{k}(\mathfrak{u}_{S}^{-}{}^{\sim})\hat{\otimes}V$ , we define the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma)}$ in $x$ as $(\operatorname{\mathrm{id}}\otimes p_{\Gamma}\otimes\operatorname{\mathrm{id}})(x)\in W\hat{\otimes}V$ .

We say that $(m_{\Lambda})_{\Lambda\leq N-N_{S}}$ is a weak solution of (9) for $\beta=\alpha^{\boldsymbol{i}}_{l}$ if the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma)}$ in the RHS is $0$ for all $\Gamma$ with $\Gamma\geq l.$

Lemma 4.5.

Fix a positive integer $0\leq l\leq N-N_{S}$ .

(i)

If $(m_{\Lambda})_{l<\Lambda\leq N-N_{S}}$ is given, this extends a unique family $(m_{\Lambda})_{\Lambda\leq N-N_{S}}$ which is a weak solution of (9) for all $\alpha^{\boldsymbol{i}}_{m}$ with $1\leq m\leq l$ .
(ii)

If $(m_{\Lambda})_{\Lambda\leq N-N_{S}}$ is a weak solution of (9) for all $\alpha^{\boldsymbol{i}}_{m}$ with $1\leq m\leq l$ , it actually satisfies (9) for $\alpha^{\boldsymbol{i}}_{m}$ with $1\leq m\leq l$ .

Let $\Pi\colon Q\longrightarrow Q_{\Delta\setminus S}$ be the projection with respect to a basis $\Delta$ .

Proof.

(i) Fix a multi-index $\Gamma$ with $l\not<\Gamma\leq\lvert R^{+}\setminus R_{S}^{+}\rvert$ and consider a decomposition $\Gamma=\Gamma^{\prime}+\delta_{i}$ such that $\gamma_{1}=\gamma_{2}=\cdots=\gamma_{i-1}=0$ and $\gamma_{i}=\gamma^{\prime}_{i}+1$ . In this case $i\leq l$ holds.

We look at the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime})}$ in the RHS of (9) with $\beta=\alpha^{\boldsymbol{i}}_{i}$ . Note that $\Delta(\hat{E}_{\boldsymbol{i},i})$ is a linear combination of $K_{\Lambda^{r}\cdot\alpha^{\boldsymbol{i}}}^{-1}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{l}}\otimes\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}$ with $\Lambda^{r}\leq i\leq\Lambda^{l}$ and $\Lambda^{l}\cdot\alpha^{\boldsymbol{i}}+\Lambda^{r}\cdot\alpha^{\boldsymbol{i}}=\alpha^{\boldsymbol{i}}_{i}$ as we can see in (5).

If $\Pi(\Lambda^{l}\cdot\alpha^{\boldsymbol{i}})\neq 0$ holds, $p_{\Gamma^{\prime}}((\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{u}_{S}^{-})})\pi_{S}((\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{p}_{S})})\neq 0$ implies $\Pi(\Lambda\cdot\alpha^{\boldsymbol{i}})=\Pi(\Gamma^{\prime}\cdot\alpha^{\boldsymbol{i}})+\Pi(\Lambda^{r}\cdot\alpha^{\boldsymbol{i}})<\Pi(\Gamma\cdot\alpha^{\boldsymbol{i}})$ .

If $\Pi(\Lambda^{l}\cdot\alpha^{\boldsymbol{i}})=0$ and $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{i-1}=0$ holds, $\Pi(\Lambda\cdot\alpha^{\boldsymbol{i}})=\Pi(\Gamma\cdot\alpha^{\boldsymbol{i}})$ is necessary for $p_{\Gamma^{\prime}}((\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{u}_{S}^{-})})\pi_{S}((\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{p}_{S})})\neq 0$ by the discussion above. Moreover $\Lambda^{r}<i$ holds when $\Lambda^{l}\neq 0$ , since $\Lambda^{l}\cdot\alpha^{\boldsymbol{i}}+\Lambda^{r}\cdot\alpha^{\boldsymbol{i}}=\alpha^{\boldsymbol{i}}_{i}$ . Then Proposition 3.4 (ii) implies $p_{\Gamma^{\prime}}((\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{u}_{S}^{-})})\pi_{S}((\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{p}_{S})})\in h^{\lvert\Lambda^{l}\rvert}U_{h}^{k}(\mathfrak{l}_{S}{}^{\sim})$ . On the other hand $\Lambda^{r}=\delta_{i}$ holds when $\Lambda^{l}=0$ , since $i\leq\Lambda^{r}$ and $\Lambda^{\prime}\cdot\alpha^{\boldsymbol{i}}\neq\alpha^{\boldsymbol{i}}_{i}$ when $i<\Lambda^{\prime}$ by [MR1169886, Theorem, p.662]. Hence we have to look at the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime})}$ in the following summation:

\displaystyle\sum_{\Lambda}K_{\beta}^{-1}m_{\Lambda,(W)}\otimes[\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i}}^{(\Lambda)}]_{q,(\mathfrak{u}_{S}^{-})}\otimes\pi_{S}([\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i}}^{(\Lambda)}]_{q,(\mathfrak{p}_{S})})m_{\Lambda,(V)}.

Consider a decomposition $\Lambda=\lambda_{i}\delta_{i}+\Lambda_{>i}$ with $\Lambda_{>i}>i$ . Then we have

\displaystyle[\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i}}^{(\Lambda)}]_{q}=[\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i},\beta}^{(\lambda_{i})}]_{q}\hat{F}_{\boldsymbol{i}}^{(\Lambda_{>i})}+q_{\beta}^{2\lambda_{i}}\hat{F}_{\boldsymbol{i},i}^{(\lambda_{i})}[\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i}}^{(\Lambda_{>i})}]_{q}.

When $\Lambda=\Gamma$ , the first term produces $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime})}$ and the second term does not. Moreover we can determine its coefficient completely since we have

\displaystyle[\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i},\beta}^{(r)}]_{q}=\begin{cases}q_{\beta}^{2r}\hat{F}_{\boldsymbol{i},\beta}^{(r-1)}(q_{\beta}^{1-r}-q_{\beta}^{r-1}e_{-2\beta}\hat{K}_{-2\beta})&(\beta\in R_{0}^{+}),\\ q_{\beta}^{2r}\hat{F}_{\boldsymbol{i},\beta}^{(r-1)}(q_{\beta}^{1-r}e_{2\beta}-q_{\beta}^{r-1}\hat{K}_{-2\beta})&(\beta\not\in R_{0}^{+}).\end{cases}

Also note that the coefficient of $K_{\beta}^{-1}\otimes\hat{E}_{\boldsymbol{i},\beta}$ in $\Delta(\hat{E}_{\boldsymbol{i},\beta})$ is $1$ by (5).

When $\Lambda\neq\Gamma$ , the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime})}$ in $[\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i},\beta}^{(\lambda_{i})}]_{q}\hat{F}_{\boldsymbol{i}}^{(\Lambda_{>i})}$ is $0$ and that in $\hat{F}_{\boldsymbol{i},i}^{(\lambda_{i})}[\hat{E}_{\boldsymbol{i},\beta},\hat{F}_{\boldsymbol{i}}^{(\Lambda_{>i})}]_{q}$ is in $hU_{h}^{k}(\mathfrak{l}_{S})$ by Proposition 3.4 (i).

Take $\nu\in Q_{\Delta\setminus S}^{+}$ and set $B_{l,\nu}:=\{\Gamma\mid\Pi(\Gamma\cdot\alpha^{\boldsymbol{i}})=\nu,\,l\not<\Gamma\leq\lvert R^{+}\setminus R_{S}^{+}\rvert\}$ . Then $n_{l,\nu}:=\lvert B_{l,\nu}\rvert<\infty$ . We also consider an enumeration $\{\Gamma_{i}\}_{i}$ on $B_{l,\nu}$ along the lexicographic order. Then the consideration above implies that we can find a $n_{l,\nu}\times n_{l,\nu}$ -matrix $A_{l,\nu}$ and $(u_{i})_{i}\in(W\hat{\otimes}V)^{n_{l,\nu}}$ such that

•

the $i$ -th entry of $A_{l,\nu}(m_{\Gamma_{i}})_{i}+(u_{i})_{i}$ is the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime}_{i})}$ in the LHS of (9),
•

each entry of $A_{l,\nu}$ is in $U_{h}^{k}(\mathfrak{g}\times\mathfrak{l}_{S})$ ,
•

each $u_{i}$ is in $U_{h}^{k}(\mathfrak{g}\times\mathfrak{l}_{S})$ -submodule generated by $m_{\Lambda}$ with $\Pi(\Lambda\cdot\alpha^{\boldsymbol{i}})<\nu$ ,

•

the matrix $A_{l,\nu}$ is of the following form modulo $h$ :

\displaystyle\begin{pmatrix}\sigma_{1}(1-e_{-2\sigma_{1}\beta_{1}})&\ast&\cdots&\ast\\ 0&\sigma_{2}(1-e_{-2\sigma_{2}\beta_{2}})&\cdots&\ast\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\sigma_{n_{l,\nu}}(1-e_{-2\sigma_{n_{l,\nu}}\beta_{n_{l,\nu}}})\end{pmatrix},

where $\sigma_{i}=1$ when $\beta_{i}\in R^{+}\cap R_{0}^{+}$ and $\sigma_{i}=-1$ when $\beta_{i}\in R^{+}\setminus R_{0}^{+}$ .

Since $1-e_{2\sigma_{i}\beta_{i}}$ is also invertible for all $i$ , $A_{l,\nu}$ is invertible. Hence we can solve the equations $A_{l,\nu}(m_{\Gamma_{i}})_{i}+(u_{i})_{i}=0$ inductively on $\nu$ and obtain a unique weak solution.

(ii) We show the statement by induction on $l$ . If $l=1$ , there is nothing to prove. For induction step, we assume that the statement holds for $l-1$ with $l\geq 2$ . Then any weak solution of $(\ref{eq:recursion formula})$ for all $\alpha^{\boldsymbol{i}}_{m}$ with $1\leq m\leq l$ is actually a solution for all $1\leq m<l$ . Now we define $m^{\prime}_{\Lambda}$ as the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Lambda)}$ in the LHS of (9) for $\alpha^{\boldsymbol{i}}_{l}$ . What we would like to show is $m_{\Lambda}=0$ for all $\Lambda$ . For this, it suffices to show that $(m^{\prime}_{\Lambda})_{\Lambda}$ is asolution of (9) for $\alpha^{\boldsymbol{i}}_{m}$ with $1\leq m\leq l-1$ since the uniqueness part of (i) and $m^{\prime}_{\Lambda}=0$ for $l<\Lambda\leq N-N_{S}$ implies $m^{\prime}_{\Lambda}=0$ for all $\Lambda$ . This follows from Proposition 3.3, which states that $\hat{E}_{\boldsymbol{i},m}\hat{E}_{\boldsymbol{i},l}$ can be expressed as a linear combination of $\hat{E}_{\boldsymbol{i},l}\hat{E}_{\boldsymbol{i},m}$ and other PBW vectors of the form $\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}$ with $m<\Lambda^{+}<l$ . ∎

Proof of Proposition 4.4.

This is a special case of Lemma 4.5, namely the case of $l=N-N_{S}$ . ∎

Definition 4.6.

Set $V=U_{h}^{k}(\mathfrak{l}_{S})$ and $W=U_{h}^{k}(\mathfrak{g})$ . A generalized $S$ -maximal vector $(U_{\Lambda})_{\Lambda}$ with $U_{0}=1$ is called the $S$ -maximizer.

The $S$ -maximizer is a universal solution of (9) in the following sense.

Proposition 4.7.

Let $(U_{\Lambda})_{\Lambda}$ be the $S$ -maximizer and $(m_{\Lambda})_{\Lambda}$ is a generalized $S$ -maximal vector. Then $m_{\Lambda}=U_{\Lambda}m_{0}$ holds for any $\Lambda$ .

Proof.

The statements immediately follows from the definition of generalized $S$ -maximal vector and the uniqueness part of Proposition 4.4. ∎

4.3. Construction of the 2-cocycles

Next we discuss a further property of the $S$ -maximizer. We begin with the following preparation.

Lemma 4.8.

Let $\mathcal{O}$ be the subalgebra of $U_{q}^{\mathcal{A}}(\mathfrak{g})$ generated by $\{\widetilde{F}_{\boldsymbol{i},\alpha}\}_{\alpha\in R^{+}}$ , $\{\widetilde{E}_{\boldsymbol{i},\alpha}\}_{\alpha\in R}$ and $\{K_{\mu}\}_{\mu\in P}$ .

(i)

A family $\{\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}K_{\mu}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}\}_{\Lambda^{\pm},\mu}$ is an $\mathcal{A}$ -basis of $\mathcal{O}$ .
(ii)

For any $x,y\in\mathcal{O}$ , $[x,y]_{q^{\pm 1}}\in(q-1)\mathcal{O}$ .

Proof.

(i) This follows from Lemma 3.10 by specializing $e_{-2\alpha}$ to $1$ for all $\alpha\in R^{+}$ .

(ii) We may assume both of $x,y$ are generators in the statement. Moreover we may assume $x,y$ are not $K_{\mu}$ for any $\mu$ since it is easy to see the statement in such a case.

We may assume that $x=\widetilde{E}_{\boldsymbol{i},\alpha}$ with $\alpha\in\Delta$ and $y=\widetilde{F}_{\boldsymbol{i},\beta}$ with $\beta\in R^{+}$ , using Lemma 3.2. In particular $y\in U_{q}^{\mathcal{A}}(\mathfrak{n}^{-})$ . Hence $[\acute{E}_{\boldsymbol{i},\alpha},y]_{q}$ is an element of $U_{q}^{\mathcal{A}}(\mathfrak{b}^{-})$ . On the other hand $\mathrm{span}_{\mathcal{A}}\{\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}K_{\mu}\acute{E}_{\boldsymbol{i}}^{(\Lambda^{+})}\}$ is closed under the multiplication. Hence $[\acute{E}_{\boldsymbol{i},\alpha},y]_{q}$ is in the intersection of these subalgebras, which is spanned by $\{\widetilde{F}_{\boldsymbol{i}}^{\Lambda^{-}}K_{\mu}\}_{\Lambda^{-},\mu}$ . This means $[x,y]\in(q_{\alpha}-q_{\alpha}^{-1})\mathcal{O}$ since $x=(q_{\alpha}-q_{\alpha}^{-1})\acute{E}_{\boldsymbol{i},\alpha}$ . ∎

Proposition 4.9.

Let $(U_{\Lambda})_{\Lambda}$ be the $S$ -maximizer. Then $U_{\Lambda}\in h^{\lvert\Lambda\rvert}U_{h}^{k}(\mathfrak{g}\times\mathfrak{l}_{S})$ for any $\Lambda$ .

Proof.

We construct a weak solution of (9) for all $\beta\in R^{+}\setminus R_{S}^{+}$ by another twisted parabolic induction using $U_{h,\chi}^{k}({}^{\sim}\mathfrak{g})$ i.e. $U_{h,\chi}^{k}({}^{\sim}\mathfrak{g})\otimes_{U_{h,\chi}^{k}({}^{\sim}\mathfrak{p}_{S})}V$ . Note that this can be naturally embedded in $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ .

In this case we consider the following equation for each $\beta\in R^{+}\setminus R_{0}^{+}$ :

(10)

\displaystyle\sum_{\Lambda}\check{E}_{\boldsymbol{i},\beta,(1)}\widetilde{m}_{\Lambda,(W)}\otimes(\check{E}_{\boldsymbol{i},\beta,(2)}\check{F}_{\boldsymbol{i}}^{\Lambda})_{(\mathfrak{u}_{S}^{-})}\otimes\pi_{S}((\check{E}_{\boldsymbol{i},\beta,(2)}\check{F}_{\boldsymbol{i}}^{\Lambda})_{(\mathfrak{p}_{S})})\widetilde{m}_{\Lambda,(V)}=0.

Then we can define the notion of a weak solution of (10) in a similar manner to that of (9). If this equation has a weak solution for any $m=\widetilde{m}_{0}$ , we can see that the statement is valid since we can obtain a weak solution $(m_{\Lambda})_{\Lambda}$ of (9) for all $\beta\in R^{+}\setminus R_{S}^{+}$ by setting $m_{\Lambda}:=(q_{\alpha^{\boldsymbol{i}}}-q_{\alpha^{\boldsymbol{i}}}^{-1})^{\Lambda}[\Lambda]_{q_{\alpha^{\boldsymbol{i}}}}!\widetilde{m}_{\Lambda}$ . Then Lemma 4.5 (ii) implies that this is actually a generalized $S$ -maximal vector. In particular we can see $m_{\Lambda}\in h^{\lvert\Lambda\rvert}W\hat{\otimes}V$ .

The outline of the proof is almost the same as that of the proof of Lemma 4.5 with $l=N-N_{S}$ . The only different point is that $A_{\nu}$ , the matrix of the coefficient of $\check{F}_{\boldsymbol{i}}^{\Gamma^{\prime}}$ in the term of $\Lambda$ with $\Pi(\Lambda\cdot\alpha^{\boldsymbol{i}})=\Pi(\Gamma\cdot\alpha^{\boldsymbol{i}})=\nu$ , is a lower triangular matrix with invertible diagonal entries modulo $h$ . In the following, we prove this fact.

Fix $\Gamma$ and take $\Gamma^{\prime}$ and $\beta=\alpha^{\boldsymbol{i}}_{i}$ in the same way as Proof of Lemma 4.5. Noting that $U_{h}(\mathfrak{g})$ is cocommutative modulo $h$ , we can see that $\Delta(\check{E}_{\boldsymbol{i},\beta})=\check{E}_{\boldsymbol{i},\beta}\otimes 1+K_{\beta}^{-1}\otimes\check{E}_{\boldsymbol{i},\beta}$ modulo $h$ and the first term does not affect to the entries of $A_{\nu}$ . Hence it suffices to look at the coefficient of $\check{F}_{\boldsymbol{i}}^{\Gamma^{\prime}}$ in the following:

\displaystyle\sum_{\Lambda}K_{\beta}^{-1}\widetilde{m}_{\Lambda,(W)}\otimes[\check{E}_{\boldsymbol{i},\beta},\check{F}_{\boldsymbol{i}}^{\Lambda}]_{q,(\mathfrak{u}_{S}^{-})}\otimes\pi_{S}([\check{E}_{\boldsymbol{i},\beta},\check{F}_{\boldsymbol{i}}^{\Lambda}]_{q,(\mathfrak{p}_{S})})\widetilde{m}_{\Lambda,(V)}.

We consider the following decomposition of a quantum commutator:

(11)		$\displaystyle{}_{q}=[\check{E}_{\boldsymbol{i},\beta},\check{F}_{\boldsymbol{i}}^{\Lambda_{<i}}]_{q}\check{F}_{\boldsymbol{i}}^{\lambda_{i}}\check{F}_{\boldsymbol{i}}^{\Lambda_{>i}}$	$\displaystyle+q^{(\beta,\Lambda_{<i}\cdot\alpha^{\boldsymbol{i}})}\check{F}_{\boldsymbol{i}}^{\Lambda_{<i}}[\check{E}_{\boldsymbol{i},\beta},\check{F}_{\boldsymbol{i},i}^{\lambda_{i}}]_{q}\check{F}_{\boldsymbol{i}}^{\Lambda_{>i}}$
(11)			$\displaystyle+q^{(\beta,\Lambda_{<i}\cdot\alpha^{\boldsymbol{i}}+\lambda_{i}\alpha^{\boldsymbol{i}}_{i})}\check{F}_{\boldsymbol{i}}^{\Lambda_{<i}}\check{F}_{\boldsymbol{i},i}^{\lambda_{i}}[\check{E}_{\boldsymbol{i},\beta},\check{F}_{\boldsymbol{i}}^{\Lambda_{>i}}]_{q}$

Assume $\Lambda_{<i}\neq 0$ . It is easy to see that the second and third terms do not affect the coefficient of $\check{F}_{\boldsymbol{i}}^{\Gamma^{\prime}}$ . We show that even the first term does not modulo $h$ . By Proposition 3.4 and Lemma 4.8 we have the following expansion:

\displaystyle[\check{E}_{\boldsymbol{i},\beta},\check{F}_{\boldsymbol{i}}^{\Lambda_{<i}}]_{q}=\sum_{\begin{subarray}{c}\Lambda^{-}<i\leq\Lambda^{+},\mu\\ \beta-\Lambda^{+}\cdot\alpha^{\boldsymbol{i}}=\Lambda_{i}\cdot\alpha^{\boldsymbol{i}}-\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}\end{subarray}}C_{\Lambda^{\pm},\mu}\check{F}_{\boldsymbol{i}}^{\Lambda^{-}}K_{\mu}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}

Then Lemma 4.8 allows us to expand the first term as follows modulo $h$ :

\displaystyle\sum_{\begin{subarray}{c}\Lambda^{-}<i\leq\Lambda^{+},\mu\\ \beta-\Lambda^{+}\cdot\alpha^{\boldsymbol{i}}=\Lambda_{i}\cdot\alpha^{\boldsymbol{i}}-\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}\end{subarray}}C_{\Lambda^{\pm},\mu}\check{F}_{\boldsymbol{i}}^{\Lambda^{-}}\check{F}_{\boldsymbol{i}}^{\lambda_{i}}\check{F}_{\boldsymbol{i}}^{\Lambda_{>i}}K_{\mu}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}.

In this summation $\check{F}_{\boldsymbol{i}}^{\Gamma^{\prime}}$ appears as the $U_{h,\chi}^{k}({}^{\sim}\mathfrak{u}_{S}^{-})$ -part only when $\Lambda^{-}=0$ , in which case $\Lambda^{+}\neq 0$ since $\Lambda^{+}\cdot\alpha^{\boldsymbol{i}}-\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}=\beta-\Lambda_{<i}\cdot\alpha^{\boldsymbol{i}}\neq 0$ by [MR1169886, Theorem, p. 662]. If $\Lambda^{+}>N-N_{S}$ holds additionally, $U_{h}^{k}(\mathfrak{l}_{S})$ -part of such a term is $0\operatorname{\mathrm{mod}}h$ since $\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}\in hU_{h}^{k}(\mathfrak{l}_{S})$ . If $\Lambda^{-}\not>N-N_{S}$ holds, such a term is removed by $\pi_{S}$ . Hence we can ignore $\Lambda$ with $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{i-1}=0$ .

Next assume $\Lambda_{<i}=0$ . The case of $\Lambda=\Gamma$ can be treated in the same way as the proof of Lemma 4.5. When $\Lambda\neq\Gamma$ , the second term does not affect the coefficient of $\check{F}_{\boldsymbol{i}}^{\Gamma^{\prime}}$ and the third term does only when $\lambda_{i}=\gamma_{i}-1<\gamma_{i}$ .

Therefore $A_{\nu}$ is of the following form modulo $h$ :

(12)

\displaystyle\begin{pmatrix}[\gamma_{1,i_{1}}]_{q_{\beta_{1}}}\sigma_{1}(1-e_{-2\sigma_{1}\beta_{1}})&\cdots&0\\ \vdots&\ddots&\vdots\\ \ast&\cdots&[\gamma_{l,i_{n_{\nu}}}]_{q_{\beta_{n_{\nu}}}}\sigma_{n_{\nu}}(1-e_{-2\sigma_{n_{\nu}}\beta_{n_{\nu}}})\end{pmatrix},

where $\Gamma_{k}=\Gamma^{\prime}_{k}+\delta_{i_{k}}$ and $\beta_{k}=\alpha^{\boldsymbol{i}}_{i_{k}}$ are taken as before. Since $[\gamma_{k,i_{k}}]_{q_{\beta_{k}}}$ is invertible in our setting, we can see the existence of a weak solution of (10) for all $\beta\in R^{+}\setminus R_{S}^{+}$ . ∎

Corollary 4.10.

Let $V$ be a $U_{h}^{k}(\mathfrak{l}_{S})$ -module and $W$ be a $U_{h}^{k}(\mathfrak{g})$ -module. Then there is a unique $S$ -maximal vector of the form $m_{(W)}\hat{\otimes}(1\otimes m_{(V)})+\cdots$ for any $m\in W\hat{\otimes}V$ .

Proof.

By the definition of the completion, we can consider $\sum_{\Lambda}(U_{\Lambda}m)_{(W)}\otimes\hat{F}_{\boldsymbol{i}}^{(\Lambda)}\otimes(U_{\Lambda}m)_{(V)}$ , which is the $S$ -maximizer. ∎

Remark 4.11.

If $V$ and $W$ are finite integrable, there is no need to take the completion.

Remark 4.12.

Since $1\otimes v$ is $S$ -maximal in $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ , the uniqueness implies that an $S$ -maximal vector in $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ must be of the form $1\otimes v$ .

Definition 4.13.

Let $\operatorname{\mathrm{Rep}}_{h,\chi}^{\mathrm{f}}\mathfrak{l}_{S}$ be the full subcategory of $U_{h,\chi}^{k}(\mathfrak{g}{}^{\sim})\text{-}\mathrm{Mod}$ consisting of objects which are isomorphic to $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ for some $V\in\operatorname{\mathrm{Rep}}^{\mathrm{f}}_{h}\mathfrak{l}_{S}$ .

Proposition 4.14.

The following facts hold:

(i)

The functor $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}$ restricts to an equivalence of category between $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{l}_{S}$ and $\operatorname{\mathrm{Rep}}_{h,\chi}^{\mathrm{f}}\mathfrak{l}_{S}$ .
(ii)

For $V\in\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{l}_{S}$ and $W\in\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{g}$ , there is a natural isomorphism from $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W\otimes V)$ to $W\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ such that $1\otimes(w\otimes v)\longmapsto w\otimes(1\otimes v)+\cdots$ .

Therefore $\operatorname{\mathrm{Rep}}_{h,\chi}^{\mathrm{f}}\mathfrak{l}_{S}$ naturally becomes a semisimple left $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{g}$ -module category.

Proof.

Using the universal property of $(R_{0}^{+},\chi)$ -parabolic induction (Lemma 3.17), we can deduce (i) from Remark 4.12.

Next we prove (ii). By the universal property and Corollary 4.10 we can see that a homomorphism stated in (ii) actually exists and is unique. Hence it suffices to show that this is an isomorphism. Consider the following composition of homomorphisms:

	$\displaystyle W\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$	$\displaystyle\longrightarrow W\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W^{*}\otimes W\otimes V)$
		$\displaystyle\longrightarrow W\otimes W^{*}\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W\otimes V)\longrightarrow\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W\otimes V),$

where $k\llbracket h\rrbracket\longrightarrow W^{*}\otimes W$ is given by $1\longmapsto\sum_{i}e^{i}\otimes e_{i}$ with $(e_{i})_{i}$ is a basis of $W$ and $(e^{i})_{i}$ is its dual basis of $W^{*}$ , and $W\otimes W^{*}\longrightarrow k\llbracket h\rrbracket$ is given by $w\otimes f\longmapsto f(K_{2\rho}w)$ where $\rho$ is the half sum of positive roots. Then the composition of this homomorphism and $\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W\otimes V)\longrightarrow W\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ maps $1\otimes(w\otimes v)$ to $1\otimes(w\otimes v)\operatorname{\mathrm{mod}}h$ and is isomorphism. This fact implies the injectivity of the homomorphism in the statement. We can see surjectivity in a similar way. ∎

We rewrite $\operatorname{\mathrm{Rep}}_{h,\chi}^{\mathrm{f}}\mathfrak{l}_{S}$ as a $2$ -cocycle twist of $\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{l}_{S}$ . For $V\in\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{l}_{S}$ and $W,W^{\prime}\in\operatorname{\mathrm{Rep}}_{h}^{\mathrm{f}}\mathfrak{g}$ , we consider the following isomorphism:

(13)		$\displaystyle\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W\otimes W^{\prime}\otimes V)$	$\displaystyle\cong W\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W^{\prime}\otimes V)$
(13)			$\displaystyle\cong(W\otimes W^{\prime})\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V\cong\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}(W\otimes W^{\prime}\otimes V).$

Proposition 4.14 (i) implies that this automorphism is induced from an automorphism on $W\otimes W^{\prime}\otimes V$ . Actually we have a stronger statement. The centralizer of $U_{h}^{k}(\mathfrak{l}_{S})$ embedded in $U_{h}^{k}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})$ via $(\Delta\otimes\operatorname{\mathrm{id}})\Delta$ is denoted by $U_{h}^{k}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})^{\mathfrak{l}_{S}}$ .

Proposition 4.15.

There is a unique $\Psi_{h,\chi}^{k}\in U_{h}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})^{\mathfrak{l}_{S}}$ such that the isomorphism (13) is induced from the mulitplication by $\Psi_{h,\chi}$ . Moreover $\Psi_{h,\chi}$ satisfies the cocyle identities

(14)			$\displaystyle(\operatorname{\mathrm{id}}\otimes\Delta\otimes\operatorname{\mathrm{id}})(\Psi_{h,\chi})(1\otimes\Psi_{h,\chi})=(\Delta\otimes\operatorname{\mathrm{id}}\otimes\operatorname{\mathrm{id}})(\Psi_{h,\chi})(\operatorname{\mathrm{id}}\otimes\operatorname{\mathrm{id}}\otimes\Delta)(\Psi_{h,\chi}),$
(15)			$\displaystyle(\varepsilon\otimes\operatorname{\mathrm{id}}\otimes\operatorname{\mathrm{id}})(\Psi_{h,\chi})=(\operatorname{\mathrm{id}}\otimes\varepsilon\otimes\operatorname{\mathrm{id}})(\Psi_{h,\chi})=1$

and has the following expansion:

(16)		$\displaystyle\Psi_{h,\chi}=1-h$	$\displaystyle\left\{\sum_{\beta\in R^{+}\cap R_{0}^{+}\setminus R_{S}^{+}}\frac{2d_{\beta}}{1-\chi_{-2\beta}}E_{\boldsymbol{i},\beta}\otimes F_{\boldsymbol{i},\beta}\otimes 1\right.$
(17)			$\displaystyle\hskip 50.00008pt\left.+\sum_{\beta\in R^{+}\setminus R_{0}^{+}}\frac{2d_{\beta}\chi_{2\beta}}{\chi_{2\beta}-1}E_{\boldsymbol{i},\beta}\otimes F_{\boldsymbol{i},\beta}\otimes 1\right\}+\cdots.$

Proof.

Fix $u\in W\otimes W^{\prime}\otimes V$ . Let $u^{\prime}$ be the image of $1\otimes u$ at $W\otimes W^{\prime}\otimes\mathrm{ind}_{\mathfrak{l}_{S},q}^{\mathfrak{g},\chi}V$ in (13) and $\Psi_{W,W^{\prime},V}\in\mathrm{End}_{U_{h}^{k}(\mathfrak{l}_{S})}(W\otimes W^{\prime}\otimes V)$ be the isomorphism corresponding to (13). Then we have

\displaystyle u^{\prime}=\Psi_{W,W^{\prime},V}(u)_{(W\otimes W^{\prime})}\otimes(1\otimes\Psi_{W,W^{\prime},V}(u)_{(V)})+\cdots.

On the other hand, by straightforward calculation, we also have

	$\displaystyle u^{\prime}=\sum_{\Lambda}$	$\displaystyle((\operatorname{\mathrm{id}}\otimes\Delta)(U_{\Lambda})u)_{(W)}$
		$\displaystyle\otimes\hat{F}_{\boldsymbol{i}}^{(\Lambda)}(((\operatorname{\mathrm{id}}\otimes\Delta)(U_{\Lambda})u)_{(W^{\prime})}\otimes(1\otimes((\operatorname{\mathrm{id}}\otimes\Delta)(U_{\Lambda})u)_{(V)})+\cdots).$

Hence we can obtain $\Psi_{h,\chi}$ as follows:

(18)

\displaystyle\Psi_{h,\chi}=\sum_{\Lambda}(1\otimes\hat{F}_{\boldsymbol{i},(1)}^{\Lambda}\otimes\pi_{S}(\hat{F}_{\boldsymbol{i},(2)}^{\Lambda}))(\operatorname{\mathrm{id}}\otimes\Delta)(U_{\Lambda}).

Note that this summation is well-defined since $U_{\Lambda}\in h^{\lvert\Lambda\rvert}U_{h}^{k}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})$ .

The cocycle identities (14), (15) can be obtained as consequences of direct calculations. To prove the expansion, we use the formula (18). Since $U_{\Lambda}\in h^{\lvert\Lambda\rvert}U_{h}^{k}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})$ and

\displaystyle\Delta(\hat{F}_{\boldsymbol{i},\beta})=\begin{cases}F_{\boldsymbol{i},\beta}\otimes 1+K_{\beta}^{-1}\otimes F_{\boldsymbol{i},\beta}&\operatorname{\mathrm{mod}}h\quad(\beta\in R^{+}\cap R_{0}^{+}),\\ \chi_{2\beta}F_{\boldsymbol{i},\beta}\otimes 1+K_{\beta}^{-1}\otimes F_{\boldsymbol{i},\beta}&\operatorname{\mathrm{mod}}h\quad(\beta\in R^{+}\setminus R_{0}^{+}),\end{cases}

it suffices to calculate the following summation:

\displaystyle\sum_{\beta\in R^{+}\cap R_{0}^{+}}(1\otimes F_{\boldsymbol{i},\beta}\otimes 1)(\operatorname{\mathrm{id}}\otimes\Delta)(U_{\beta})+\sum_{\beta\in R^{+}\setminus R_{0}^{+}}\chi_{2\beta}(1\otimes F_{\boldsymbol{i},\beta}\otimes 1)(\operatorname{\mathrm{id}}\otimes\Delta)(U_{\beta}),

where $U_{\beta}:=U_{\delta_{k}}$ with $\beta=\alpha^{\boldsymbol{i}}_{k}$ . To determine $U_{\beta}$ modulo $h$ for each $\beta=\alpha^{\boldsymbol{i}}_{k}$ , we go back to (12) and show that entries in the $i$ -th row of $A_{\Pi(\beta)}$ , where $\Gamma_{i}=\delta_{k}$ , are $0$ modulo $h$ except the diagonal entry. Let $\Lambda$ be a multi-index with $\Pi(\Lambda\cdot\alpha^{\boldsymbol{i}})=\Pi(\beta)$ and $\Lambda\geq k$ and $\Lambda\neq\delta_{i_{k}}$ . Then we have $\Lambda>k$ , in which case we can use Proposition 3.4 to obtain the following expansion:

\displaystyle[\check{E}_{\boldsymbol{i},k},\check{F}_{\boldsymbol{i}}^{\Lambda}]=\sum_{\Lambda^{+}<k\leq\Lambda^{-}}C_{\Lambda^{\pm}}\check{F}_{\boldsymbol{i}}^{\Lambda^{-}}\check{E}_{\boldsymbol{i}}^{(\Lambda^{+})}.

Since $\pi_{S}(\check{E}_{\boldsymbol{i}}^{\Lambda^{+}})=0$ whenever $\Lambda^{+}\neq 0$ , it suffices to look at the terms with $\Lambda^{+}$ . Moreover $\Gamma_{i}=0+\delta_{k}$ implies that we may additionally assume $\Pi(\Lambda^{-}\cdot\alpha^{\boldsymbol{i}})=0$ . Such a term is $0$ modulo $h$ in $U_{h}^{k}(\mathfrak{p}_{S})$ if $\Lambda^{-}\neq 0$ , which is automatically satisfied since $\Lambda\cdot\alpha^{\boldsymbol{i}}\neq\alpha^{\boldsymbol{i}}_{k}$ by [MR1169886, Theorem, p.662] and [MR1890629, Chapter VI, Proposition 19].

Therefore we have

\displaystyle U_{\beta}=-\sigma\frac{q_{\beta}-q_{\beta}^{-1}}{1-\chi_{-2\sigma\beta}}E_{\boldsymbol{i},\beta}K_{\beta}^{-1}\otimes 1=-\sigma\frac{2d_{\beta}}{1-\chi_{-2\sigma\beta}}E_{\boldsymbol{i},\beta}\otimes 1\quad\operatorname{\mathrm{mod}}h

where $\sigma=1$ when $\beta\in R_{0}^{+}$ and $\sigma=-1$ when $\beta\not\in R_{0}^{+}$ and obtain the expansion (16). ∎

4.4. Construction of quantum semisimple coadjoint orbits

Let $G$ be a $1$ -connected complex Lie group with the Lie algebra $\mathfrak{g}$ and $L_{S}$ be its connected complex Lie subgroup with the Lie algebra $\mathfrak{l}_{S}$ . In this subsection we use $\Psi_{h,\chi}$ to construct a quantum semisimple coadjoint orbit, which is a deformation quantization of $\mathcal{O}(L_{S}\backslash G)$ with an action of $U_{h}(\mathfrak{g})$ . Let $\operatorname{\mathrm{Irr}}_{h}\mathfrak{g}$ be the set of equivalence classes of irreducible finite integrable $U_{h}^{k}(\mathfrak{g})$ modules. We also fix a representative $\pi$ of each class and sometimes identify $\operatorname{\mathrm{Irr}}_{h}\mathfrak{g}$ as the set of all representatives.

For a $U_{h}^{k}(\mathfrak{g})$ -module $V$ , its $U_{h}^{k}(\mathfrak{l}_{S})$ -invariant part is denoted by $V^{\mathfrak{l}_{S}}$ .

Set $\mathcal{O}_{h,\chi}^{k}(L_{S}\backslash G)$ as follows:

(19)

\displaystyle\mathcal{O}_{h,\chi}^{k}(L_{S}\backslash G)=\bigoplus_{\pi\in\operatorname{\mathrm{Irr}}_{h}\mathfrak{g}}\operatorname{\mathrm{Hom}}_{U_{h}^{k}(\mathfrak{l}_{S})}(V_{\pi},k\llbracket h\rrbracket)\otimes V_{\pi}.

Note that there is a unique linear map $p_{V}\colon\operatorname{\mathrm{Hom}}_{U_{h}^{k}(\mathfrak{l}_{S})}(V,k\llbracket h\rrbracket)\otimes V\longrightarrow\mathcal{O}_{h,\chi}^{k}(L_{S}\backslash G)$ for each finite integrable module $V$ which is natural in the sense that $p_{V}(f\circ T\otimes v)=p_{W}(f\otimes T(v))$ for any $f\in\operatorname{\mathrm{Hom}}_{U_{h}^{k}}(\mathfrak{l}_{S},k\llbracket h\rrbracket),v\in V$ and $T\in\operatorname{\mathrm{Hom}}_{\mathfrak{g}_{h}}(V,W)$ . For ease to read, we use $f\otimes v$ to represent $p_{V}(f\otimes v)$ .

Using this we construct a product $\ast_{h,\chi}$ on $\mathcal{O}_{h,\chi}^{k}(L_{S}\backslash G)$ defined as follows by using $\Psi_{h,\chi,0}=(\operatorname{\mathrm{id}}\otimes\operatorname{\mathrm{id}}\otimes\varepsilon)(\Psi_{h,\chi})$ , which is thought as a $U_{h}^{k}(\mathfrak{l}_{S})$ -homomorphism:

\displaystyle(f\otimes v)\ast_{h,\chi}(g\otimes w)=(f\otimes g)\Psi_{h,\chi,0}^{-1}\otimes(v\otimes w)

This product is actually associative. To see this consider the product of three elements in different order:

	$\displaystyle((f\otimes u)\ast_{h,\chi}(g\otimes v))$	$\displaystyle\ast_{h,\chi}(h\otimes w)$
		$\displaystyle=(f\otimes g\otimes h)(\Psi_{h,\chi,0}^{-1}\otimes 1)(\Delta\otimes\operatorname{\mathrm{id}})(\Psi_{h,\chi,0}^{-1})\otimes(u\otimes v\otimes w),$
	$\displaystyle(f\otimes u)\ast_{h,\chi}((g\otimes v)$	$\displaystyle\ast_{h,\chi}(h\otimes w))$
		$\displaystyle=(f\otimes g\otimes h)(1\otimes\Psi_{h,\chi,0}^{-1})(\operatorname{\mathrm{id}}\otimes\Delta)(\Psi_{h,\chi,0}^{-1})\otimes(u\otimes v\otimes w).$

Since $hx=\varepsilon(x)h$ for all $x\in U_{h}^{k}(\mathfrak{l}_{S})$ , we can see that $(f\otimes g\otimes h)(\Psi_{h,\chi,0}^{-1}\otimes 1)=(f\otimes g\otimes h)\Psi_{h,\chi}^{-1}$ . Then the associativity follows from the cocycle identity (14).

When $k=\mathbb{C}$ , $\mathcal{O}_{h,\chi}(L_{S}\backslash G)$ has a natural structure of deformation quantizations of $\mathcal{O}(L_{S}\backslash G)$ . To determine the semi-classical limit of $\mathcal{O}_{h,\chi}(L_{S}\backslash G)$ , we recall the identification of $U_{h}(\mathfrak{g})$ and the twist of $U(\mathfrak{g})\llbracket h\rrbracket$ . For any subset $S\subset\Delta$ we can take an isomorphism $T\colon U(\mathfrak{g})\llbracket h\rrbracket\longrightarrow U_{h}(\mathfrak{g})$ as topological $\mathbb{C}\llbracket h\rrbracket$ -algebra such that

(i)

$T(U(\mathfrak{l}_{S})\llbracket h\rrbracket)=U_{h}(\mathfrak{l}_{S})$ .
(ii)

There is $F\in U(\mathfrak{g}\times\mathfrak{g})\llbracket h\rrbracket$ such that $F=1\operatorname{\mathrm{mod}}h$ and $(T\otimes T)^{-1}\circ\Delta\circ T(x)=F\Delta(x)F^{-1}$

The standard $r$ -matrix is given by the following formula:

\displaystyle r=\sum_{\alpha\in R^{+}}d_{\alpha}E_{\alpha}\wedge F_{\alpha},

here $\mathfrak{g}\wedge\mathfrak{g}$ is embedded into $\mathfrak{g}\otimes\mathfrak{g}$ by $x\wedge y\longmapsto\frac{1}{2}(x\otimes y-y\otimes x)$ . Then the anti-symmetric part of the first coefficient of $F$ is given by $-r$ .

Under this identification of $U_{h}(\mathfrak{g})$ with $U(\mathfrak{g})\llbracket h\rrbracket$ , $V$ is finite integrable $U_{h}(\mathfrak{g})$ -module if and only if it is isomorphic to $L\otimes\mathbb{C}\llbracket h\rrbracket$ where $L$ is finite dimensional representation of $\mathfrak{g}$ . In particular we can identify $\operatorname{\mathrm{Irr}}_{h}\mathfrak{g}$ with $\operatorname{\mathrm{Irr}}\mathfrak{g}$ .

Using this picture of $U_{h}(\mathfrak{g})$ we can compare $\mathcal{O}_{h,\chi}(L_{S}\backslash G)$ with $\mathcal{O}(L_{S}\backslash G)\llbracket h\rrbracket$ . At first one should note that we can decompose $\mathcal{O}(L_{S}\backslash G)$ as follows:

	$\displaystyle\mathcal{O}(L_{S}\backslash G)=\bigoplus_{[\pi]\in\operatorname{\mathrm{Irr}}\mathfrak{g}}\operatorname{\mathrm{Hom}}_{\mathfrak{l}_{S}}(L_{\pi},\mathbb{C})\otimes L_{\pi},$
	$\displaystyle(f\otimes v)(g\otimes w)=(f\otimes g)\otimes(v\otimes w).$

Here the isomorphism is given by $f\otimes v\longmapsto f(\pi(\textendash)v)$ .

Hence we have a canonical identification of $\mathcal{O}_{h,\chi}(L_{S}\backslash G)$ with $\mathcal{O}(L_{S}\backslash G)\llbracket h\rrbracket$ . To interpret $\ast_{h,\chi}$ as a product on $\mathcal{O}(L_{S}\backslash G)\llbracket h\rrbracket$ , one should note that $\operatorname{\mathrm{Hom}}_{\mathfrak{g}}(W,W^{\prime}\otimes W^{\prime\prime})\cong\operatorname{\mathrm{Hom}}_{U_{h}(\mathfrak{g})}(W,W^{\prime}\otimes W^{\prime\prime})$ via $T\longmapsto F\circ T$ . Therefore we have

\displaystyle(f\otimes v)\ast_{h,\chi}(g\otimes w)=(f\otimes g)\Psi_{h,\chi,0}^{-1}F\otimes F^{-1}(v\otimes w).

Now we can obtain the semi-classical limit of $\mathcal{O}_{h,\chi}(L_{S}\backslash G)$ by a direct calculation. For $\theta\in\mathfrak{g}\wedge\mathfrak{g}$ , the right (resp. left) invariant bi-vector field corresponding to $\theta$ is denoted by $L_{*}(\theta)$ (resp. $R_{*}(\theta)$ ).

Proposition 4.16.

The algebra $\mathcal{O}_{h,\chi}(L_{S}\backslash G)$ is an equivariant deformation quantization of $\mathcal{O}(L_{S}\backslash G)$ . The associated Poisson structure is given by

\displaystyle\{F,G\}=(L_{*}(\pi)+R_{*}(r))(F,G)

where

\displaystyle\pi=\sum_{\beta\in R^{+}\cap R_{0}^{+}}d_{\beta}\frac{1+\chi_{-2\beta}}{1-\chi_{-2\beta}}E_{\beta}\wedge F_{\beta}+\sum_{\beta\in R^{+}\setminus R_{0}^{+}}d_{\beta}\frac{\chi_{2\beta}+1}{\chi_{2\beta}-1}E_{\beta}\wedge F_{\beta}.

5. Universal families of $2$ -cocycles

5.1. The toric variety associated to a root system

For general treatment of toric varieties, see [MR1234037, telen2022]. Let us recall the definition of the fan associated to a root system. For a root system $R$ , $Q^{\vee}$ denotes the dual lattice $\operatorname{\mathrm{Hom}}_{\mathbb{Z}}(Q,\mathbb{Z})$ . We also set $\mathfrak{h}_{\mathbb{R}}^{*}$ (resp. $\mathfrak{h}_{\mathbb{R}}^{**}$ ) as $Q\otimes_{\mathbb{Z}}\mathbb{R}$ (resp. $Q^{\vee}\otimes_{\mathbb{Z}}\mathbb{R}$ ), which are thought as an $\mathbb{R}$ -linear subspace of $\mathfrak{h}^{*}$ (resp. $\mathfrak{h}^{**}$ ).

Then the fan associated to $R$ is the collection $\Sigma_{R}$ of the following cones:

\displaystyle\sigma_{R_{0}^{+},S}:=\{\lambda\in\mathfrak{h}^{*}_{\mathbb{R}}\mid\lambda|_{R_{0}^{+}}\geq 0,\lambda|_{S_{0}}=0\},

where $R_{0}^{+}$ is a positive system and $S_{0}$ is a set of its simple roots. It is not difficult to see that this collection is actually a fan i.e. satisfies the following conditions:

(i)

For any $\sigma\in\Sigma_{R}$ , $\sigma\cap-\sigma=\{0\}$ holds.
(ii)

For any $\sigma\in\Sigma_{R}$ and $m\in\sigma^{\vee}=\{x\in\mathfrak{h}_{\mathbb{R}}\mid\lambda(x)\geq 0\text{ for }\lambda\in\sigma\}$ , the face $\sigma_{m}:=\{\lambda\in\sigma\mid\lambda(m)=0\}$ is also contained in $\Sigma_{R}$ .
(iii)

For any $\sigma,\sigma^{\prime}\in\Sigma_{R}$ , $\sigma\cap\sigma^{\prime}$ is of the form $\sigma_{m}$ for some $m\in\sigma^{\vee}$ .

Let $k$ be a commutative ring. The affine toric scheme $U_{\sigma}(k)$ associated to $\sigma\in\Sigma_{R}$ is defined as $\operatorname{\mathrm{Spec}}k[\sigma^{\vee}\cap 2Q]$ , here we use $2Q$ instead of $Q$ by a notational reason. Since any face of $\sigma$ is of the form $\sigma_{m}$ with $m\in\sigma^{\vee}\cap 2Q$ , a face $\tau$ of $\sigma$ gives a canonical open subscheme, namely $\operatorname{\mathrm{Spec}}k[\sigma^{\vee}\cap 2Q]_{e_{m}}=\operatorname{\mathrm{Spec}}k[\tau^{\vee}\cap 2Q]$ for $\tau=\sigma_{m}$ with $m\in\sigma^{\vee}\cap 2Q$ . Then the toric scheme associated to $\Sigma_{R}$ over $k$ , denoted by $X_{R}(k)$ , can be obtained by gluing $\{U_{\sigma}\}_{\sigma\in\Sigma_{R}}$ along with the gluing data $\{U_{\sigma\cap\tau}\rightarrow U_{\sigma}\}_{\sigma,\tau\in\Sigma_{R}}$ .

5.2. The moduli space of equivariant Poisson brackets on $L_{S}\backslash G$

Let $\mathcal{O}(X_{L_{S}\backslash G})$ be the quotient of $\mathbb{C}[\varphi_{\alpha}]_{\alpha\in R\setminus R_{S}}$ divided by the following relations:

(i)

$\varphi_{-\alpha}=-\varphi_{\alpha}$ for $\alpha\in R\setminus R_{S}$ ,
(ii)

$\varphi_{\alpha}\varphi_{\beta}+1=\varphi_{\alpha+\beta}(\varphi_{\alpha}+\varphi_{\beta})$ when $\alpha,\beta,\alpha+\beta\in R\setminus R_{S}$ ,
(iii)

$\varphi_{\alpha}=\varphi_{\beta}$ when $\alpha,\beta\in R\setminus R_{S}$ and $\alpha-\beta\in R_{S}$ .

As pointed out in [MR1817512, MR2141466, MR2349621], this is the moduli of Poisson structures on $L_{S}\backslash G$ equivariant with respect to the Poisson-Lie group structure on $G$ . Actually we can associate an equivariant Poisson bracket $\{\textendash,\textendash\}_{\varphi}$ to each $\varphi\in X_{L_{S}\backslash G}$ as follows:

\displaystyle\{F,G\}_{\varphi}=(L_{*}(\pi_{\varphi})+R_{*}(r))(F,G),\text{ where }\pi_{\varphi}=\sum_{\beta\in R^{+}\setminus R_{S}^{+}}d_{\beta}\varphi_{\beta}E_{\beta}\wedge F_{\beta}.

In the following we describe a canonical immersion from $X_{L_{S}\backslash G}(k)$ into the toric scheme $X_{R}(k)$ for an arbitrary commutative ring $k$ .

Let $\mathcal{O}(X_{L_{S}\backslash G}(k))$ be the quotient of $k[\psi_{\alpha}]_{\alpha\in R\setminus R_{S}}$ divided by the following relations:

(i_ψ)

$\psi_{\alpha}+\psi_{-\alpha}=1$ for $\alpha\in R\setminus R_{S}$ ,
(ii_ψ)

$\psi_{\alpha}\psi_{\beta}=\psi_{\alpha+\beta}(\psi_{\alpha}+\psi_{\beta}-1)$ when $\alpha,\beta,\alpha+\beta\in R\setminus R_{S}$ ,
(iii_ψ)

$\psi_{\alpha}=\psi_{\beta}$ when $\alpha,\beta\in R\setminus R_{S}$ satisfy $\alpha-\beta\in\operatorname{\mathrm{span}}_{\mathbb{Z}}S$ .

Then we set $X_{L_{S}\backslash G}(k)=\operatorname{\mathrm{Spec}}\mathcal{O}(X_{L_{S}\backslash G}(k))$ . Note that these relations coincide with (i), (ii), (iii) under the transformation $\varphi_{\alpha}=2\psi_{\alpha}-1$ if $2\in k^{\times}$ . In particular we can canonically identify $X_{L_{S}\backslash G}(\mathbb{C})$ with $X_{L_{S}\backslash G}$ .

Lemma 5.1.

For a positive system $R_{0}^{+}$ , we set $f_{R_{0}^{+}}=\prod_{\alpha\in R_{0}^{+}\setminus R_{S}}\psi_{\alpha}$ and $U(R_{0}^{+}):=\{\mathfrak{p}\in X_{L_{S}\backslash G}(k)\mid f_{R_{0}^{+}}\not\in\mathfrak{p}\}$ . Then $\{U(R_{0}^{+})\}_{R_{0}^{+}\supset R_{S}^{+}}$ is an open covering of $X_{L_{S}\backslash G}(k)$ .

Proof.

Take a prime ideal $\mathfrak{p}$ . What we have to show is the existence of a positive system $R_{0}^{+}$ containing $R_{S}^{+}$ such that $\psi_{\alpha}\not\in\mathfrak{p}$ for all $\alpha\in R_{0}^{+}\setminus R_{S}^{+}$ , which is equivalent to $\psi_{\alpha}\neq 0$ in $\mathcal{O}(X_{L_{S}\backslash G}(k))/\mathfrak{p}$ . Since this algebra is a domain, the following lemma can be applied. ∎

Lemma 5.2 (c.f. [MR2141466, Lemma 11]).

Let $k$ be a domain and consider $\psi=\{\psi_{\alpha}\}_{\alpha}\in k^{R\setminus R_{S}}$ satisfying (i_ψ), (ii_ψ), (iii_ψ). Then there is a positive system $R_{0}^{+}$ contained in $R_{S}^{+}\cup\{\alpha\in R\setminus R_{S}\mid\psi_{\alpha}\neq 0\}$ .

Proof.

At first we show that $P:=R_{S}\cup\{\alpha\in R\setminus R_{S}\mid\psi_{\alpha}\neq 0\}$ contains a positive system. By [MR1890629, Chapter VI, Proposition 20], it suffices to show that the following properties hold:

(a)

For any $\alpha\in R$ , either of $\alpha\in P$ or $-\alpha\in P$ holds.
(b)

For any $\alpha,\beta\in P$ , $\alpha+\beta\in P$ whenever $\alpha+\beta\in R$ .

The condition (a) follows from (i_ψ). To see (b), we assume either of $\alpha$ or $\beta$ is in $R_{S}$ at first. Then (iii_ψ) shows (b) in this case. Next we assume that both of them are not in $R_{S}$ and $\alpha+\beta\in R$ . Moreover we also assume $\alpha+\beta\not\in R_{S}$ since there is nothing to prove when $\alpha+\beta\in R_{S}$ . In this case, using (ii_ψ), we have

\psi_{\alpha+\beta}(\psi_{\alpha}+\psi_{\beta}-1)=\psi_{\alpha}\psi_{\beta}\neq 0

since $k$ is a domain. Hence we have $\psi_{\alpha+\beta}\neq 0$ , which implies $\alpha+\beta\in P$ .

Take a positive system $R_{1}^{+}$ of $R$ contained in $P$ . Then $R_{1}^{+}\cap R_{S}$ is also a positive system of $R_{S}$ . Hence there is $w$ , a product of $s_{\alpha}$ with $\alpha\in R_{S}$ , such that $w(R_{1}^{+}\cap R_{S})=R^{+}\cap R_{S}$ . Now $R_{0}^{+}:=w(R_{1}^{+})$ meets the required conditions since $w$ preserves a decompostion $R=R_{S}\sqcup(R\setminus R_{S})$ and $\psi_{w(\alpha)}=\psi_{\alpha}$ holds for $\alpha\in R\setminus R_{S}$ . ∎

Lemma 5.3.

Let $\mathcal{O}(U(R_{0}^{+}))$ be the localization of $\mathcal{O}(X_{L_{S}\backslash G}(k))$ by $f_{R_{0}^{+}}$ and $\mathcal{B}_{R_{0}^{+}}$ be the localization of $k[2Q_{0}^{-}]/(e_{-2\alpha}-1)_{\alpha\in R_{S}^{+}}$ by $(1-e_{-2\alpha})_{\alpha\in R_{0}^{+}\setminus R_{S}^{+}}$ . Then there is a canonical isomorphism between these algebras, as constructed in the proof below.

Proof.

We give the constructions of $u\colon\mathcal{O}(U(R_{0}^{+}))\longrightarrow\mathcal{B}_{R_{0}^{+}}$ and $v\colon\mathcal{B}_{R_{0}^{+}}\longrightarrow\mathcal{O}_{X_{L_{S}\backslash G}}(U(R_{0}^{+}))$ which are mutually inverse. It suffices to give the images of the generators $\{\psi_{\alpha}\}_{\alpha\in R_{0}^{+}\setminus R_{S}^{+}}$ and $\{e_{-2\alpha}\}_{\alpha\in R_{0}^{+}\setminus R_{S}^{+}}$ :

(20)

\displaystyle u(\psi_{\alpha})=\frac{1}{1-e_{-2\alpha}},\quad v(e_{-2\alpha})=1-\frac{1}{\psi_{\alpha}}.

It can be checked by direct calculations that these elements satisfy the required relations. ∎

Now we have a locally closed immersion $X_{L_{S}\backslash G}(k)\subset X_{R}(k)$ .

Proposition 5.4.

There is a canonical immersion from $X_{L_{S}\backslash G}(k)$ to $X_{R}(k)$ , as constructed in the proof below.

Proof.

We can obtain a closed subscheme of $X_{R}(k)$ by gluing $\operatorname{\mathrm{Spec}}k[\sigma^{\vee}\cap 2Q]/(e_{-2\alpha}-1)_{\alpha\in\sigma^{\vee}\cap Q_{S}}$ and also obtain its open subscheme by gluing $\operatorname{\mathrm{Spec}}\mathcal{B}_{R_{0}^{+}}$ over all $R_{0}\supset R_{S}^{+}$ . Hence it suffices to check the compatibility of the isomorphisms constructed in Lemma 5.3. Since $\operatorname{\mathrm{Spec}}\mathcal{B}_{R_{0}^{+}}\cap\operatorname{\mathrm{Spec}}\mathcal{B}_{R_{1}^{+}}=\operatorname{\mathrm{Spec}}\mathcal{B}_{R_{0}^{+},R_{1}^{+}}$ , where $\mathcal{B}_{R_{0}^{+},R_{1}^{+}}$ is the localization of $k[2Q_{0}^{-}+2Q_{1}^{-}]/(1-e_{-2\alpha})_{\alpha\in R_{S}^{+}}$ by $(1-e_{-2\alpha})_{\alpha\in R_{0}^{+}\cup R_{1}^{+}\setminus R_{S}^{+}}$ , it suffices to construct an isomorphism between $\mathcal{B}_{R_{0}^{+},R_{1}^{+}}$ and $\mathcal{O}(X_{L_{S}\backslash G}(k))_{f_{R_{0}^{+}}f_{R_{1}^{+}}}$ which makes the following diagram commutative:

This can be proven in the completely same way as the proof of Lemma 5.3. ∎

5.3. The sheaf of deformed quantum enveloping algebra on $X_{R}(k)$

In this subsection we construct a sheaf of algebras on the toric scheme using deformed quantum enveloping algebras. For this purpose we have to compare deformed quantum envelpoing algebras based on different positive systems.

We begin with some preparations on root systems. We say that $L\subset R^{+}$ is admissible if it is of the form $R^{+}\setminus R_{0}^{+}$ for some positive system $R_{0}^{+}$ , or equivalently of the form $R^{+}\cap R_{1}^{-}$ for some positive system $R_{1}^{+}$ . The admissibility can be characterized by the following conditions ([MR1169886, Theorem, p. 663]):

(i)

$\alpha+\beta\in L$ for $\alpha,\beta\in L$ .
(ii)

If $\alpha,\beta\in R^{+}$ and $\alpha+\beta\in L$ , either of $\alpha\in L$ or $\beta\in L$ holds.

Note that $R^{+}\setminus L$ is also admissible when $L$ is admissible and that $L$ contains a simple root if $L\neq\emptyset$ .

Lemma 5.5.

Let $L\subset R^{+}$ be an admissible subset. Then there is a unique subset $S\subset\Delta$ such that $\operatorname{\mathrm{span}}_{\mathbb{Z}}L\cap R^{+}=R_{S}^{+}$ .

In the following proof, $\operatorname{\mathrm{supp}}\alpha:=\{\varepsilon\in\Delta\mid c_{\varepsilon}\neq 0\}$ where $\alpha=\sum_{\varepsilon\in\Delta}c_{\varepsilon}\varepsilon\in Q$ .

Proof.

By induction on $\lvert L\rvert$ . If $\lvert L\rvert=0,1$ , there is nothing to prove since $L=,\{\varepsilon\}$ for some $\varepsilon\in\Delta$ respectively. Next we proceed to the induction step. At first we show the following claim: if $A\subset R^{+}$ and $S\subset\Delta$ satisfy $\operatorname{\mathrm{span}}_{\mathbb{Z}}A\cap R^{+}=R_{S}^{+}$ , $\operatorname{\mathrm{span}}_{\mathbb{Z}}A\cup\{\varepsilon\}=R_{S\cup\{\varepsilon\}}^{+}$ also holds for any $\varepsilon\in\Delta$ . This can be checked easily if $\varepsilon\in S$ . Hence we may assume $\varepsilon\in\Delta\setminus S$ . Since $S$ is a union of $\{\operatorname{\mathrm{supp}}\alpha\}_{\alpha\in A}$ , it is easy to see that $\operatorname{\mathrm{span}}_{\mathbb{Z}}(A\cup\{\varepsilon\})\cap R^{+}\subset R_{S\cup\{\varepsilon\}}^{+}$ . The other inclusion is trivial.

Now we take a simple root $\varepsilon\in L$ . Then $L\setminus\{\varepsilon\}$ is admissible in $s_{\varepsilon}(R^{+})$ . Hence there is a subset $S_{0}\subset\Delta$ with $\operatorname{\mathrm{span}}_{\mathbb{Z}}(L\setminus\{\varepsilon\})\cap s_{\varepsilon}(R^{+})=s_{\varepsilon}(R^{+})_{s_{\varepsilon}(S_{0})}$ . Since $-\varepsilon$ is simple in $s_{\varepsilon}(R^{+})$ , the claim above implies that $\operatorname{\mathrm{span}}_{\mathbb{Z}}L\cap s_{\varepsilon}(R^{+})=s_{\varepsilon}(R^{+})_{s_{\varepsilon}(S_{0})\cup\{-\varepsilon\}}$ . Since $\operatorname{\mathrm{span}}_{\mathbb{Z}}L$ is invariant under $s_{\varepsilon}$ , we can obtain the statement for $L$ with $S=S_{0}\cup\{\varepsilon\}$ . ∎

Corollary 5.6.

Let $R_{0}^{+}$ and $R_{1}^{+}$ be positive systems of $R$ . Then there is $m\in\mathfrak{h}_{\mathbb{R}}^{**}$ with the following conditions:

(i)

$\{\alpha\in R\mid m(\alpha)>0\}\subset R_{0}^{+}\cap R_{1}^{+}$ ,
(ii)

$\{\alpha\in R\mid m(\alpha)=0\}=\operatorname{\mathrm{span}}_{\mathbb{Z}}R_{0}^{+}\triangle R_{1}^{+}$ ,
(iii)

$\{\alpha\in R\mid m(\alpha)<0\}\subset R_{0}^{-}\cap R_{1}^{-}$ .

Proof.

We may assume $R_{1}^{+}=R^{+}$ . Take $S\subset\Delta$ by using Lemma 5.5 for $L=R^{+}\setminus R_{0}^{+}$ . Then $m=\sum_{\varepsilon\in\Delta\setminus S}\varepsilon^{\vee}$ satisfies the condition. ∎

Lemma 5.7.

Let $R_{0}^{+}$ and $R_{1}^{+}$ be positive systems of $R$ . There are reduced expressioins $s_{\boldsymbol{i}}$ and $s_{\boldsymbol{j}}$ of $w_{0}$ , compatible with $R_{0}^{+}$ and $R_{1}^{+}$ respectively, and positive integers $1\leq k<l\leq N$ with the following property:

(i)

For $1\leq n\leq k$ , $\alpha^{\boldsymbol{i}}_{n}=\alpha^{\boldsymbol{j}}_{n}\in R^{+}\setminus(R_{0}^{+}\cup R_{1}^{+})$ .
(ii)

For $k<n\leq l$ , $\alpha^{\boldsymbol{i}}_{n},\alpha^{\boldsymbol{j}}_{n}\in\operatorname{\mathrm{span}}_{\mathbb{Z}}R_{0}^{+}\triangle R_{1}^{+}$ .
(iii)

For $l<n\leq N$ , $\alpha^{\boldsymbol{i}}_{n}=\alpha^{\boldsymbol{j}}_{n}\in R^{+}\cap R_{0}^{+}\cap R_{1}^{+}$ .

Proof.

By using $m\in\mathfrak{h}_{\mathbb{R}}^{**}$ in Corollary 5.6, we can see the admissiblilty of the following sets.

\displaystyle L^{\pm}:=R^{+}\cap R_{0}^{\pm}\cap R_{1}^{\pm}\setminus V=\{\alpha\in R^{+}\mid\pm m(\alpha)>0\}.

Take $w^{\pm}\in W$ such that $L^{\pm}=R^{+}\setminus w^{\pm}(R^{+})$ and also take $w,w^{\prime}\in W$ such that $R_{0}^{+}=w(R^{+})$ and $R_{1}^{+}=w^{\prime}(R^{+})$ . Consider a reduced expression of $w^{-}$ . This expression can be extended to reduced expressions $w$ and $w^{\prime}$ . Similarly a reduced expression of $w^{+}$ can be extended to reduced expressions of $ww_{0}$ and $w^{\prime}w_{0}$ . Then we obtain enumerations $\{\alpha^{i}_{n}\}$ on $R^{+}\setminus R_{i}^{+}$ and $\{\beta^{i}_{m}\}$ on $R^{+}\cap R_{i}^{+}$ from these reduced expressions. Then, as a consequence of [MR1169886, Theorem, p.662], we can find reduced expressions $s_{\boldsymbol{i}}$ and $s_{\boldsymbol{j}}$ which satisfy the following conditions respectively:

\displaystyle\alpha^{\boldsymbol{i}}_{n}=\begin{cases}\alpha^{0}_{n}&(1\leq n\leq\ell(w)),\\ \beta^{0}_{N-n+1}&(\ell(w)<n\leq N),\end{cases}\quad\alpha^{\boldsymbol{j}}_{n}=\begin{cases}\alpha^{1}_{n}&(1\leq n\leq\ell(w^{\prime})),\\ \beta^{1}_{N-n+1}&(\ell(w^{\prime})<n\leq N).\end{cases}

By construction these reduced expressions satisfy the required conditions. ∎

Remark 5.8.

Let $w^{-}$ be as defined in the proof above. Then $(w^{-})^{-1}(R^{+}\cap\operatorname{\mathrm{span}}_{\mathbb{Z}}R_{0}^{+}\triangle R_{1}^{+})$ is of the form $R_{S}^{+}$ for some $S\subset\Delta$ since the image is $\{\alpha\in R^{+}\mid m(w^{-}(\alpha))=0\}$ and its complement in $R^{+}$ is $\{\alpha\in R^{+}\mid m(w^{-}(\alpha))>0\}$ . This fact implies that the linear span of $\{F_{\boldsymbol{i}}^{(\Lambda)}\}_{k<\Lambda\leq l}$ coincides with the linear span of $\{F_{\boldsymbol{j}}^{(\Lambda)}\}_{k<\Lambda\leq l}$ .

Proposition 5.9.

Let $k$ be an $\mathcal{A}$ -algebra and $\chi\colon 2Q_{0}^{-}+2Q_{1}^{-}\longrightarrow k$ be a character. Then there is a canonical isomorphism between $U_{q,\chi_{0}}^{k}(\mathfrak{g}{}^{\sim})$ and $U_{q,\chi_{1}}^{k}(\mathfrak{g}{}^{\sim})$ as constructed in the proof, where $\chi_{i}$ is the restriction of $\chi$ on $2Q_{i}^{-}$ .

Proof.

Set $U_{i}=U_{q,e_{i}}^{\mathcal{A}[2Q_{0}^{-}+2Q_{1}^{+}]}(\mathfrak{g}{}^{\sim})$ . It suffices to show $U_{0}=U_{1}$ in $U_{q}^{\mathcal{A}[P]}(\mathfrak{g})$ , where $e\colon 2Q_{0}-+2Q_{1}\longrightarrow\mathcal{A}[2Q_{0}^{-}+2Q_{1}^{-}]$ is the canonical character.

Take a reduced expressions $s_{\boldsymbol{i}}$ and $s_{\boldsymbol{j}}$ of $w_{0}$ as Lemma 5.7. Since $e_{2\alpha}$ with $\alpha\in R^{+}\cap\operatorname{\mathrm{span}}_{\mathbb{Z}}R_{0}^{+}\triangle R_{1}^{+}$ is invertible in $\mathcal{A}[2Q_{0}^{-}+2Q_{1}^{-}]$ , $U_{0}$ is generated by the following elements:

\displaystyle\{(q_{\alpha}-q_{\alpha}^{-1})\acute{E}_{\boldsymbol{i},\alpha}\}_{\alpha\in R^{+}},\quad\{e_{-\lambda}\acute{K}_{\lambda}\}_{\lambda\in P},\quad\begin{cases}e_{2\alpha^{\boldsymbol{i}}_{n}}\acute{F}_{\boldsymbol{i},n}^{(r)}&(1\leq n\leq k),\\ \acute{F}_{\boldsymbol{i},n}^{(r)}&(k<n\leq l),\\ \acute{F}_{\boldsymbol{i},n}^{(r)}&(l<n\leq N).\end{cases}

The same thing can be said for $U_{1}$ . By the property of $s_{\boldsymbol{i}}$ and $s_{\boldsymbol{j}}$ , only different generators are $\acute{F}_{\boldsymbol{i},n}^{(r)}$ and $\acute{F}_{\boldsymbol{j},n}^{(r)}$ with $k<n\leq l$ . But the algebras generated by these elements are independent of $\boldsymbol{i}$ and $\boldsymbol{j}$ as stated in the remark above. Hence we have $U_{0}=U_{1}$ . ∎

Remark 5.10.

Take another positive system $R_{2}^{+}$ and consider a character $\chi$ from $2Q_{0}^{-}+2Q_{1}^{-}+2Q_{2}^{-}$ . In this case we have three isomorphisms $U_{q,\chi_{0}}^{k}(\mathfrak{g}{}^{\sim})\cong U_{q,\chi_{1}}^{k}(\mathfrak{g}{}^{\sim})$ , $U_{q,\chi_{1}}^{k}(\mathfrak{g}{}^{\sim})\cong U_{q,\chi_{2}}^{k}(\mathfrak{g}{}^{\sim})$ and $U_{q,\chi_{0}}^{k}(\mathfrak{g}{}^{\sim})\cong U_{q,\chi_{2}}^{k}(\mathfrak{g}{}^{\sim})$ . By construction of the isomorphisms, we can see the commutativity of the following diagram:

Now we can define a sheaf of deformed quantum enveloping algebras on $X_{R}(k)$ and $X_{L_{S}\backslash G}(k)$ .

Definition 5.11.

Let $k$ be an $\mathcal{A}$ -algebra. We define $\mathscr{U}_{q,X_{R}}^{k}(\mathfrak{g}{}^{\sim})$ , a sheaf of algebras on $X_{R}(k)$ , by gluing $\{U_{q,e}^{k[2Q_{0}^{-}]}(\mathfrak{g}{}^{\sim})\}_{R_{0}^{+}}$ by the isomorpshims in Proposition 5.9. We also define $\mathscr{U}_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ as the inverse image of $\mathscr{U}_{q,X_{R}}^{k}(\mathfrak{g}{}^{\sim})$ by the immersion $X_{L_{S}\backslash G}(k)\longrightarrow X_{R}(k)$ defined in Proposition 5.4.

Remark 5.12.

By construction of the isomorphisms in Proposition 5.9, we can obtain an injection from $f^{*}\widetilde{U_{q}^{k}(\mathfrak{l}_{S}{}^{\sim})}$ to $\mathscr{U}_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ where $f\colon X_{L_{S}\backslash G}(k)\longrightarrow\operatorname{\mathrm{Spec}}k$ is the canonical morphism.

Remark 5.13.

Since $X_{L_{S}\backslash G}(k)$ is an affine $k$ -scheme and $\mathscr{U}_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ is quasi-coherent, we can reconstruct the sheaf from its global sections $U_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim}):=\mathscr{U}_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})(X_{L_{S}\backslash G}(k))$ .

By construction of $\mathscr{U}_{q,X_{R}}^{k}(\mathfrak{g}{}^{\sim})$ , we have the local freeness of $U_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ . For $S\subset\Delta$ with $\lvert\Delta\setminus S\rvert=1$ , we can say a stronger statement: there is a “PBW basis” for $U_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ , constructed as follows.

At first one should note that $X_{L_{S}\backslash G}$ is covered by $U(R^{+})$ and $U(R_{0}^{+})$ where $R_{0}^{+}=R_{S}^{+}\cup(R^{-}\setminus R_{S}^{-})$ . Take a reduced expression $s_{\boldsymbol{i}}$ of $w_{0}$ compatible with $R_{0}^{+}$ . By definition of deformed quantum enveloping algebra, we have $\hat{E}_{\boldsymbol{i}}$ and $\hat{K}_{\lambda}$ as elements of $U_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ . For $\hat{F}_{\boldsymbol{i},\alpha}$ , we define a global section by gluing $\psi_{\alpha}\hat{F}_{\boldsymbol{i},\alpha}$ on $U(R^{+})$ and $-\psi_{\alpha}\hat{F}_{\boldsymbol{i},\alpha}$ on $U(R_{0}^{+})$ . Then these global sections behave as “quantum root vectors” in $U_{q,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ .

For general $\mathfrak{l}_{S}\subset\mathfrak{g}$ , it seems to be still possible to show the freeness. For example we can define $F_{\alpha+\beta}\in U_{q,X_{H\backslash SL_{3}}}^{k}(\mathfrak{sl}_{3}^{\sim})$ as follows:

\displaystyle F_{\alpha+\beta}=\psi_{\alpha+\beta}F_{\beta}F_{\alpha}-\psi_{\alpha+\beta}(q\psi_{\alpha}+q^{-1}\psi_{-\alpha})F_{\alpha}F_{\beta}.

But we do not have any idea to obtain good quantum root vectors in general case. Even in the case of $\mathfrak{sl}_{3}$ , we do not know which element should be taken as $F_{\alpha+\beta}$ .

5.4. Explicit construction of all quantizations

Let $k$ be a $\mathbb{Q}$ -algebra. In this case we can consider $\mathscr{U}_{h,X_{R}}^{k}(\mathfrak{g}{}^{\sim})$ and $\mathscr{U}_{h,X_{L_{S}\backslash G}}^{k}(\mathfrak{g}{}^{\sim})$ on $X_{R}(k)$ and $X_{L_{S}\backslash G}(k)$ respectively. Also note that we can use $\varphi_{\alpha}:=2\psi_{\alpha}-1$ as generators of $\mathcal{O}(X_{L_{S}\backslash G}(k))$ since $2\in k^{\times}$ .

Let us recall the construction of the $2$ -cocycle $\Psi_{h,\chi}^{k}$ . This can be characterized as a unique element inducing the isomorphism (13), which only depends on the $Q^{-}$ -grading of the $(R_{0}^{+},\chi)$ -twisted parabolically induced modules. Since the identification in Proposition 5.9 preserves the triangular decomposition and the $Q$ -grading, we can see that $\Psi_{h,\chi}^{k}$ does not depend on the choice of $R_{0}^{+}$ . This leads us to the following conclusion. Note that there is a canonical homomorphism from $\mathcal{O}(X_{L_{S}\backslash G}(k))$ to $k$ when we take a character $\chi\colon 2Q_{0}^{-}\longrightarrow k$ with $R_{S}^{+}\subset R_{0}^{+},\chi_{-2\alpha}=1$ for $\alpha\in R_{S}^{+}$ and $\chi_{-2\alpha}-1\in k^{\times}$ for $\alpha\in R_{0}^{+}\setminus R_{S}^{+}$ .

Theorem 5.14.

There is a unique $2$ -cocycle $\Psi_{h,\varphi}^{k}\in U_{h}^{\mathcal{O}(X_{L_{S}\backslash G}(k))}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})^{\mathfrak{l}_{S}}$ which coincides with $\Psi_{h,\chi}^{k}$ under the homomorphism above for all $(k,\chi)$ . Moreover it has the following expansion:

(21)

\displaystyle\Psi_{h,\varphi}=1-h\sum_{\alpha\in R^{+}\setminus R_{S}^{+}}d_{\alpha}(\varphi_{\alpha}+1)E_{\alpha}\otimes F_{\alpha}\otimes 1+\cdots.

As an immediate corollary, we can prove [MR1817512, Proposition 5.3] in a constructible way.

Corollary 5.15.

There is an algebraic family $\{\mathcal{O}_{h,\varphi}(L_{S}\backslash G)\}_{\varphi\in X_{L_{S}\backslash G}}$ of equivariant deformation quantization of $\mathcal{O}(L_{S}\backslash G)$ with the associated Poisson bracket $\{\textendash,\textendash\}_{\varphi}$ .

By using [MR1817512, Proposition 5.6] we also obtain the classification theorem. A formal path on $X_{L_{S}\backslash G}$ is $(\varphi_{\alpha}(h))_{\alpha\in R\setminus R_{S}}\in\mathbb{C}\llbracket h\rrbracket^{R\setminus R_{S}}$ satisfying the relations (i), (ii), (iii). Then we can consider a homomorphism $\mathcal{O}(X_{L_{S}\backslash G})\longrightarrow\mathbb{C}\llbracket h\rrbracket$ by setting $\varphi_{\alpha}\longmapsto\varphi_{\alpha}(h)$ and obtain a deformation quantization $\mathcal{O}_{h,\varphi(h)}(L_{S}\backslash G)$

Corollary 5.16.

Let $\mathcal{O}_{h}(L_{S}\backslash G)$ be an equivariant deformation quantization of $\mathcal{O}(L_{S}\backslash G)$ . Then there is a unique formal path $\varphi(h)$ on $X_{L_{S}\backslash G}$ such that $\mathcal{O}_{h}(L_{S}\backslash G)$ is equivalent to $\mathcal{O}_{h,\varphi(h)}(L_{S}\backslash G)$ as an equivariant deformation quantization.

6. Comparison theorem

Let $S\subset S_{0}\subset\Delta$ be subsets. The Levi subalgebra of $\mathfrak{g}$ (resp. Levi subgroup of $G$ ) associated to $S_{0}$ is denoted by $\mathfrak{l}_{0}$ (resp. $L_{0}$ ). Then, by the results in the previous section, we have the $2$ -cocycle $\Psi_{0}\in U_{h}^{\mathcal{O}(X_{L_{S}\backslash L_{0}})}(\mathfrak{l}_{0}\times\mathfrak{l}_{0}\times\mathfrak{l}_{S})^{\mathfrak{l}_{S}}$ . Then we can thought $\Psi_{0}$ as a $2$ -cocycle in $U_{h}^{\mathcal{O}(X_{L_{S}\backslash L_{0}})}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})^{\mathfrak{l}_{S}}$ and obtain a deformation quantization by the evaluation at an element of $X_{L_{S}\backslash L_{0}}$ . In light of Corollary 5.16, it is natural to compare this quantization and quantizations coming from $\Psi\in U_{h}^{\mathcal{O}(X_{L_{S}\backslash G})}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})^{\mathfrak{l}_{S}}$ . For such a purpose we consider a homomorphism $\pi\colon\mathcal{O}(X_{L_{S}\backslash G}(k))\longrightarrow\mathcal{O}(X_{L_{S}\backslash L_{0}}(k))$ defined by

\displaystyle\pi(\varphi_{\alpha})=\begin{cases}\varphi_{\alpha}&(\alpha\in R_{S_{0}}\setminus R_{S}),\\ -1&(\alpha\in R^{+}\setminus R_{S_{0}}^{+}),\\ 1&(\alpha\in R^{-}\setminus R_{S_{0}}^{-}).\end{cases}

Proposition 6.1.

Let $\Psi\in U_{h}^{\mathcal{O}(X_{L_{S}\backslash G}(k))}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})^{\mathfrak{l}}$ be the $2$ -cocycle constructed in the previous section. Then $\pi_{*}\Psi=\Psi_{0}$ .

We prove this fact as a corollary of the fully faithfulness of a functor introduced in the following discussion, which holds in the integral setting.

Let $R_{0}^{+}$ be a positive system of $R$ containing $R_{S}^{+}\cup(R^{-}\setminus R_{S_{0}}^{-})$ . Take a reduced expression $s_{\boldsymbol{i}}$ compatible with $R_{0}^{+}$ . Then we can introduce the notion of generalized $S_{0}$ -maximal vector in the similar manner to Definition 4.2.

We say that a $U_{q}^{k}(\mathfrak{g})$ -module $V$ is locally strongly $U_{q}^{k}(\mathfrak{n}^{+})$ -finite if, for any $v\in V$ , $U_{q}^{k}(\mathfrak{n}^{+})_{\mu}v=0$ for all $\mu\in Q^{+}$ with finitely many exceptions.

Theorem 6.2.

Let $(k,\chi)$ be an $\mathcal{A}[2Q_{0}^{-}]$ -algebra with $\chi_{-2\alpha}=0$ for $\alpha\in R^{-}\setminus R_{S_{0}}^{-}$ and $\chi_{-2\alpha}=1$ for $\alpha\in R_{S}^{+}$ .

(i)

For any $V\in U_{q,\chi}^{k}\text{-}\mathrm{Mod}(\mathfrak{l}_{0}{}^{\sim})$ and $W\in U_{q}^{k}(\mathfrak{g})\text{-}\mathrm{Mod}$ , each $m\in W\otimes V$ has a unique generalized $S_{0}$ -maximal vector $(m_{\Lambda})_{\Lambda}$ with $m_{0}=m$ .
(ii)

The functor $\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}\colon U_{q,\chi}^{k}(\mathfrak{l}_{0}{}^{\sim})\text{-}\mathrm{Mod}\longrightarrow U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})\text{-}\mathrm{Mod},V\longmapsto U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})\otimes_{U_{q,\chi}^{k}(\mathfrak{p}_{0}{}^{\sim})}V$ is fully faithful.
(iii)

If a $U_{q,\chi}^{k}(\mathfrak{g}{}^{\sim})$ -module $W$ is locally strongly $U_{q}^{k}(\mathfrak{n}^{+})$ -finite, there is a unique homomorphism $\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}(W\otimes V)\longrightarrow W\otimes\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}V$ which sends $m$ to a $S_{0}$ -maximal vector of the form $m_{(W)}\otimes(1\otimes m_{(V)})+\cdots$ . Moreover this is an isomorphism.
(iv)

Let $W,W^{\prime}$ be locally strongly $U_{q}^{k}(\mathfrak{b}^{+})$ -finite $U_{q}^{k}(\mathfrak{g})$ -module and $V$ be a $U_{q,\chi}^{k}(\mathfrak{l}_{0}{}^{\sim})$ -module. Then the following diagram commutes.

Proof.

(i) The proof is very similar to that of Proposition 4.4, hence that of Lemma 4.5. The only different point is to show the upper diagonality and the invertibility of diagonal entries in the usual sense.

Fix a multi-index $\Gamma$ and consider its decomposition $\Gamma=\Gamma^{\prime}+\delta_{i}$ such that $\gamma_{1}=\gamma_{2}=\cdot=\gamma_{i-1}=0$ . Then we look at the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime})}$ in (9) with $\beta=\alpha^{\boldsymbol{i}}_{i}$ :

\displaystyle\sum_{\Lambda}\hat{E}_{\boldsymbol{i},\beta,(1)}m_{\Lambda,(W)}\otimes(\hat{E}_{\boldsymbol{i},\beta,(2)}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{u}_{S}^{-})}\otimes\pi_{S}((\hat{E}_{\boldsymbol{i},\beta,(2)}\hat{F}_{\boldsymbol{i}}^{(\Lambda)})_{(\mathfrak{p}_{S})})m_{\Lambda,(V)}.

By (5), we can replace $\Delta(\hat{E}_{\boldsymbol{i},\beta})$ by $K_{\Lambda^{r}\cdot\alpha^{\boldsymbol{i}}}^{-1}\otimes\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}$ with $\Lambda^{r}\leq i\leq\Lambda^{l}$ . Moreover it suffices to consider the terms with $\Pi(\Lambda^{l}\cdot\alpha^{\boldsymbol{i}})=0$ and $\Lambda^{r}\neq\delta_{i}$ to show the upper diagonality. Note that $\Lambda^{r}<i$ holds in this case.

Consider $\Lambda$ such that $\Lambda\in B_{l,\Pi(\Gamma\cdot\alpha^{\boldsymbol{i}})}$ and $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{i-1}=0$ . Then we have the following expansion by Proposition 3.4 (i):

\displaystyle\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{r}}\acute{F}_{\boldsymbol{i}}^{(\Lambda)}=\sum_{\begin{subarray}{c}\Lambda^{+}<i\leq\Lambda^{-}\\ \Lambda^{r}\cdot\alpha^{\boldsymbol{i}}-\Lambda^{+}\cdot\alpha^{\boldsymbol{i}}=\Lambda\cdot\alpha^{\boldsymbol{i}}-\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}\in Q^{+}\end{subarray}}C_{\Lambda^{\pm}}\acute{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\widetilde{E}_{\boldsymbol{i}}^{\Lambda^{+}}.

Hence we also have the following:

\displaystyle\hat{E}_{\boldsymbol{i}}^{\Lambda^{r}}\hat{F}_{\boldsymbol{i}}^{(\Lambda)}=\sum_{\begin{subarray}{c}\Lambda^{+}<i\leq\Lambda^{-}\\ \Lambda^{r}\cdot\alpha^{\boldsymbol{i}}-\Lambda^{+}\cdot\alpha^{\boldsymbol{i}}=\Lambda\cdot\alpha^{\boldsymbol{i}}-\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}\in Q^{+}\end{subarray}}c_{\Lambda^{\pm}}\hat{F}_{\boldsymbol{i}}^{(\Lambda^{-})}\hat{E}_{\boldsymbol{i}}^{\Lambda^{+}}.

If $\Lambda^{+}\neq 0$ , such a term is removed by $\pi_{S_{0}}$ since $\Lambda^{+}<i$ . It means that we only have to look at the term with $\Lambda^{+}=0$ and $\Lambda^{-}=\Gamma^{\prime}$ to determine the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime})}$ . On the other hand, the coefficient of such a term can be calculated as follows:

\displaystyle c_{0,\Gamma^{\prime}}=C_{0,\Gamma^{\prime}}\chi_{2\Lambda\cdot\alpha^{\boldsymbol{i}}-2\Lambda^{-}\cdot\alpha^{\boldsymbol{i}}}=C_{0,\Gamma^{\prime}}\chi_{2\Lambda^{r}\cdot\alpha^{\boldsymbol{i}}}=0.

Hence our $\Lambda$ does not affect the coefficient of $\hat{F}_{\boldsymbol{i}}^{(\Gamma^{\prime})}$ . This shows the upper diagonality of $A_{l,\nu}$ .

To show the invertibility, consider the case of $\Lambda=\Gamma$ . By the consideration above we may assume $\Lambda^{r}=\delta_{i}$ . Then a completely same argument in the proof of Lemma 4.5 works to prove the invertibility of the diagonal entry. Then the upper triangularity implies the invertibility of $A_{l,\nu}$ .

(ii) This is a corollary of (i).

(iii) To see the former half of the statement, it suffices to show that $m_{\Lambda}=0$ except finitely many $\Lambda$ for any generalized $S_{0}$ -maximal vector $(m_{\Lambda})_{\Lambda}$ since it implies the existence and uniqueness of required $S_{0}$ -maximal vectors and also the existence and uniqueness of the required homomorphism.

By setting $W=U_{q}^{k}(\mathfrak{g})$ and $V=U_{q,\chi}^{k}(\mathfrak{l}_{0}{}^{\sim})$ , we can obtain the $S_{0}$ -maximizer $(U_{\Lambda})_{\Lambda}$ . Moreover the consideration above implies that $U_{\Lambda}\in\sum_{\Pi(\nu)=\Pi(\Lambda\cdot\alpha^{\boldsymbol{i}})}U_{q}^{k}(\mathfrak{b}^{+})_{\nu}\otimes U_{q,\chi}^{k}(\mathfrak{l}_{0}{}^{\sim})$ . Hence our assumption on $W$ implies that $m_{\Lambda}=U_{\Lambda}m_{0}=0$ except for finitely many $\Lambda$ .

Next we show the bijectivity. Note that we have $\Delta(\hat{F}_{\boldsymbol{i}}^{\Lambda})=K_{\Lambda\cdot\alpha^{\boldsymbol{i}}}^{-1}\otimes\hat{F}_{\boldsymbol{i}}^{\Lambda}$ . Hence the image of $w\otimes(\hat{F}_{\boldsymbol{i}}^{(\Gamma)}\otimes v)$ is

\displaystyle\sum_{\Lambda}K_{\Gamma\cdot\alpha^{\boldsymbol{i}}}^{-1}U_{\Lambda,(U_{q}^{k}(\mathfrak{b}^{+}))}w\otimes\hat{F}_{\boldsymbol{i}}^{\Gamma}\hat{F}_{\boldsymbol{i}}^{\Lambda}\otimes U_{\Lambda,(U_{q,\chi}^{k}(\mathfrak{l}_{0}{}^{\sim}))}v.

Now we regard the homomorphism as an endomorphism $T$ on $U_{q,\chi}^{k}(\mathfrak{u}_{S_{0}}^{-}{}^{\sim})\otimes W\otimes V$ . Then $U_{q,\chi}^{k}(\mathfrak{u}_{S_{0}}^{-}{}^{\sim})\otimes U_{q}^{k}(\mathfrak{b}^{+})w\otimes V$ is stable under this map. Moreover our assumption implies the existence of a positive integer $n$ such that $(T-\operatorname{\mathrm{id}})^{n}=0$ on this module. This fact shows the bijectivity.

(iv) By (i) we have an automorphism $T$ on $W\otimes W^{\prime}\otimes V$ such that $\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}T$ coincides with the following composition:

	$\displaystyle\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}(W\otimes W^{\prime}\otimes V)$	$\displaystyle\longrightarrow W\otimes\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}(W^{\prime}\otimes V)$
		$\displaystyle\longrightarrow W\otimes W^{\prime}\otimes\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}V\longrightarrow\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}(W\otimes W^{\prime}\otimes V).$

Then the image of $1\otimes(w\otimes w^{\prime}\otimes v)$ at $W\otimes W^{\prime}\otimes\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}V$ is of the form $T(w\otimes w^{\prime}\otimes v)_{(W\otimes W^{\prime})}\otimes(1\otimes T(w\otimes w^{\prime}\otimes v)_{(V)})+\cdots$ . On the other hand, by using the $S_{0}$ -maximizer, we can calculate the image and see that it is of the form $w\otimes w^{\prime}\otimes(1\otimes v)+\cdots$ . Hence we have $T=\operatorname{\mathrm{id}}$ and the diagram commutes. ∎

Proof of Proposition 6.1.

By the construction of the cocycles, it suffices to show the commutativity of the following diagram:

Since the both images of $1\otimes(w\otimes v)$ at the upper right corner is of the form $w\otimes(1\otimes v)+\cdots$ , it suffices to show that these are $S_{0}$ -maximal vectors. The image under the upper isomorphism is trivially $S$ -maximal. To see the maximality of another image, we show more general statement: Let $W$ be a locally strongly $U_{q}^{k}(\mathfrak{n}^{+})$ -finite module and $V$ be a $U_{q,\chi}^{k}(\mathfrak{l}_{0}{}^{\sim})$ -module. If $m\in W\otimes V$ is $S$ -maximal, the $S_{0}$ -maximal vector $\widetilde{m}=m_{(W)}\otimes(1\otimes m_{(V)})+\cdots\in W\otimes\mathrm{ind}_{\mathfrak{l}_{0},q}^{\mathfrak{g},\chi}V$ is also $S$ -maximal.

Take $\hat{E}_{\boldsymbol{i},\beta}$ with $\beta\in R_{S_{0}}^{+}\setminus R_{S}^{+}$ . Then $\hat{E}_{\boldsymbol{i},\beta}\widetilde{m}$ is still of the form $(\hat{E}_{\boldsymbol{i},\beta}m)_{(W)}\otimes(1\otimes(\hat{E}_{\boldsymbol{i},\beta}m)_{(V)})+\cdots$ . By a discussion similar to Lemma 3.17, $\hat{E}_{\boldsymbol{i},\beta}\widetilde{m}$ is again $S_{0}$ -maximal. This implies $\hat{E}_{\boldsymbol{i},\beta}\widetilde{m}=0$ as a consequence of the uniqueness for a maximal vector and $\hat{E}_{\boldsymbol{i},\beta}m=0$ . ∎

At last we interpret this comparison theorem as a statement on deformation quantizations. Let us recall the $U_{h}^{k}(\mathfrak{g})$ -bimodule structure on $\mathcal{O}_{h}^{k}(G)$ , induced from the embedding $\mathcal{O}_{h}^{k}(G)\subset U_{h}^{k}(\mathfrak{g})^{*}$ . Then, for a $U_{h}^{k}(\mathfrak{l}_{0})$ -module algebra $B$ , we define the induced $U_{h}^{k}(\mathfrak{g})$ -module algebra $\mathrm{ind}_{L_{0,h}}^{G_{h}}B$ as follows:

\displaystyle\mathrm{ind}_{L_{0,h}}^{G_{h}}B=\{a\in\mathcal{O}_{h}^{k}(G)\otimes B\mid(r(a)\otimes\operatorname{\mathrm{id}})(x)=(\operatorname{\mathrm{id}}\otimes l(x))(a)\text{ for all }x\in U_{h}^{k}(\mathfrak{l}_{0})\},

where $r(x)$ (resp. $l(x)$ ) is the right (resp. left) multiplication by $x\in U_{h}^{k}(\mathfrak{l}_{0})$ .

Corollary 6.3.

Let $(k\llbracket h\rrbracket,\chi)$ be an $\mathcal{A}[2Q_{0}^{-}]$ -algebra same with $\chi_{-2\alpha}=0$ for $\alpha\in R^{-}\setminus R_{S_{0}}^{-}$ and $\chi_{-2\alpha}=1$ for $\alpha\in R_{S}^{+}$ . Then $\mathcal{O}_{h,\chi}^{k}(L_{S}\backslash G)\cong\mathrm{ind}_{L_{0,h}}^{G_{h}}\mathcal{O}_{h,\chi}(L_{S}\backslash L_{0})$ .

Proof.

The $2$ -cocycle arising from $U_{h}^{k}(\mathfrak{l}_{S}{}^{\sim})\subset U_{h,\chi}^{k}(\mathfrak{l}_{0}{}^{\sim})$ (resp. $U_{h,\chi}(\mathfrak{g}{}^{\sim})$ ) is denoted by $\Psi_{0}$ (resp. $\Psi$ ). By Proposition 6.1, we have $\Psi_{0}=\Psi$ throught the embedding $U_{h}^{k}(\mathfrak{l}_{0}\times\mathfrak{l}_{0}\times\mathfrak{l}_{S})\subset U_{h}^{k}(\mathfrak{g}\times\mathfrak{g}\times\mathfrak{l}_{S})$ .

In light of the spectral decomposition (19) and the Peter-Weyl theorem for $\mathcal{O}_{h}^{k}(G)$ , $\mathrm{ind}_{L_{0,h}}^{G_{h}}\mathcal{O}_{h,\chi}(L_{S}\backslash L_{0})$ has the following expression:

	$\displaystyle\mathrm{ind}_{L_{0,h}}^{G_{h}}\mathcal{O}_{h,\chi}(L_{S}\backslash L_{0})\cong\bigoplus_{\pi\in\operatorname{\mathrm{Irr}}_{h}\mathfrak{g},\rho\in\operatorname{\mathrm{Irr}}_{h}\mathfrak{l}_{0}}\operatorname{\mathrm{Hom}}_{U_{h}^{k}(\mathfrak{l}_{S})}(V_{\rho},k\llbracket h\rrbracket)\otimes\operatorname{\mathrm{Hom}}_{U_{h}^{k}(\mathfrak{l}_{0})}(V_{\pi},V_{\rho})\otimes V_{\pi},$
	$\displaystyle(f\otimes T\otimes v)\ast(g\otimes S\otimes w)=(f\otimes g)\Psi_{0}^{-1}\otimes(T\otimes S)\otimes(v\otimes w).$

Note that we have

\displaystyle(f\otimes g)\Psi_{0}^{-1}(T\otimes S)=(f\otimes g)(T\otimes S)\Psi^{-1}=(f\circ T\otimes g\circ S)\Psi^{-1}.

On the other hand, we have

\displaystyle\operatorname{\mathrm{Hom}}_{U_{h}^{k}(\mathfrak{l}_{S})}(V_{\pi},k\llbracket h\rrbracket)\cong\bigoplus_{\rho\in\operatorname{\mathrm{Irr}}_{h}\mathfrak{l}_{0}}\operatorname{\mathrm{Hom}}_{U_{h}^{k}(\mathfrak{l}_{S})}(V_{\rho},k\llbracket h\rrbracket)\otimes\operatorname{\mathrm{Hom}}_{U_{h}^{k}(\mathfrak{l}_{0})}(V_{\pi},V_{\rho}),

in which $f\otimes T$ corresponds to $f\circ T$ . Combining these facts we obtain the desired isomorphism. ∎

Acknowlegements. The author is grateful to Yasuyuki Kawahigashi for comments on this paper and his invaluable supports. He also thanks Yuki Arano for encouraging him to write this paper.

Polynomial families of quantum semisimple coajoint orbits via deformed quantum enveloping algebras

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Contents of this paper

2. Preliminaries

2.1. Quantum enveloping algebras of semisimple Lie algebras

2.2. The Drinfeld pairing and the quantum Killing form

2.3. The restricted integral form

3. Deformed quantum enveloping algebras

3.1. More on quantum PBW bases

Remark 3.1.

Lemma 3.2.

Proof.

Proposition 3.3 (c.f. [MR1116413, Proposition 5.5.2]).

Proposition 3.4.

Proof.

Proposition 3.5.

Proof.

3.2. Deformed quantum enveloping algebra

Definition 3.6.

Remark 3.7.

Remark 3.8.

Proposition 3.9.

Proof.

Lemma 3.10.

Proof.

Proposition 3.11.

Proof.

Proposition 3.12.

3.3. Twisted parabolic induction via deformed QEA

Lemma 3.13.

Remark 3.14.

Proof.

Definition 3.15.

Remark 3.16.

Lemma 3.17.

Lemma 3.18.

Proof.

4. SS-maximal vectors in the formal setting

4.1. Notation

Definition 4.1.

4.2. Generalized maximal vector

Definition 4.2.

Remark 4.3.

Proposition 4.4.

Lemma 4.5.

Proof.

Proof of Proposition 4.4.

Definition 4.6.

Proposition 4.7.

Proof.

4.3. Construction of the 2-cocycles

Lemma 4.8.

Proof.

Proposition 4.9.

Proof.

Corollary 4.10.

Proof.

Remark 4.11.

Remark 4.12.

Definition 4.13.

Proposition 4.14.

Proof.

Proposition 4.15.

Proof.

4.4. Construction of quantum semisimple coadjoint orbits

Proposition 4.16.

5. Universal families of 22-cocycles

5.1. The toric variety associated to a root system

5.2. The moduli space of equivariant Poisson brackets on LS\GL_{S}\backslash G

Lemma 5.1.

Proof.

Lemma 5.2 (c.f. [MR2141466, Lemma 11]).

Proof.

Lemma 5.3.

Proof.

Proposition 5.4.

Proof.

4. $S$ -maximal vectors in the formal setting

5. Universal families of $2$ -cocycles

5.2. The moduli space of equivariant Poisson brackets on $L_{S}\backslash G$

5.3. The sheaf of deformed quantum enveloping algebra on $X_{R}(k)$