# Representation theory of very non-standard quantum ${{\mathfrak{so}}}(2N-1)$

Stefan Kolb  Stefan Kolb and Jake Stephens, School of Mathematics, Statistics
and Physics, Newcastle University, Herschel Building, Newcastle upon Tyne NE1
7RU, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jake
Stephens

###### Abstract.

We classify the finite dimensional representations of the quantum symmetric
pair coideal subalgebra ${\mathcal{B}}_{\mathbf{c}}$ of type $DII$
corresponding to the symmetric pair
$({\mathfrak{so}}(2N),{\mathfrak{so}}(2N-1))$. For
${\mathcal{B}}_{\mathbf{c}}$ defined over an arbitrary field $k$ and $q\in k$
not a root of unity we establish a one-to-one correspondence between finite
dimensional, simple ${\mathcal{B}}_{\mathbf{c}}$-modules and dominant integral
weights for ${\mathfrak{so}}(2N-1)$. We use specialisation to show that the
category of finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-modules is
semisimple if $\mathrm{char}(k)=0$ and $q$ is transcendental over
$\mathbb{Q}$. In this case the characters of simple
${\mathcal{B}}_{\mathbf{c}}$-modules are given by Weyl’s character formula.
This means in particular that the quantum symmetric pair of type $DII$ can be
used to obtain Gelfand-Tsetlin bases for irreducible representations of the
Drinfeld-Jimbo quantum group $U_{q}({\mathfrak{so}}(2N))$.

###### Key words and phrases:

Quantum groups, quantum symmetric pairs, Gelfand-Tsetlin bases

###### 2020 Mathematics Subject Classification:

17B37

##  1\. Introduction

###  1.1. Motivation and background

Let ${\mathfrak{g}}_{n}$ denote the general linear Lie algebra
${\mathfrak{gl}}(n)$ of complex $n\times n$ matrices or the complex orthogonal
Lie algebra ${\mathfrak{so}}(n)$. There are natural embeddings of Lie algebras
${\mathfrak{g}}_{n}\hookrightarrow{\mathfrak{g}}_{n+1}$. In their seminal work
[GT50a], [GT50b] I. Gelfand and M. Tsetlin developed explicit formulas for
representations of ${\mathfrak{g}}_{n}$ in terms of combinatorial data. These
Gelfand-Tsetlin patters can be interpreted as sequences
$\lambda_{n},\lambda_{n-1},\dots,\lambda_{2},(\lambda_{1})$, where
$\lambda_{\ell}$ is a dominant integral weight for ${\mathfrak{g}}_{\ell}$,
such that the simple ${\mathfrak{g}}_{\ell}$-module of highest weight
$\lambda_{\ell}$ is contained inside the simple
${\mathfrak{g}}_{\ell+1}$-module of highest weight $\lambda_{\ell+1}$ under
restriction. As restriction from ${\mathfrak{g}}_{\ell+1}$ to
${\mathfrak{g}}_{\ell}$ is multiplicity free, the sequence
$\lambda_{n},\lambda_{n-1},\dots,\lambda_{2},(\lambda_{1})$ determines a basis
of the simple ${\mathfrak{g}}_{n}$-module of highest weight $\lambda_{n}$ up
to scalar factors.

With the advent of quantum groups in the 1980s, it was natural to ask if
similar bases exist for representations of Drinfeld-Jimbo quantised enveloping
algebras $U_{q}({\mathfrak{g}}_{n})$. For
${\mathfrak{g}}_{n}={\mathfrak{gl}}(n)$ this was settled early on [Jim86],
[UTS90]. However, for orthogonal Lie algebras there is no embedding of
$U_{q}({\mathfrak{so}}(n-1))$ into $U_{q}({\mathfrak{so}}(n))$ and therefore
Gelfand-Tsetlin bases for $U_{q}({\mathfrak{so}}(n))$ were long deemed non-
existent. This observation was the starting point for A. Gavrilik and A.
Klimyk’s invention of non-standard $U^{\prime}_{q}({\mathfrak{so}}(n))$ in
[GK91]. These algebras support inclusions
$U^{\prime}_{q}({\mathfrak{so}}(n-1))\subset
U^{\prime}_{q}({\mathfrak{so}}(n))$ and Gavrilik and Klimyk constructed finite
dimensional, irreducible representations of
$U^{\prime}_{q}({\mathfrak{so}}(n))$ in terms of Gelfand-Tsetlin patterns for
${\mathfrak{so}}(n)$.

M. Noumi observed in [Nou96] that Gavrilik and Klimyk’s
$U^{\prime}_{q}({\mathfrak{so}}(n))$ can be realised as a (right) coideal
subalgebra of Drinfeld-Jimbo $U_{q}({\mathfrak{sl}}(n))$, and he established
the pair of algebras
$(U_{q}({\mathfrak{sl}}(n)),U^{\prime}_{q}({\mathfrak{so}}(n)))$ as one of the
first examples of a quantum symmetric pair. Noumi and collaborators
constructed quantum symmetric pairs for all classical symmetric Lie algebras,
that is, for all involutive Lie algebra automorphisms
$\theta:{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ with ${\mathfrak{g}}$ of
classical type [Nou96], [NS95], [NDS97]. Their construction proceeds case by
case and relies on explicit solutions of the reflection equation.

Independently, G. Letzter devised a uniform construction of quantum symmetric
pairs for all semisimple symmetric Lie algebras [Let99]. Recall that
involutive automorphisms $\theta$ of a semisimple, complex Lie algebra
${\mathfrak{g}}$ are classified up to conjugation in terms of Satake diagrams
$(X,\tau)$, see also Section 2.1. Here $X$ is a subset of the set of nodes of
the Dynkin diagram of ${\mathfrak{g}}$ and $\tau$ is a diagram automorphism
preserving the subset $X$. For each Satake diagram, Letzter defined a right
coideal subalgebra ${\mathcal{B}}\subset{U_{q}(\mathfrak{g})}$ which is a
quantum group analogue of $U({\mathfrak{k}})$ for
${\mathfrak{k}}=\\{x\in{\mathfrak{g}}\,|\,\theta(x)=x\\}$. We refer to
$({U_{q}(\mathfrak{g})},{\mathcal{B}})$ as a quantum symmetric pair (QSP) and
to the algebra ${\mathcal{B}}$ as a QSP-coideal subalgebra of
${U_{q}(\mathfrak{g})}$. For ${\mathfrak{g}}$ of classical type, Letzter’s
construction reproduces the coideal subalgebras constructed by Noumi and
collaborators, see [Let99, Section 6].

The pair of Lie algebras $({\mathfrak{so}}(n),{\mathfrak{so}}(n-1))$ is a
symmetric pair. If $n=2N$ is even, then the corresponding Satake diagram is of
type $DII$, see Figure 1(A).

$1$$2$$N-1$$N$ (A) $n=2N$, type $DII$

$1$$2$$N$$N$ (B) $n=2N+1$, type $BII$

Figure 1. Satake diagrams for $({\mathfrak{so}}(n),{\mathfrak{so}}(n-1))$

Quantum symmetric pairs provide a corresponding coideal subalgebra
${\mathcal{B}}_{2N-1}\subset U_{q}({\mathfrak{so}}(2N))$ which is a quantum
group analogue of $U({\mathfrak{so}}(2N-1))$. The algebra
${\mathcal{B}}_{2N-1}$ is different from Gavrilik and Klimyk’s non-standard
$U_{q}^{\prime}({\mathfrak{so}}_{2N-1})$ and hence we also refer to it as
‘very non-standard’ quantum ${\mathfrak{so}}(2N-1)$. By Letzter’s construction
we obtain a chain of algebras

(1.1) $\displaystyle\dots\subset
U_{q}({\mathfrak{so}}(2N-2))\subset{\mathcal{B}}_{2N-1}\subset
U_{q}({\mathfrak{so}}(2N))\subset\dots$

In the odd case $n=2N+1$, the Satake diagram corresponding to the symmetric
pair $({\mathfrak{so}}(2N+1),{\mathfrak{so}}(2N))$ is of type $BII$, see
Figure 1(B). Again, quantum symmetric pairs provide a coideal subalgebra
${\mathcal{B}}_{2N}\subset U_{q}({\mathfrak{so}}(2N+1))$ which is a quantum
group analogue of $U({\mathfrak{so}}(2N))$, and we obtain a chain of algebras

(1.2) $\displaystyle\dots\subset
U_{q}({\mathfrak{so}}(2N-1))\subset{\mathcal{B}}_{2N}\subset
U_{q}({\mathfrak{so}}(2N+1))\subset\dots$

It is natural to expect that the chains of algebras (1.1) and (1.2) lead to
Gelfand-Tsetlin bases for representations of $U_{q}({\mathfrak{so}}(n))$. To
this end, we need to understand the representation theory of the QSP-coideal
subalgebras ${\mathcal{B}}_{2N-1}$ and ${\mathcal{B}}_{2N}$. The two
situations are quite different as they correspond to different Satake
diagrams. In the present paper we consider the quantum symmetric pair
$(U_{q}({\mathfrak{so}}(2N)),{\mathcal{B}}_{2N-1})$.

###  1.2. Representations of QSP-coideal subalgebras

If $({\mathfrak{g}},{\mathfrak{k}})$ is a symmetric pair with semisimple
${\mathfrak{g}}$ then ${\mathfrak{k}}$ is a reductive Lie algebra. Hence it is
natural to expect that any QSP-coideal subalgebra ${\mathcal{B}}$ has a nice
category of finite-dimensional representations. However, to this date, finite
dimensional representations of ${\mathcal{B}}$ have been classified only for a
fairly limited class of examples. To put the results of the present paper into
context, we provide a brief overview of known results.

In type $AI$, corresponding to the symmetric pair
$({\mathfrak{sl}}(n),{\mathfrak{so}}(n))$, finite dimensional representations
of ${\mathcal{B}}$ were classified inductively by N. Iorgov and A. Klimyk
[IK05]. There appear two types of irreducible representations, those of
classical type which allow specialisation $q\rightarrow 1$ and those of
nonclassical type which don’t. The classification relies on the explicit
realisation of these representations given in [GK91]. These realisations
involve square roots of $q$-numbers and hence [IK05] work over the field
$k={\mathbb{C}}$ with $q\in{\mathbb{C}}$ not a root of unity. The induction
argument in [IK05] also shows that the category of finite dimensional
${\mathcal{B}}$-modules is semisimple in this case.

The classification in type $AI$ was revisited by H. Wenzl who developed a
Verma module approach in this case [Wen20]. In particular, Wenzl identifies
large proper submodules of Verma modules and can show by an
${\mathfrak{sl}}(2)$-argument similar to the classical case that the
corresponding quotients are finite dimensional if the highest weight is
dominant integral [Wen20, Section 5.3]. This is possible because the rank two
case provides ${\mathfrak{sl}}(2)$-triples in type $AI$.

In type $AII$, corresponding to the symmetric pair
$({\mathfrak{sl}}(2N)),\mathfrak{sp}(2N))$, irreducible representations of
${\mathcal{B}}$ were classified by A. Molev in [Mol06]. Molev works
essentially with Noumi’s construction of ${\mathcal{B}}$ and again follows a
Verma module approach. To show that simple quotients of Verma modules for
dominant integral weights are finite dimensional, Molev uses Gelfand-Tsetlin
bases for $U_{q}({\mathfrak{gl}}(2N))$ to construct non-vanishing
homomorphisms from Verma modules for ${\mathcal{B}}$ to finite dimensional
$U_{q}({\mathfrak{gl}}(2N))$-modules. Moreover, Molev uses specialisation to
show that in type $AII$ the characters of simple ${\mathcal{B}}$-modules are
given by Weyl’s character formula.

A unifying approach to the classification of finite dimensional
${\mathcal{B}}$-modules was proposed by H. Watanabe in [Wat21]. To include all
types, he works over the algebraic closure $\overline{{\mathbb{C}}(q)}$.
Watanabe defines a class of modules of classical weight which excludes the
nonclassical modules of type $AI$ and which is not always closed under tensor
products with 1-dimensional ${U_{q}(\mathfrak{g})}$-modules. For this class
Watanabe develops a highest weight theory. To make this theory work, one needs
to establish a triangular decomposition of ${\mathcal{B}}$ in terms of
analogues of root vectors in ${\mathcal{B}}$, which is quite challenging in
general. In [Wat21] this program is performed for several classes of examples,
reproving the results of [Mol06], parts of the classification in [IK05] and
[Wen20], and leading to a classification of simple modules of classical weight
in type AIII corresponding to the symmetric pairs
$({\mathfrak{sl}}(2n),\mathfrak{s}({\mathfrak{gl}}(n)\oplus{\mathfrak{gl}}(n)))$
and
$({\mathfrak{sl}}(2n+1),\mathfrak{s}({\mathfrak{gl}}(n)\oplus{\mathfrak{gl}}(n+1)))$.
To show that simple quotients of Verma modules for dominant integral weights
are finite dimensional, Watanabe uses specialisation early on. Specialisation
is also employed to show that any finite dimensional ${\mathcal{B}}$-module is
completely reducible in those cases.

###  1.3. Results

In the present paper, we classify all finite dimensional simple
${\mathcal{B}}$-modules in type $DII$ corresponding to the symmetric pair
$({\mathfrak{so}}(2N),{\mathfrak{so}}(2N-1))$. We aim to emulate the approach
in Jantzen’s textbook [Jan96, Chapter 5]. In particular, we work over any
field $k$ with $q\in k$ not a root of unity. We follow the conventions in
[Kol14] and write ${\mathcal{B}}={\mathcal{B}}_{\mathbf{c}}$ to indicate the
parameter dependence of the QSP-coideal subalgebra of type $DII$.

Let $U^{0}\subset U_{q}({\mathfrak{so}}(2N))$ be the subalgebra generated by
the group-like elements $K_{i}^{\pm 1}$ for $i=1,\dots,N$. In type $DII$, the
number of black dots of the Satake diagram $|X|=N-1$ coincides with the rank
of ${\mathfrak{k}}={\mathfrak{so}}(2N-1)$. Hence, the QSP-coideal subalgebra
${\mathcal{B}}_{\mathbf{c}}$ has a natural analogue of a Cartan subalgebra,
namely $U^{0}_{X}:={\mathcal{B}}_{\mathbf{c}}\cap U^{0}=k\langle K_{i}^{\pm
1}\,|\,i\in X\rangle$. This only happens for symmetric pairs of type $AII$,
$DII$ and $EIV$ and simplifies the situation significantly.

The algebra ${\mathcal{B}}_{\mathbf{c}}$ is generated by a single element
$B_{1}$ and the subalgebra ${\mathcal{M}}_{X}=U_{q}({\mathfrak{g}}_{X})$
corresponding to the set $X$ of black dots in the Satake diagram. Taking
inspiration from the classical case, we define root vectors for
${\mathcal{B}}_{\mathbf{c}}$ in terms of the generator $B_{1}$ and the Lusztig
braid automorphism $T_{i}$ for $i\in X$. The root vectors are ordered by a
choice of longest element in the Weyl group $W$. Moreover, they split up into
positive and negative root vectors depending on their behaviour under
conjugation by $U^{0}_{X}$. Let ${\mathcal{B}}_{\mathbf{c}}^{+}$ and
${\mathcal{B}}_{\mathbf{c}}^{-}$ be the span of all ordered monomials in the
positive and negative root vectors of ${\mathcal{B}}_{\mathbf{c}}$,
respectively. We obtain a triangular decomposition for
${\mathcal{B}}_{\mathbf{c}}$.

Theorem A. (Theorem 4.6, Corollary 4.7) The multiplication map

$\displaystyle{\mathcal{B}}_{\mathbf{c}}^{-}\otimes
U^{0}_{X}\otimes{\mathcal{B}}_{\mathbf{c}}^{+}\rightarrow{\mathcal{B}}$

is a linear isomorphism. Moreover, the ordered monomials in the positive
(negative) root vectors form a basis of ${\mathcal{B}}_{\mathbf{c}}^{+}$
(${\mathcal{B}}_{\mathbf{c}}^{-}$).

Any finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-module $V$ can be
considered as a ${\mathcal{M}}_{X}$-module by restriction, and hence the
action of $U^{0}_{X}$ on $V$ is diagonalisable. The notion of type of a finite
dimensional ${\mathcal{M}}_{X}$-module [Jan96, 5.2] lifts to a notion of type
of a finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-module. As in the quantum
group case, it suffices to consider finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-modules of type $\mathbf{1}$, as all others can
be obtained by tensoring by a one-dimensional
$U_{q}({\mathfrak{so}}(2N))$-module, see Proposition 5.1.

Let $P_{2N-1}$ denote the weight lattice of ${\mathfrak{so}}(2N-1)$ and let
$P_{2N-1}^{+}$ be the subset of dominant integral weights. Finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-modules decompose into weight spaces and hence
contain a highest weight vector of weight $\lambda\in P_{2N-1}$ on which
${\mathcal{B}}_{\mathbf{c}}^{+}$ acts trivially. We need to show that
$\lambda$ is dominant. Let $\gamma_{1},\dots,\gamma_{N-1}$ be the simple roots
of ${\mathfrak{so}}(2N-1)$ with $\gamma_{N-1}$ being the short root. For
$i=1,\dots,N-2$ the triple $\\{E_{i+1},F_{i+1},K_{i+1}^{\pm
1}\\}\subset{\mathcal{B}}_{\mathbf{c}}$ generates a copy of
$U_{q}({\mathfrak{sl}}(2))$ corresponding to the root $\gamma_{i}$, and hence
$\lambda(h_{\gamma_{i}})\geq 0$ where $h_{\gamma_{i}}$ is the corresponding
coroot. We denote the positive and negative root vectors corresponding to the
short root $\gamma_{N-1}$ by $B_{\beta_{N-1}}$ and $B_{\beta_{N}}$,
respectively. We show in Proposition 5.10 that the triple
$\\{B_{\beta_{N-1}},B_{\beta_{N}},(K_{N}K_{N-1}^{-1})^{\pm 1}\\}$ acts on a
suitable subspace $H(V)\subset V$ as a copy of $U_{q}({\mathfrak{sl}}(2))$.
Moreover, $H(V)$ contains the highest weight vector. This implies that
$\lambda(h_{\gamma_{N-1}})\geq 0$ and hence $\lambda\in P^{+}_{2N-1}$ is
dominant.

Using the root vectors for ${\mathcal{B}}_{\mathbf{c}}$ we define a Verma
module $M(\lambda)$ for each $\lambda\in P_{2N-1}$, see Section 5.5. The Verma
module $M(\lambda)$ is generated by a highest weight vector $v_{\lambda}$ and
we show in Theorem 5.13 that the linear map

$\displaystyle{\mathcal{B}}_{\mathbf{c}}^{-}\rightarrow M(\lambda),\qquad
b\mapsto bv_{\lambda}$

is a linear isomorphism. Any finite dimensional simple
${\mathcal{B}}_{\mathbf{c}}$-module is a quotient of $M(\lambda)$ by a proper
maximal submodule $N(\lambda)\subset M(\lambda)$ for some $\lambda\in
P^{+}_{2N-1}$. To complete the classification of finite dimensional, simple
${\mathcal{B}}_{\mathbf{c}}$-modules, it remains to show that

$\displaystyle L(\lambda)=M(\lambda)/N(\lambda)$

is finite dimensional for all $\lambda\in P_{2N-1}^{+}$. We first consider the
submodule

$\displaystyle\widetilde{N}(\lambda)=\sum_{i=1}^{N-2}{\mathcal{B}}_{\mathbf{c}}F_{i+1}^{n_{i}+1}v_{\lambda}+{\mathcal{B}}_{\mathbf{c}}B_{\beta_{N}}^{n_{N-1}+1}v_{\lambda}$

with $n_{i}=\lambda(h_{\gamma_{i}})\geq 0$. As in the quantum group case, we
show that $\widetilde{N}(\lambda)\subset M(\lambda)$ is a proper submodule.
However, it is unclear how to adapt the Weyl group invariance argument from
[Jan96, 5.9], since $\\{B_{\beta_{N-1}},B_{\beta_{N}},(K_{N}K_{N-1}^{-1})^{\pm
1}\\}$ acts as an ${\mathfrak{sl}}(2)$-triple only on the subspace $H(V)$.
Instead we consider the submodule

$\displaystyle\accentset{\approx}{N}(\lambda)=\sum_{i=1}^{N-2}{\mathcal{B}}_{\mathbf{c}}F_{i+1}^{n_{i}+1}v_{\lambda}+{\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda}$

with $n_{\lambda}=\lambda(h_{\gamma_{N-2}}+h_{\gamma_{N-1}})$ which contains
$\widetilde{N}(\lambda)$, see Remark 5.19. By Proposition 5.15 we know that
$\accentset{\approx}{N}(\lambda)$ is also a proper submodule of $M(\lambda)$.
Using a filtered-graded argument, we show in Section 5.8 that
$M(\lambda)/\accentset{\approx}{N}(\lambda)$ is finite dimensional. This gives
us the desired classification.

Theorem B. (Theorem 5.22) Let $k$ be any field and $q\in k$ not a root of
unity. For each $\lambda\in P_{2N-1}^{+}$ the simple
${\mathcal{B}}_{\mathbf{c}}$-module $L(\lambda)$ has finite dimension. Each
finite-dimensional simple ${\mathcal{B}}_{\mathbf{c}}$-module (of type ${\bf
1}$) is isomorphic to exactly one $L(\lambda)$ with $\lambda\in P_{2N-1}^{+}$.

We would like to know the characters of the simple
${\mathcal{B}}_{\mathbf{c}}$-modules $L(\lambda)$ and we would like to show
that all finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-modules are
semisimple. At this point, we can only achieve this via specialisation, as in
[Jan96, 5.15, 5.17], [Mol06], [Wat21]. We obtain the following result.

Theorem C. (Corollary 6.8, Theorem 6.11) Assume that $k$ is a field of
characteristic zero and that $q$ is transcendental over $\mathbb{Q}$. Then
every finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-module is semisimple.
Moreover, for any $\lambda\in P^{+}_{2N-1}$ the submodules
$N(\lambda),\widetilde{N}(\lambda)$ and $\accentset{\approx}{N}(\lambda)$
coincide, and the dimensions of the weight spaces $L(\lambda)_{\mu}$ for
$\mu\in P_{2N-1}$ are given by Weyl’s character formula.

Theorem C implies in particular that finite dimensional
$U_{q}({\mathfrak{so}}(2N))$-modules decompose into irreducible
${\mathcal{B}}_{\mathbf{c}}$-modules as in the classical case $q=1$. And
similarly, finite-dimensional ${\mathcal{B}}_{\mathbf{c}}$-modules decompose
into irreducible $U_{q}({\mathfrak{so}}(2N-2))$-modules as in the classical
case $q=1$. Hence, if $\mathrm{char}(k)=0$ and $q$ is transcendental over
$\mathbb{Q},$ then the chain of inclusions (1.1) provides a Gelfand-Tsetlin
basis of any simple $U_{q}({\mathfrak{so}}(2N))$-module, up to scalar factors.

###  1.4. Very recent developments

The theory of quantum symmetric pairs is presently developing at rapid speed.
While we were putting final touches to the present paper, two preprints
appeared which have bearing on our results. In [Wat24] H. Watanabe proposed a
notion of integrable modules for QSP-coideal subalgebras. He shows that
simple, integrable modules over QSP-coideal subalgebras of finite type are
finite dimensional [Wat24, Section 5.1]. In the spirit of our paper to emulate
the constructions of [Jan96, Chapter 5] as closely as possible, it would be
desirable to show that the quotient
$M(\lambda)/\accentset{\approx}{N}(\lambda)$ for $\lambda\in P^{+}_{2N-1}$ is
integrable in the sense of Watanabe’s paper.

M. Lu, R. Yang and W. Zhang proposed notions of root vectors and PBW bases for
QSP-coideal subalgebras of finite type [LYZ24]. In type $DII$ the root vectors
given in [LYZ24, 3.7] coincide with the root vectors given in Section 4.2.
Moreover, the second part of Theorem A, Theorem 4.6, is a special case of
[LYZ24, Theorem 4.4]. We note that the definition of the root vectors in type
$DII$ and the PBW-Theorem 4.6 already appeared in the second named author’s
PhD thesis [Ste23].

Acknowledgements. The first named author is grateful to Johnathan Brundan and
Ben Webster for helpul comments at an early stage of this project.

##  2\. The symmetric pair $({\mathfrak{so}}(2N),{\mathfrak{so}}(2N-1))$

In this preliminary section we identify the orthogonal Lie algebra
${\mathfrak{so}}(2N-1)$ explicitly as the Lie subalgebra pointwise fixed under
a maximally split involution on ${\mathfrak{so}}(2N)$. This allows us to
identify simple root vectors of ${\mathfrak{so}}(2N-1)$ inside
${\mathfrak{so}}(2N)$, which will in turn inform the choice of root vectors in
the quantum setting in Section 4.

###  2.1. Symmetric pairs and Satake diagrams

Let ${\mathfrak{g}}$ be a semisimple complex Lie algebra with Cartan
subalgebra ${\mathfrak{h}}$, root system $\Phi$, a fixed choice of simple
roots $\Pi=\\{\alpha_{i}\,|\,i\in I\\}$, and corresponding triangular
decomposition
${\mathfrak{g}}={\mathfrak{n}}^{+}\oplus{\mathfrak{h}}\oplus{\mathfrak{n}}^{-}$.
Let $e_{i},f_{i},h_{i}$ for $i\in I$ denote the Chevalley generators of
${\mathfrak{g}}$. For any subset $X\subset I$ let ${\mathfrak{g}}_{X}$ denote
the Lie subalgebra of ${\mathfrak{g}}$ generated by $e_{i},f_{i},h_{i}$ for
all $i\in X$. Let $Q_{X}=\sum_{i\in X}{\mathbb{Z}}\alpha_{i}$ be the
corresponding root lattice and $Q_{X}^{+}=\sum_{i\in
X}{\mathbb{N}}_{0}\alpha_{i}$ the positive cone. For $X=I$ we write $Q$ and
$Q^{+}$ instead of $Q_{I}$ and $Q_{I}^{+}$, respectively.

Let $\theta:{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ be an involutive Lie
algebra automorphism, i.e. $\theta^{2}={\mbox{id}}_{\mathfrak{g}}$, and
${\mathfrak{k}}=\\{x\in{\mathfrak{g}}\,|\,\theta(x)=x\\}$ the pointwise fixed
Lie subalgebra of ${\mathfrak{g}}$. We refer to
$({\mathfrak{g}},{\mathfrak{k}})$ as a symmetric pair. Up to conjugation by a
Lie algebra automorphism of ${\mathfrak{g}}$, the involution $\theta$ has the
following properties, see [Dix96, 1.13], also [Let02, Section 7]:

  1. (1)

$\theta({\mathfrak{h}})={\mathfrak{h}}$,

  2. (2)

There exists $X\subset I$ such that
$\theta|_{{\mathfrak{g}}_{X}}={\mbox{id}}_{{\mathfrak{g}}_{X}}$,

  3. (3)

$\theta(e_{i})\in{\mathfrak{n}}^{-}$ and $\theta(f_{i})\in{\mathfrak{n}}^{+}$
if $i\in I\setminus X$, where $X$ is the set given in (2).

Following [Let02], we call the involution $\theta$ maximally split if
properties (1), (2) and (3) hold. From now on we assume that $\theta$ is
maximally split. By condition (1), the map $\theta$ induces an involution
$\Theta:{\mathfrak{h}}^{\ast}\rightarrow{\mathfrak{h}}^{\ast}$ which leaves
the root system $\Phi$ invariant. By construction we have
$\theta({\mathfrak{g}}_{\alpha})={\mathfrak{g}}_{\Theta(\alpha)}$ for all
roots $\alpha\in\Phi$. Moreover, if $\theta$ is maximally split, then there
exists an involutive diagram automorphism $\tau:I\rightarrow I$ such that
$\Theta(-\alpha_{i})-\alpha_{\tau(i)}\in Q^{+}_{X}$ for all $i\in I\setminus
X$, see also [Let02, (7.5)]. The pair $(X,\tau)$ is called a Satake diagram
and determines the involution $\theta$ uniquely up to conjugation, see also
[Kol14, Theorem 2.7]. The Satake diagram $(X,\tau)$ is recorded in the Dynkin
diagram of ${\mathfrak{g}}$ by colouring all nodes in $X$ black and indicating
the diagram automorphism by a double-pointed arrow. See [Ara62, p. 32/33] for
a list of all Satake diagrams for simple Lie algebras ${\mathfrak{g}}$.

###  2.2. Root system and Chevalley generators for ${\mathfrak{so}}(n)$

For any $n\in{\mathbb{N}}$ let ${\mathfrak{gl}}(n)$ denote the Lie algebra of
complex $n\times n$-matrices. For any $m\times n$ matrix $M$ we write $M^{t}$
to denote its transpose matrix. The complex orthogonal Lie algebra
${\mathfrak{so}}(n)$ can be defined as

(2.1)
$\displaystyle{\mathfrak{gl}}_{S}(n)=\\{M\in{\mathfrak{gl}}(n)\,|\,M=-SM^{t}S^{-1}\\}$

for any non-degenerate symmetric $n\times n$-matrix $S$. The Lie algebras
${\mathfrak{gl}}_{S}(n)$ for different choices of $S$ are all isomorphic by
the classification of complex symmetric bilinear forms. For the purpose of
this paper, it is convenient to choose

(2.2) $\displaystyle S=S_{n}=\begin{pmatrix}&&1\\\ &\iddots&\\\
1&&\end{pmatrix}=\sum_{i=1}^{n}E_{i,n-i+1}$

where $E_{i,j}$ denotes the elementary $n\times n$ matrix with entry $1$ in
the $ij$-th position and zero entries elsewhere. For this choice of $S$, the
set ${\mathfrak{h}}_{n}$ of diagonal matrices in ${\mathfrak{gl}}_{S}(n)$ is a
Cartan subalgebra. Moreover, the definition of $S=S_{n}$ is uniform for even
and odd $n$. Throughout this paper, the symbol ${\mathfrak{so}}(n)$ for
$n\in{\mathbb{N}}$ will always denote the Lie algebra ${\mathfrak{gl}}_{S}(n)$
for $S$ given by (2.2).

For any $m\times n$ matrix $A$ we write $A^{\tau}=S_{n}A^{t}S_{m}$ to denote
the matrix obtained by reflection in the antidiagonal. With this notation we
have

$\displaystyle{\mathfrak{so}}(n)={\mathfrak{gl}}_{S}(n)=\\{M\in{\mathfrak{gl}}(n)\,|\,M=-M^{\tau}\\}.$

######  Remark 2.1.

Let ${\mathbb{K}}$ be any field of characteristic $0$. Then (2.1) with $S$
given by (2.2) defines a Lie algebra ${\mathfrak{gl}}_{n}(S)_{\mathbb{K}}$
over ${\mathbb{K}}$. The set ${\mathfrak{h}}_{n}$ of diagonal matrices in
${\mathfrak{gl}}_{S}(n)_{\mathbb{K}}$ is a splitting Cartan subalgebra. Up to
a change of basis this example is discussed in [Jac62, IV, Theorems 7 and 9].
Moreover, finite dimensional representations of
${\mathfrak{gl}}_{S}(n)_{\mathbb{K}}$ are completely reducible, irreducible
representations are given by dominant integral weights, and the characters of
irreducible ${\mathfrak{gl}}_{n}(S)$-modules are given by Weyl’s character
formula, see [Jac62]. We will need these statements in Section 6.3 when we
prove analogous results for very non-standard quantum ${\mathfrak{so}}(2N-1)$
over a field of characteristic $0$ and $q$ transcendental over $\mathbb{Q}$,
see also [Jan96, Remark 5.14].

For later reference, we describe root systems and Chevalley generators of
${\mathfrak{so}}(n)$ for even and for odd $n$ separately. Moreover, it is
convenient to give block decompositions of ${\mathfrak{so}}(n)$ for even and
odd $n$ which will allow us to realise explicit embeddings of
${\mathfrak{so}}(n-1)$ into ${\mathfrak{so}}(n)$.

Case $n=2N$ even: In this case, we can write

(2.3)
$\displaystyle{\mathfrak{so}}(2N)={\mathfrak{gl}}_{S}(2N)=\left\\{\begin{pmatrix}A&B\\\
C&-A^{\tau}\end{pmatrix}\Bigg{|}\,B=-B^{\tau},C=-C^{\tau}\right\\}.$

where $A,B,C\in{\mathfrak{gl}}(N)$. The elements
$h_{i}=E_{i,i}-E_{n-i+1,n-i+1}$ for $i=1,\dots,N$ form a basis of the Cartan
subalgebra ${\mathfrak{h}}_{2N}$. Let $\\{\varepsilon_{i}\,|\,i=1,\dots N\\}$
be the dual basis of ${\mathfrak{h}}_{2N}^{*}$. The root system of
${\mathfrak{so}}(2N)$ is given by
$\\{\pm(\varepsilon_{i}\pm\varepsilon_{j})\,|\,1\leq i<j\leq N\\}$. We choose
the simple roots

(2.4)
$\displaystyle\alpha_{i}=\begin{cases}\varepsilon_{i}-\varepsilon_{i+1}&\mbox{for
$1\leq i\leq N-1$,}\\\ \varepsilon_{N-1}+\varepsilon_{N}&\mbox{for
$i=N$.}\end{cases}$

Corresponding Chevalley generators of ${\mathfrak{so}}(2N)$ are given by

(2.5) $\displaystyle e_{\alpha_{i}}$
$\displaystyle=E_{i,i+1}{-}E_{n-i,n-i+1},$ $\displaystyle f_{\alpha_{i}}$
$\displaystyle=E_{i+1,i}{-}E_{n-i+1,n-i},$ $\displaystyle h_{\alpha_{i}}$
$\displaystyle=h_{i}{-}h_{i+1}$

for $1\leq i\leq N-1$, and

(2.6) $\displaystyle e_{\alpha_{N}}$
$\displaystyle{=}E_{N-1,N+1}{-}E_{N,N+2},$ $\displaystyle f_{\alpha_{N}}$
$\displaystyle{=}E_{N+1,N-1}{-}E_{N+2,N},$ $\displaystyle h_{\alpha_{N}}$
$\displaystyle{=}h_{N-1}{+}h_{N}.$

Case $n=2N+1$ odd: In this case, we can write

(2.7)
$\displaystyle{\mathfrak{so}}(2N+1)={\mathfrak{gl}}_{S}(2N+1)=\left\\{\begin{pmatrix}A&b&B\\\
d&0&-b^{\tau}\\\
C&-d^{\tau}&-A^{\tau}\end{pmatrix}\Bigg{|}\,B=-B^{\tau},C=-C^{\tau}\right\\}$

where $b\in{\mathbb{C}}^{N}$ is a column vector, $d\in{\mathbb{C}}^{N}$ is a
row vector and $A,B,C$ are $N\times N$-matrices as before. As in the even
case, the elements $h_{i}=E_{i,i}-E_{n-i+1,n-i+1}$ for $i=1,\dots,N$ form a
basis of ${\mathfrak{h}}_{2N+1}$, and we still denote the dual basis by
$\\{\varepsilon_{i}\,|\,i=1,\dots,N\\}$. The root system of
${\mathfrak{so}}(2N+1)$ is given by
$\\{\pm\varepsilon_{\ell},\pm(\varepsilon_{i}\pm\varepsilon_{j})\,|\,\ell=1,\dots,N,\,1\leq
i<j\leq N\\}$. We choose the simple roots

$\displaystyle\gamma_{i}=\begin{cases}\varepsilon_{i}-\varepsilon_{i+1}&\mbox{for
$1\leq i\leq N-1$,}\\\ \varepsilon_{N}&\mbox{for $i=N$.}\end{cases}$

Corresponding Chevalley generators
$e_{\gamma_{i}},f_{\gamma_{i}},h_{\gamma_{i}}$ of ${\mathfrak{so}}(2N+1)$ are
given by (2.5) for $1\leq i\leq N-1$, and

$\displaystyle e_{\gamma_{N}}$ $\displaystyle=E_{N,N+1}{-}E_{N+1,N+2},$
$\displaystyle f_{\gamma_{N}}$ $\displaystyle=2(E_{N+1,N}{-}E_{N+2,N+1}),$
$\displaystyle h_{\gamma_{N}}$ $\displaystyle=2h_{N}.$

In the case $n=2N+1$, it helps to define

(2.8) $\displaystyle b_{i}$ $\displaystyle=E_{i,N+1}{-}E_{N+1,2N-i+2},$
$\displaystyle d_{i}$ $\displaystyle=E_{N+1,i}{-}E_{2N-i+2,N+1}$ for
$i=1,\dots,N$.

With this notation, we have $e_{\gamma_{N}}=b_{N}$ and
$f_{\gamma_{N}}=2d_{N}$. Moreover, note that

(2.9) $\displaystyle[f_{\gamma_{i}},b_{i}]$ $\displaystyle=b_{i+1},$
$\displaystyle[e_{\gamma_{i}},b_{i+1}]$ $\displaystyle=b_{i},$
$\displaystyle[f_{\gamma_{i}},d_{i+1}]$ $\displaystyle=-d_{i},$
$\displaystyle[e_{\gamma_{i}},d_{i}]$ $\displaystyle=-d_{i+1}$

for $1\leq i\leq N-1$. It is convenient to consider the Chevalley generators
(2.6) of the copy of ${\mathfrak{so}}(2N)$ inside ${\mathfrak{so}}(2N+1)$. We
have

(2.10) $\displaystyle[f_{\alpha_{N}},b_{N-1}]$ $\displaystyle=-d_{N},$
$\displaystyle[f_{\alpha_{N}},b_{N}]$ $\displaystyle=d_{N-1},$ (2.11)
$\displaystyle[e_{\alpha_{N}},d_{N-1}]$ $\displaystyle=b_{N},$
$\displaystyle[e_{\alpha_{N}},d_{N}]$ $\displaystyle=-b_{N-1}.$

The above formulas (2.10), (2.11) and (2.9) reflect the fact that the matrices
$\\{b_{i},d_{i}\,|\,i=1,\dots,N\\}$ form a basis of the vector representation
of the Lie subalgebra ${\mathfrak{so}}(2N)\subset{\mathfrak{so}}(2N+1)$ under
the adjoint action. This is also evident from the block matrix representation
(2.7) of ${\mathfrak{so}}(2N+1)$.

###  2.3. Realising ${\mathfrak{so}}(2N-1)\subset{\mathfrak{so}}(2N)$ as a
symmetric pair

In the realisations (2.3), (2.7), there is an immediate embedding of Lie
algebras $\varphi:{\mathfrak{so}}(2N)\hookrightarrow{\mathfrak{so}}(2N+1)$
given by

$\displaystyle\varphi\left(\begin{pmatrix}A&B\\\
C&-A^{\tau}\end{pmatrix}\right)=\begin{pmatrix}A&{\mathbf{0}}&B\\\
{\mathbf{0}}^{t}&0&{\mathbf{0}}^{t}\\\ C&{\mathbf{0}}&-A^{\tau}\end{pmatrix}$

where ${\mathbf{0}}\in{\mathbb{C}}^{N}$ denotes the zero column vector. The
embedding ${\mathfrak{so}}(2N-1)\subset{\mathfrak{so}}(2N)$ is slightly less
immediate. In the following, we will realise ${\mathfrak{so}}(2N-1)$ for
$N\geq 2$ as the fixed Lie subalgebra for a maximally split involution on
${\mathfrak{so}}(2N)$. Consider the $2N\times 2N$-matrix

(2.12) $\displaystyle J=\begin{pmatrix}\begin{array}[]{c|c|c}1&&\\\
\hline\cr&S_{2N-2}&\\\ \hline\cr&&1\end{array}\end{pmatrix}$

where all empty blocks are filled with zero-matrices of the appropriate size.
As $J$ and the matrix $S_{2N}$ in (2.2) commute, the map

(2.13)
$\displaystyle\theta:{\mathfrak{so}}(2N)\rightarrow{\mathfrak{so}}(2N),\qquad\theta(M)=-JM^{t}J^{-1}$

is well-defined. More explicitly, any $M\in{\mathfrak{so}}(2N)$ can be written
as a block matrix

(2.14) $\displaystyle M=\begin{pmatrix}a&-s^{\tau}&0\\\ t&P&s\\\
0&-t^{\tau}&-a\end{pmatrix}$

where $a\in{\mathbb{C}}$, $s,t\in{\mathbb{C}}^{2N-2}$ are column vectors and
$P\in{\mathfrak{so}}(2N-2)$. A direct calculation shows that

(2.15) $\displaystyle\theta(M)=\begin{pmatrix}-a&-t^{\tau}&0\\\ s&P&t\\\
0&-s^{\tau}&a\end{pmatrix}.$

We consider the fixed Lie subalgebra

$\displaystyle{\mathfrak{so}}(2N)^{\theta}=\\{M\in{\mathfrak{so}}(2N)\,|\,\theta(M)=M\\}={\mathfrak{gl}}_{S}(2N)\cap{\mathfrak{gl}}_{J}(2N).$

Comparing (2.14) and (2.15), we see that

(2.16)
$\displaystyle{\mathfrak{so}}(2N)^{\theta}=\left\\{\begin{pmatrix}0&-s^{\tau}&0\\\
s&P&s\\\
0&-s^{\tau}&0\end{pmatrix}\,\Bigg{|}\,P\in{\mathfrak{so}}(2N-2)\right\\}$

where $s\in{\mathbb{C}}^{2N-2}$ is a column vector.

######  Lemma 2.2.

The map $\eta:{\mathfrak{so}}(2N-1)\rightarrow{\mathfrak{so}}(2N)$ defined by

$\displaystyle\eta\left(\begin{pmatrix}A&b&B\\\ d&0&-b^{\tau}\\\
C&-d^{\tau}&-A^{\tau}\end{pmatrix}\right)=\begin{pmatrix}0&\frac{1}{2}d&-b^{\tau}&0\\\
b&A&2B&b\\\
-\frac{1}{2}d^{\tau}&\frac{1}{2}C&-A^{\tau}&-\frac{1}{2}d^{\tau}\\\
0&\frac{1}{2}d&-b^{\tau}&0\end{pmatrix}$

is an injective homomorphism of Lie algebras with image
${\mathfrak{so}}(2N)^{\theta}$.

###### Proof.

It follows from (2.7) and (2.16) that $\eta$ is an injective linear map with
image ${\mathfrak{so}}(2N)^{\theta}$. A direct calculation shows that the map
$\eta$ respects the Lie brackets. ∎

The above Lemma shows that $({\mathfrak{so}}(2N),{\mathfrak{so}}(2N-1))$ is a
symmetric pair. We can identify the Chevalley generators of
${\mathfrak{so}}(2N-1)$ inside ${\mathfrak{so}}(2N)$ via the embedding $\eta$.
Observe first that

(2.17) $\displaystyle\eta(e_{\gamma_{i}})$ $\displaystyle{=}e_{\alpha_{i+1}},$
$\displaystyle\eta(f_{\gamma_{i}})$ $\displaystyle{=}f_{\alpha_{i+1}},$
$\displaystyle\eta(h_{\gamma_{i}})$ $\displaystyle{=}h_{\alpha_{i+1}}$

for $i=1,\dots,N-2$. Again, considering the Chevalley generators
$e_{\alpha_{N-1}},f_{\alpha_{N-1}}$ of ${\mathfrak{so}}(2N-2)$ inside
${\mathfrak{so}}(2N-1)$ we have

(2.18) $\displaystyle\eta(e_{\alpha_{N-1}})$ $\displaystyle=2e_{\alpha_{N}},$
$\displaystyle\eta(f_{\alpha_{N-1}})$
$\displaystyle=\frac{1}{2}f_{\alpha_{N}}.$

Next, recall the elements $b_{i}\in{\mathfrak{so}}(2N-1)$ defined by (2.8) for
$i=1,\dots,N-1$. We have

(2.19) $\displaystyle f_{\alpha_{1}}+\theta(f_{\alpha_{1}})=\eta(b_{1})$

and hence

$\displaystyle\eta(e_{\gamma_{N-1}})$ $\displaystyle=\eta(b_{N-1})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:2Nin2N1}}}{{=}}\eta\big{(}[f_{\gamma_{N-2}},[f_{\gamma_{N-3}},\dots,[f_{\gamma_{1}},b_{1}]\dots]]\big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:etaxi}}}{{=}}[f_{\alpha_{N-1}},[f_{\alpha_{N-2}},\dots,[f_{\alpha_{2}},\eta(b_{1})]\dots]]$
(2.20)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:f1+tf1}}}{{=}}[f_{\alpha_{N-1}},[f_{\alpha_{N-2}},\dots,[f_{\alpha_{2}},f_{\alpha_{1}}+\theta(f_{\alpha_{1}})]\dots]]$
$\displaystyle\eta(f_{\gamma_{N-1}})$ $\displaystyle=2\eta(d_{N-1})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:bc1}}}{{=}}-2\eta\big{(}[f_{\alpha_{N-1}},b_{N-2}]\big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:2Nin2N1}}}{{=}}-2\eta\big{(}[f_{\alpha_{N-1}},[f_{\gamma_{N-3}},\dots,[f_{\gamma_{1}},b_{1}]\dots]]\big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:etaxi},\eqref{eq:etaxN-1}}}{{=}}-[f_{\alpha_{N}},[f_{\alpha_{N-2}},\dots,[f_{\alpha_{2}},\eta(b_{1})]\dots]]$
(2.21)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:f1+tf1}}}{{=}}-[f_{\alpha_{N}},[f_{\alpha_{N-2}},\dots,[f_{\alpha_{2}},f_{\alpha_{1}}+\theta(f_{\alpha_{1}})]\dots]].$

Moreover, we have

(2.22)
$\displaystyle\eta(h_{\gamma_{N-1}})=\eta\big{(}2(E_{N-1,N-1}-E_{N+1,N+1})\big{)}=2h_{N}=h_{\alpha_{N}}-h_{\alpha_{N-1}}.$

We will use Equations (2.20) and (2.21) in Section 4.2 to identify
$q$-analogues of $e_{\gamma_{N-1}}$ and $f_{\gamma_{N-1}}$ inside the QSP-
subalgebra ${\mathcal{B}}_{\mathbf{c}}$.

###  2.4. The Satake diagram of the symmetric pair
$({\mathfrak{so}}(2N),{\mathfrak{so}}(2N-1))$

The involution $\theta$ of ${\mathfrak{so}}(2N)$ defined by (2.13) satisfies
properties (1), (2), (3) in Section 2.1 with $X=\\{2,3,\dots,N\\}$. Indeed, by
(2.15) we have

$\displaystyle\theta(e_{1})$
$\displaystyle=\theta(E_{1,2}-E_{N+2,N+1})=-E_{2N-1,1}+E_{2N,2}$ (2.23)
$\displaystyle=(-1)^{N+1}[f_{2},[f_{3},{\dots}[f_{N-2},[f_{N},[f_{N-1},[f_{N-2},{\dots}[f_{2},f_{1}]{\dots}]]]]{\dots}]],$
$\displaystyle\theta(f_{1})$
$\displaystyle=\theta(E_{2,1}-E_{2N,2N-1})=-E_{1,2N-1}+E_{2,2N}$ (2.24)
$\displaystyle=(-1)^{N+1}[e_{2},[e_{3},{\dots}[e_{N-2},[e_{N},[e_{N-1},[e_{N-2},{\dots}[e_{2},e_{1}]{\dots}]]]]{\dots}]]$

where to shorten notation we wrote $e_{i},f_{i}$ instead of $e_{\alpha_{i}}$
and $f_{\alpha_{i}}$, respectively. Formulas (2.23) and (2.24) show that
condition (3) in Section 2.1 holds with $X=\\{2,3,\dots,N\\}$, and hence the
involution $\theta$ defined by (2.13) is maximally split. On the level of
roots we obtain

$\displaystyle\Theta(-\alpha_{1})-\alpha_{1}=2(\alpha_{2}+\dots+\alpha_{N-2})+\alpha_{N-1}+\alpha_{N}\in
Q_{X}^{+}$

as expected, and the involution $\theta$ corresponds to the Satake diagram of
type $DII$ depicted in Figure 1(A).

##  3\. Very non-standard quantum ${\mathfrak{so}}(2N-1)$

The theory of quantum symmetric pairs provides a coideal subalgebra
${\mathcal{B}}_{\mathbf{c}}$ of $U_{q}({\mathfrak{so}}(2N))$ corresponding to
the Satake diagram in Figure 1(A). The algebra ${\mathcal{B}}_{\mathbf{c}}$ is
a quantum group analogue of $U({\mathfrak{so}}(2N-1))$. We recall the
definition of ${\mathcal{B}}_{\mathbf{c}}$ in the conventions of [Kol14] and
its presentation in terms of generators and relations as given in [BK15].

###  3.1. Preliminaries on quantum groups

For the remainder of this paper we fix ${\mathfrak{g}}={\mathfrak{so}}(2N)$.
Let $k$ be a field and fix an element $q\in k\setminus\\{0\\}$ which is not a
root of unity. Let $U={U_{q}(\mathfrak{g})}$ be the corresponding Drinfeld-
Jimbo quantum enveloping algebra defined over $k$ in terms of generators
$E_{i},F_{i},K_{i}^{\pm 1}$ for $i=1,\dots,N$ and relations given in [Jan96,
4.3]. For simplicity we write $I=\\{1,2,\dots,N\\}$. Let $U^{+},U^{-}$ and
$U^{0}$ denote the subalgebras of $U$ generated by $\\{E_{i}\,|\,i\in I\\}$,
$\\{F_{i}\,|\,i\in I\\}$ and $\\{K_{i}^{\pm 1}\,|\,i\in I\\}$, respectively.
By [Jan96, 4.21] the algebra $U$ has a triangular decomposition in the sense
that the multiplication map

(3.1) $\displaystyle U^{-}\otimes U^{0}\otimes U^{+}\rightarrow U$

is a linear isomorphism. Recall the set of simple roots
$\Pi=\\{\alpha_{i}\,|\,i\in I\\}$ given by (2.4), the root lattice
$Q={\mathbb{Z}}\Pi$ and the positive cone $Q^{+}={\mathbb{N}}_{0}\Pi$. For any
$\mu\in Q$ let $U_{\mu}=\\{u\in
U\,|\,K_{i}uK_{i}^{-1}=q^{\mu(h_{\alpha_{i}})}u\mbox{ for all $i\in I$}\\}$ be
the corresponding weight space. We also write $U^{\pm}_{\mu}=U_{\mu}\cap
U^{\pm}$. Set $\Pi^{\vee}=\\{h_{\alpha_{i}}\,|\,i\in I\\}$ and let
$Q^{\vee}={\mathbb{Z}}\Pi^{\vee}$ be the coroot lattice. We can view $U^{0}$
as the group algebra of $Q^{\vee}$ and we write $K_{h}=\prod_{i\in
I}K_{i}^{n_{i}}$ if $h=\sum_{i\in I}n_{i}h_{\alpha_{i}}$. With this notation
we have

(3.2) $\displaystyle K_{h}E_{i}K_{h}^{-1}=q^{\alpha_{i}(h)}E_{i},\quad
K_{h}F_{i}K_{h}^{-1}=q^{-\alpha_{i}(h)}F_{i}\qquad\mbox{for all $h\in
Q^{\vee},i\in I$.}$

Recall that $U$ is a Hopf algebra with coproduct ${\varDelta}$, counit
$\varepsilon$ and antipode $S$ uniquely determined by

$\displaystyle{\varDelta}(E_{i})=E_{i}\otimes 1+K_{i}\otimes
E_{i},\quad{\varDelta}(F_{i})=F_{i}\otimes K_{i}^{-1}+1\otimes
F_{i},\quad{\varDelta}(K_{i})=K_{i}\otimes K_{i}.$

Let $T_{i}:U\rightarrow U$ for $i\in I$ denote the Lusztig braid group
automorphisms on $U$ in the conventions of [Jan96, 8.14]. In particular, for
$i,j\in I$ with $a_{ij}=-1$ we have

(3.3) $\displaystyle
T_{i}(E_{j})=[E_{i},E_{j}]_{q^{-1}}=T_{j}^{-1}(E_{i}),\qquad
T_{i}(F_{j})=[F_{j},F_{i}]_{q}=T_{j}^{-1}(F_{i}).$

Let $W$ be the Weyl group of ${\mathfrak{g}}$ generated by the simple
reflections $s_{i}$ corresponding to the simple roots $\alpha_{i}$ for $i\in
I$. For any $w\in W$ with reduced expression $w=s_{i_{1}}\dots s_{i_{\ell}}$
let $T_{w}=T_{i_{1}}\dots T_{i_{\ell}}$ denote the corresponding Lusztig
automorphism of $U$.

Finally, let $\sigma:U\rightarrow U$ denote the involutive algebra
antiautomorphism defined by $\sigma(E_{i})=E_{i}$, $\sigma(F_{i})=F_{i}$ and
$\sigma(K_{i})=K_{i}^{-1}$, see [Lus94, 3.1.3].

###  3.2. A very non-standard deformation of $U({\mathfrak{so}}(2n-1))$

Let $X\subset I$ be the subset $X=\\{2,\dots,N\\}$ and let
${\mathcal{M}}_{X}\subset U$ be the subalgebra generated by all
$E_{j},F_{j},K_{j}^{\pm 1}$ for $j\in X$. By construction, the Hopf-subalgebra
${\mathcal{M}}_{X}$ is isomorphic to the Drinfeld-Jimbo quantum enveloping
algebra $U_{q}({\mathfrak{so}}(2N-2))$. Let $W_{X}\subset W$ be the parabolic
subgroup corresponding to the subset $X$ and let $w_{X}\in W_{X}$ be the
longest element. Let ${\mathcal{B}}_{\mathbf{c}}\subset U$ be the subalgebra
generated by ${\mathcal{M}}_{X}$ and the element

$\displaystyle B_{1}=F_{1}-c_{1}T_{w_{X}}(E_{1})K_{1}^{-1}.$

The algebra ${\mathcal{B}}_{\mathbf{c}}$ depends on a single parameter
$c_{1}\in k\setminus\\{0\\}$. We call ${\mathcal{B}}_{\mathbf{c}}$ the very-
nonstandard quantum deformation of $U({\mathfrak{so}}(2N-1))$. The pair
$(U,{\mathcal{B}}_{\mathbf{c}})$ is an example of a quantum symmetric pair as
constructed by G. Letzter in [Let99]. Here we follow the conventions of
[Kol14]. In particular, ${\mathcal{B}}_{\mathbf{c}}\subset U$ is a right
coideal subalgebra, that is ${\varDelta}({\mathcal{B}}_{\mathbf{c}})\subset
U\otimes{\mathcal{B}}_{\mathbf{c}}$. As in [Kol14] we also refer to the
algebra ${\mathcal{B}}_{\mathbf{c}}$ as a quantum symmetric pair (QSP) coideal
subalgebra of $U$.

######  Remark 3.1.

The algebra ${\mathcal{B}}_{\mathbf{c}}$ was first defined by Noumi and
Sugitani in [NS95, Section 3] as case ‘$BDI$ for $\ell=1$ and even $N$’. Their
construction is more in the spirit of the FRT approach to quantum groups and
relies on an explicit solution of the reflection equation which is a
$q$-analogue of the matrix $J$ in (2.12). Noumi and Sugitani denoted their
QSP-coideal subalgebra by $U^{\mathrm{tw}}_{q}({\mathfrak{k}})$. Letzter
showed in [Let99, Section 6] that ${\mathcal{B}}_{\mathbf{c}}$ as defined here
and $U^{\mathrm{tw}}_{q}({\mathfrak{k}})$ as defined in [NS95] coincide for a
suitable choice of the parameter $c_{1}$.

For $i=1,\dots,N-1$ define reduced words $\sigma_{i}$ in the Weyl group $W$ by

(3.4) $\displaystyle\sigma_{i}=s_{i}\dots s_{N-2}s_{N-1}s_{N}s_{N-2}\dots
s_{i}.$

Then $w_{X}=\sigma_{2}\dots\sigma_{N-1}$ is a reduced expression for the
longest element $w_{X}\in W_{X}$. Using Equation (3.3) we obtain

$\displaystyle T_{w_{X}}$
$\displaystyle(E_{1})=T_{\sigma_{2}}(E_{1})=T_{2}\dots
T_{N-2}T_{N-1}T_{N}T_{N-2}\dots T_{2}(E_{1})$ (3.5)
$\displaystyle=\big{[}[E_{2},\hbox
to10.00002pt{.\hss.\hss.}[E_{N-2},E_{N}]_{q^{-1}}\hbox
to10.00002pt{.\hss.\hss.}]_{q^{-1}},[E_{N-1},[E_{N-2},\hbox
to10.00002pt{.\hss.\hss.}[E_{2},E_{1}]_{q^{-1}}\hbox
to10.00002pt{.\hss.\hss.}]_{q^{-1}}]_{q^{-1}}\big{]}_{q^{-1}}.$

This expression allows us to show that the algebras
${\mathcal{B}}_{\mathbf{c}}$ for different parameters $\mathbf{c}=(c_{1})$ are
isomorphic via Hopf algebra automorphisms of $U$.

######  Proposition 3.2.

Let $c_{1},d_{1}\in k\setminus\\{0\\}$ and ${\mathcal{B}}_{\mathbf{c}}$,
${\mathcal{B}}_{\mathbf{d}}$ the corresponding very nonstandard quantum
deformations of $U({\mathfrak{so}}(2N-1))$. Then there exists a Hopf algebra
automorphism $\phi:U\rightarrow U$ such that
$\phi({\mathcal{B}}_{\mathbf{c}})={\mathcal{B}}_{\mathbf{d}}$.

###### Proof.

Define the automorphism $\phi:U\rightarrow U$ by $\phi(K)=K$ for all $K\in
U^{0}$ and

$\displaystyle\phi(E_{i})=\begin{cases}E_{i}&\mbox{if $i\neq N$,}\\\
\frac{d_{1}}{c_{1}}E_{N}&\mbox{if
$i=N$,}\end{cases}\qquad\phi(F_{i})=\begin{cases}F_{i}&\mbox{if $i\neq N$,}\\\
\frac{c_{1}}{d_{1}}F_{N}&\mbox{if $i=N$.}\end{cases}$

By Equation (3.5) we have

$\displaystyle\phi(F_{1}-c_{1}T_{w_{X}}(E_{1})K_{1}^{-1})=F_{1}-d_{1}T_{w_{X}}(E_{1})K_{1}^{-1}.$

Moreover, $\phi$ restricts to a Hopf algebra automorphism of
${\mathcal{M}}_{X}$. This shows that
$\phi({\mathcal{B}}_{\mathbf{c}})={\mathcal{B}}_{\mathbf{d}}$. ∎

######  Remark 3.3.

In [Let99] and [Kol14] the quantised enveloping algebra
${U_{q}(\mathfrak{g})}$ and its coideal subalgebra
${\mathcal{B}}_{\mathbf{c}}$ are defined over the field $k={\mathbb{K}}(q)$ of
rational functions in an indeterminate $q$ and ${\mathbb{K}}$ is assumed to be
of characteristic zero. However, all relevant results in [Kol14] not involving
specialisation also hold in the more general setting that $q\in k$ is not a
root of unity in an arbitrary field $k$. Only once we use specialisation in
Section 6 will we have to restrict to fields $k$ of characteristic $0$ and
$q\in k$ which is transcendental over $\mathbb{Q}$. This is similar to the
approach in [Jan96, Section 5] which we are trying to emulate.

###  3.3. Generators and relations for ${\mathcal{B}}_{\mathbf{c}}$

Recall the Lusztig-Kashiwara skew derivation ${}_{i}r,r_{i}:U^{+}\rightarrow
U^{+}$ defined by

(3.6) $\displaystyle
uF_{i}-F_{i}u=\frac{r_{i}(u)K_{i}-K_{i}^{-1}{}_{i}r(u)}{q-q^{-1}}\qquad\mbox{for
all $u\in U^{+}$,}$

see [Jan96, Lemma 6.17], and define

(3.7) $\displaystyle{\mathcal{Z}}_{1}=r_{1}(T_{w_{X}}(E_{1})).$

For $N=3$ one gets ${\mathcal{Z}}_{1}=(1-q^{-2})^{2}E_{2}E_{3}$. For $N\geq 4$
one calculates as in [BK15, Section 2.2] to obtain

(3.8) $\displaystyle{\mathcal{Z}}_{1}=(1-q^{-2})[T_{3}^{-1}\dots
T_{N-2}^{-1}T_{N}^{-1}T_{N-1}^{-1}T_{N-2}^{-1}\dots
T_{3}^{-1}(E_{2}),E_{2}]_{q^{-2}}.$

For $N\geq 4$ Equation (3.7) implies that
$T_{i}({\mathcal{Z}}_{1})={\mathcal{Z}}_{1}$ if $i\geq 3$. Hence Equation
(3.8) can be rewritten as

(3.9) $\displaystyle{\mathcal{Z}}_{1}=(1-q^{-2})[E_{2},T_{3}\dots
T_{N-2}T_{N}T_{N-1}T_{N-2}\dots T_{3}(E_{2})]_{q^{-2}}.$

By [Jan96, 8.26] one has

(3.10) $\displaystyle T_{i}(U^{+})\cap U^{+}=\\{x\in U^{+}\,|\,r_{i}(x)=0\\},$
(3.11) $\displaystyle T_{i}^{-1}(U^{+})\cap U^{+}=\\{x\in
U^{+}\,|\,{}_{i}r(x)=0\\}.$

By application of (3.10) and (3.11) to the expressions (3.8) and (3.9),
respectively, we hence obtain

$\displaystyle r_{2}({\mathcal{Z}}_{1})$
$\displaystyle=(1-q^{-2})^{2}T_{3}^{-1}\dots
T_{N-2}^{-1}T_{N}^{-1}T_{N-1}^{-1}T_{N-2}^{-1}\dots T_{3}^{-1}(E_{2}),$
$\displaystyle{}_{2}r({\mathcal{Z}}_{1})$
$\displaystyle=(1-q^{-2})^{2}T_{3}\dots T_{N-2}T_{N}T_{N-1}T_{N-2}\dots
T_{3}(E_{2})$

in the case $N\geq 4$. With these preparations we are ready to describe the
defining relations of the algebra ${\mathcal{B}}_{\mathbf{c}}$ explicitly.

######  Theorem 3.4 ([Let03, Theorem 7.1], see also [BK15, Theorem 3.9]).

The algebra ${\mathcal{B}}_{\mathbf{c}}$ is generated over ${\mathcal{M}}_{X}$
by the element $B_{1}$ subject only to the relations

$\displaystyle K_{i}B_{1}K_{i}^{-1}$
$\displaystyle=q^{-(\alpha_{i},\alpha_{1})}B_{1}$ for $i=2,\dots,N$,
$\displaystyle E_{i}B_{1}$ $\displaystyle=B_{1}E_{i}$ for $i=2,\dots,N$,
$\displaystyle F_{j}B_{1}$ $\displaystyle=B_{1}F_{j}$ for $j=3,\dots,N$,
(3.12) $\displaystyle B_{1}^{2}F_{2}-(q+q^{-1})B_{1}F_{2}B_{1}+F_{2}B_{1}^{2}$
$\displaystyle=$ $\displaystyle-qc_{1}\Big{(}F_{2}{\mathcal{Z}}_{1}+$
$\displaystyle\frac{qr_{2}({\mathcal{Z}}_{1})K_{2}+q^{-1}{}_{2}r({\mathcal{Z}}_{1})K_{2}^{-1}}{(q-q^{-1})^{2}}\Big{)},$
(3.13) $\displaystyle F_{2}^{2}B_{1}-(q+q^{-1})F_{2}B_{1}F_{2}+B_{1}F_{2}^{2}$
$\displaystyle=0.$

The algebra ${\mathcal{B}}_{\mathbf{c}}$ is invariant under the Lusztig
automorphisms $T_{i}$ for $i=2,\dots,N$. More explicitly, a direct calculation
shows that

(3.14) $\displaystyle T_{i}(B_{1})$ $\displaystyle=\begin{cases}B_{1}&\mbox{if
$i=3,\dots,N$,}\\\ [B_{1},F_{2}]_{q}&\mbox{if $i=2$,}\end{cases}$ (3.15)
$\displaystyle T_{i}^{-1}(B_{1})$ $\displaystyle=\begin{cases}B_{1}&\mbox{if
$i=3,\dots,N$,}\\\ [F_{2},B_{1}]_{q}&\mbox{if $i=2$,}\end{cases}$

see also [BW18, Theorem 4.2].

Recall the algebra antiautomorphism $\sigma$ from Section 3.1. There exists an
involutive algebra antiautomorphism
$\sigma^{B}:{\mathcal{B}}_{\mathbf{c}}\rightarrow{\mathcal{B}}_{\mathbf{c}}$
defined by

(3.16)
$\displaystyle\sigma^{B}|_{{\mathcal{M}}_{X}}=\sigma|_{{\mathcal{M}}_{X}}\quad\mbox{and}\quad\sigma^{B}(B_{1})=B_{1}.$

To show that $\sigma^{B}$ is well defined, it suffices to check that the right
hand side of Equation (3.12) is preserved under $\sigma$. This follows from
the relation $\sigma({\mathcal{Z}}_{1})={\mathcal{Z}}_{1}$, which was proved
in [BK15, Proposition 2.3], and the relation
$\sigma(r_{2}({\mathcal{Z}}_{1}))={}_{2}r({\mathcal{Z}}_{1})$ which is a
direct consequence of the definition of $r_{i}$ and ${}_{i}r$ via Equation
(3.6). Note also that
$K_{2}\,{}_{2}r({\mathcal{Z}}_{1})={}_{2}r({\mathcal{Z}}_{1})K_{2}$ as
${}_{2}r({\mathcal{Z}}_{1})\in({\mathcal{M}}^{+}_{X})_{w_{X}(\alpha_{1})-\alpha_{1}-\alpha_{2}}$.

###  3.4. The standard filtration on ${\mathcal{B}}_{\mathbf{c}}$

Let ${\mathcal{A}}$ be the subalgebra of $U$ generated by ${\mathcal{M}}_{X}$
and the element $F_{1}$. Let ${\mathcal{M}}_{X}^{+},{\mathcal{M}}_{X}^{-}$ and
$U^{0}_{X}$ be the subalgebras of ${\mathcal{M}}_{X}$ generated by
$\\{E_{j}\,|\,j\in X\\},\\{F_{j}\,|\,j\in X\\}$ and $\\{K_{j}^{\pm 1}\,|\,j\in
X\\}$, respectively. By the triangular decomposition of $U$, the
multiplication map

(3.17) $\displaystyle U^{-}\otimes
U^{0}_{X}{\mathcal{M}}_{X}^{+}\rightarrow{\mathcal{A}}$

is a linear isomorphism. The algebra ${\mathcal{A}}$ is
${\mathbb{N}}_{0}$-graded by the degree function $\deg$ given by
$\deg(F_{1})=1$ and $\deg(m)=0$ for all $m\in{\mathcal{M}}_{X}$. The algebra
${\mathcal{B}}_{\mathbf{c}}$, on the other hand, has an
${\mathbb{N}}_{0}$-filtration ${\mathcal{F}}_{\ast}$ defined by the degree
function on the generators given by

$\displaystyle\deg(B_{1})$ $\displaystyle=1,$
$\displaystyle\deg(m)=0\quad\mbox{for all $m\in{\mathcal{M}}_{X}$.}$

Let ${\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})$ denote the associated graded
algebra. The defining relations of ${\mathcal{B}}_{\mathbf{c}}$ in Section 3.3
imply that there is a surjective homomorphism
$\varphi:{\mathcal{A}}\rightarrow{\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})$ of
graded algebras such that

$\displaystyle\varphi(F_{1})$
$\displaystyle=B_{1}\in{\mathcal{F}}_{1}({\mathcal{B}}_{\mathbf{c}})/{\mathcal{F}}_{0}({\mathcal{B}}_{\mathbf{c}}),$
$\displaystyle\varphi(m)=m\in{\mathcal{F}}_{0}({\mathcal{B}}_{\mathbf{c}})\quad\mbox{for
$m\in{\mathcal{M}}_{X}$.}$

It follows from [Kol14, Proposition 6.2] that the map
$\varphi:{\mathcal{A}}\rightarrow{\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})$ is
an isomorphism of graded algebras.

By Equations (3.14), (3.15) the algebra isomorphisms
$T_{i}:{\mathcal{B}}_{\mathbf{c}}\rightarrow{\mathcal{B}}_{\mathbf{c}}$ for
$i\in X$ restrict to linear isomorphisms
$T_{i}:{\mathcal{F}}_{n}({\mathcal{B}}_{\mathbf{c}})\rightarrow{\mathcal{F}}_{n}({\mathcal{B}}_{\mathbf{c}})$
for any $n\in{\mathbb{N}}_{0}$. Hence the maps
$T_{i}:{\mathcal{B}}_{\mathbf{c}}\rightarrow{\mathcal{B}}_{\mathbf{c}}$ induce
algebra isomorphisms
${\mathrm{gr}}(T_{i}):{\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})\rightarrow{\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})$.
The second formula in (3.3) and Equations (3.14), (3.15) now imply the
following result.

######  Lemma 3.5.

For any $i\in X$ the algebra isomorphism
$\varphi:{\mathcal{A}}\rightarrow{\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})$ is
compatible with the Lusztig automorphism $T_{i}$ in the sense that
$\varphi\circ T_{i}={\mathrm{gr}}(T_{i})\circ\varphi$.

##  4\. A Poincaré-Birkhoff-Witt Theorem for ${\mathcal{B}}_{\mathbf{c}}$

We now use our understanding of the classical symmetric pair
$({\mathfrak{so}}(2N),{\mathfrak{so}}(2N-1))$ from Section 2 to define root
vectors for the algebra ${\mathcal{B}}_{\mathbf{c}}$. We will see in Theorem
4.6 that the ordered monomials in the root vectors provide a Poincaré-
Birkhoff-Witt basis for ${\mathcal{B}}_{\mathbf{c}}$. This statement can be
reformulated as a triangular decomposition for ${\mathcal{B}}_{\mathbf{c}}$ in
Corollary 4.7. We end the section with some preliminary results on
$q$-commutators of root vectors.

###  4.1. The Poincaré-Birkhoff-Witt Theorem for $U_{q}({\mathfrak{so}}(2N))$

Recall the reduced words $\sigma_{i}$ for $i=1,\dots,N-1$ defined by (3.4).
The word

(4.1) $\displaystyle w_{0}=\sigma_{1}\sigma_{2}\dots\sigma_{N-1}$

is a reduced expression of the longest element in $W$. Write the reduced
expression (4.1) as $w_{0}=s_{i_{1}}\dots s_{i_{N(N-1)}}$. We obtain the
$N(N-1)$ positive roots of ${\mathfrak{so}}(2N)$ as

$\displaystyle\beta_{j}=s_{i_{1}}\dots
s_{i_{j-1}}(\alpha_{i_{j}})\qquad\mbox{for $j=1,\dots,N(N-1)$.}$

Let $\tau:I\rightarrow I$ be the nontrivial diagram automorphism, i.e.
$\tau(i)=i$ for $i=1,\dots,N-2$ and $\tau(N-1)=N$, $\tau(N)=N-1$. As
$\sigma_{i}(\alpha_{j})=\alpha_{\tau(j)}$ for $i<j$, we see that $\alpha_{1}$
is contained with a nonzero coefficient only in the roots
$\beta_{1},\dots,\beta_{2(N-1)}$ but not in $\beta_{j}$ for $j>2(N-1)$. Define
root vectors $F_{\beta_{j}}\in U^{-}$ by

(4.2) $\displaystyle F_{\beta_{j}}=T_{i_{1}}\dots
T_{i_{j-1}}(F_{i_{j}})\qquad\mbox{for $j=1,\dots,N(N-1)$.}$

By the Poincaré-Birkhoff-Witt Theorem [Jan96, 8.24] the ordered monomials

(4.3) $\displaystyle F_{{\mathcal{J}}}=F_{\beta_{N(N-1)}}^{j_{N(N-1)}}\dots
F_{\beta_{2}}^{j_{2}}F_{\beta_{1}}^{j_{1}}\qquad\mbox{ for
${\mathcal{J}}=(j_{1},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{N(N-1)}$}$

form a $k$-vector space basis $U^{-}$. We want to rewrite the root vectors
$F_{\beta_{j}}$ in the case where $\beta_{j}$ contains $\alpha_{1}$ with a
non-zero coefficient.

######  Lemma 4.1.

For $j\in\\{1,\dots,2(N-1)\\}$ the root vector $F_{\beta_{j}}$ can be written
as

(4.4) $\displaystyle F_{\beta_{j}}=\begin{cases}F_{1}&\mbox{if $j=1$,}\\\
T_{j}^{-1}\dots T_{2}^{-1}(F_{1})&\mbox{if $2\leq j\leq N-1$,}\\\
T_{2N-j}^{-1}\dots T_{N}^{-1}T_{N-2}^{-1}\dots T_{2}^{-1}(F_{1})&\mbox{if
$N\leq j\leq 2(N-1)$.}\end{cases}$

###### Proof.

It follows from (3.3) that Equation (4.4) holds for $j=1,\dots,N$. For $j\geq
N+1$ we have

$\displaystyle F_{\beta_{j}}=T_{1}\dots T_{N}T_{N-2}\dots
T_{k+1}(F_{k})\qquad\mbox{where $k=2N-j-1$.}$

Using again (3.3) we hence obtain

$\displaystyle F_{\beta_{j}}=T_{1}\dots T_{N}T^{-1}_{k}\dots
T_{N-3}^{-1}(F_{N-2}).$

As $T_{1}\dots T_{N-2}T^{-1}_{k}\dots T^{-1}_{N-3}=T^{-1}_{k+1}\dots
T_{N-2}^{-1}T_{1}\dots T_{N-2}$ we obtain

$\displaystyle F_{\beta_{j}}=T^{-1}_{k+1}\dots T_{N-2}^{-1}T_{1}\dots
T_{N}(F_{N-2}).$

Now we use $T_{N-2}T_{N-1}T_{N}(F_{N-2})=T_{N-1}^{-1}T_{N}^{-1}(F_{N-2})$ and
(3.3) to obtain

$\displaystyle F_{\beta_{j}}=T^{-1}_{2N-j}\dots T^{-1}_{N-2}T^{-1}_{N}\dots
T^{-1}_{2}(F_{1})$

where we also substituted $k+1=2N-j$. This settles the remaining cases. ∎

###  4.2. Root vectors for ${\mathcal{B}}_{\mathbf{c}}$

Recall from Equation (3.14), (3.15) that the Lusztig automorphisms $T_{i}$ for
$i\in X$ leave ${\mathcal{B}}_{\mathbf{c}}$ invariant. In analogy to (4.4) we
define root vectors of ${\mathcal{B}}_{\mathbf{c}}$ for $j=1,\dots,N(N-1)$ by

(4.5) $\displaystyle B_{\beta_{j}}=\begin{cases}B_{1}&\mbox{if $j=1$,}\\\
T_{j}^{-1}\dots T_{2}^{-1}(B_{1})&\mbox{if $2\leq j\leq N-1$,}\\\
T_{2N-j}^{-1}\dots T_{N}^{-1}T_{N-2}^{-1}\dots T_{2}^{-1}(B_{1})&\mbox{if
$N\leq j\leq 2(N-1)$.}\\\ F_{\beta_{j}}&\mbox{if $j>2(N-1)$.}\end{cases}$

In view of Equation (3.15) we can write the root vectors for
${\mathcal{B}}_{\mathbf{c}}$ explicitly as iterated $q$-commutators.

######  Lemma 4.2.

For $j\in\\{1,\dots,2(N-1)\\}$ the root vectors $B_{\beta_{j}}$ can be written
as

(4.6) $\displaystyle B_{\beta_{j}}=\begin{cases}B_{1}&\mbox{if $j=1$,}\\\
[F_{j},[F_{j-1},\dots,[F_{3},[F_{2},B_{1}]_{q}]_{q}{\dots}]_{q}]_{q}&\mbox{if
$2\leq j\leq N-1$,}\\\
[F_{N},[F_{N-2},\dots,[F_{3},[F_{2},B_{1}]_{q}]_{q}{\dots}]_{q}]_{q}&\mbox{if
$j=N$,}\\\
[F_{N},[F_{N-1},\dots,[F_{3},[F_{2},B_{1}]_{q}]_{q}{\dots}]_{q}]_{q}&\mbox{if
$j=N+1$,}\\\
[T_{N}{\dots}T_{2N-j+1}(F_{2N-j}),T_{N-2}^{-1}{\dots}T_{2}^{-1}(B_{1})]_{q}&\mbox{if
$N{+}2{\leq}j{\leq}2(N{-}1)$.}\end{cases}$

###### Proof.

For $2\leq j\leq N+1$ Equation (4.6) follows from Equations (3.3), (3.15). In
the case $N+2\leq j\leq 2(N-1)$ we again write $k=2N-j-1$ and get

$\displaystyle B_{\beta_{j}}$ $\displaystyle=T_{k+1}^{-1}\dots
T_{N-2}^{-1}T_{N}^{-1}\dots T_{2}^{-1}(B_{1})$
$\displaystyle=T_{k+1}^{-1}\dots
T_{N-2}^{-1}\big{(}[T_{N}^{-1}T_{N-1}^{-1}(F_{N-2}),T^{-1}_{N-3}\dots
T_{2}^{-1}(B_{1})]_{q}\big{)}$ $\displaystyle=T_{k+1}^{-1}\dots
T_{N-3}^{-1}\big{(}[T_{N}T_{N-1}(F_{N-2}),T^{-1}_{N-2}\dots
T_{2}^{-1}(B_{1})]_{q}\big{)}$ $\displaystyle=[T_{N}\dots
T_{k+2}(F_{k+1}),T^{-1}_{N-2}\dots T_{2}^{-1}(B_{1})]_{q}$

where we made repeated use of Equation (3.3). ∎

######  Remark 4.3.

Lemma 4.2 allows us to informally identify the limit of the root vectors
$B_{\beta_{j}}$ for $q\rightarrow 1$. Recall the notation from Sections 2.2
and 2.3. By (2.19) the generator $B_{1}\in{\mathcal{B}}_{\mathbf{c}}$ is a
$q$-analogue of the root vector $b_{1}\in{\mathfrak{so}}(2N-1)$ corresponding
to the root $\varepsilon_{1}$. Similarly, the generators
$F_{i}\in{\mathcal{B}}_{\mathbf{c}}$ for $i\in X$ are $q$-analogues of the
Chevalley generators $f_{\alpha_{i}}\in{\mathfrak{so}}(2N)$ and hence of
$f_{\gamma_{i-1}}\in{\mathfrak{so}}(2N-1)$ if $i\leq N-1$. Hence the first
equation in (2.9) shows that $B_{\beta_{j}}$ for $1\leq j\leq N-1$ are
$q$-analogues of the elements $b_{j}\in{\mathfrak{so}}(2N-1)$. In particular,

$\displaystyle B_{\beta_{N-1}}=T^{-1}_{N-1}T^{-1}_{N-2}\dots
T^{-1}_{2}(B_{1})$

is a $q$-analogue of the Chevalley generator
$e_{\gamma_{N-1}}\in{\mathfrak{so}}(2N-1)$, see also Equation (2.20).
Similarly, by Equation (2.21),

$\displaystyle B_{\beta_{N}}=T^{-1}_{N}T^{-1}_{N-2}\dots T^{-1}_{2}(B_{1})$

is a $q$-analogue of $-f_{\gamma_{N-1}}\in{\mathfrak{so}}(2N-1)$. We will make
these argument precise in Section 6.2 using specialisation.

The interpretation of $B_{\beta_{N-1}}$ and $-B_{\beta_{N}}$ as $q$-analogues
of the Chevalley generators corresponding to the simple roots
$\pm\gamma_{N-1}$ is confirmed by the following Lemma which also identifies
$q$-analogues of the remaining simple root vectors. Let
$Q^{\vee}_{2N-1}={\mathbb{Z}}\\{h_{\gamma_{j}}\,|\,j=1,\dots,N-1\\}$ be the
coroot lattice of ${\mathfrak{so}}(2N-1)$. By Equations (2.17) and (2.22) the
embedding $\eta$ induces a group homomorphism $\eta:Q^{\vee}_{2N-1}\rightarrow
Q^{\vee}$.

######  Lemma 4.4.

For $i=2\dots,N-1$ and any $h\in Q^{\vee}_{2N-1}$ the following relations
hold:

(4.7) $\displaystyle K_{\eta(h)}E_{i}K_{\eta(h)}^{-1}$
$\displaystyle=q^{\gamma_{i-1}(h)}E_{i},$ $\displaystyle
K_{\eta(h)}B_{\beta_{N-1}}K_{\eta(h)}^{-1}$
$\displaystyle=q^{\gamma_{N-1}(h)}B_{\beta_{N-1}},$ (4.8) $\displaystyle
K_{\eta(h)}F_{i}K_{\eta(h)}^{-1}$ $\displaystyle=q^{-\gamma_{i-1}(h)}F_{i},$
$\displaystyle K_{\eta(h)}B_{\beta_{N}}K_{\eta(h)}^{-1}$
$\displaystyle=q^{-\gamma_{N-1}(h)}B_{\beta_{N}}.$

Moreover, we have

(4.9) $\displaystyle K_{\eta(h)}E_{N}K_{\eta(h)}^{-1}$
$\displaystyle=q^{(\gamma_{N-2}+2\gamma_{N-1})(h)}E_{i},$ (4.10)
$\displaystyle K_{\eta(h)}F_{N}K_{\eta(h)}^{-1}$
$\displaystyle=q^{-(\gamma_{N-2}+2\gamma_{N-1})(h)}F_{N}.$

###### Proof.

We check the relations (4.7), the relations (4.8) follow analogously. It
suffices to verify the relations (4.7) for $h=h_{\gamma_{j}}$ with
$j=1,\dots,N-1$. For $j=1,\dots,N-2$ we have

$\displaystyle
K_{\eta(h_{\gamma_{j}})}E_{i}K_{\eta(h_{\gamma_{j}})}^{-1}=q^{\alpha_{i}(\eta(h_{\gamma_{j}}))}E_{i}=q^{\alpha_{i}(h_{\alpha_{j+1}})}E_{i}=q^{\gamma_{i-1}(h_{\gamma_{j}})}E_{i}.$

Moreover, we have

$\displaystyle
K_{\eta(h_{\gamma_{N-1}})}E_{i}K_{\eta(h_{\gamma_{N-1}})}^{-1}=q^{\alpha_{i}(\eta(h_{\gamma_{N-1}}))}E_{i}=q^{\alpha_{i}(h_{\alpha_{N}}-h_{\alpha_{N-1}})}E_{i}=q^{\gamma_{i-1}(h_{\gamma_{N-1}})}E_{i}.$

This proves the first equation in (4.7). Next we use the definition (4.5) of
$B_{\beta_{N-1}}$ to calculate

$\displaystyle K_{\eta(h)}B_{\beta_{N-1}}K_{\eta(h)}^{-1}$
$\displaystyle=q^{-s_{N-1}s_{N-2}\dots
s_{2}(\alpha_{1})(\eta(h))}B_{\beta_{N-1}}$ (4.11)
$\displaystyle=q^{-\alpha_{1}(s_{2}\dots
s_{N-2}s_{N-1}(\eta(h)))}B_{\beta_{N-1}}.$

For $j=1,\dots,N-1$ we have

$\displaystyle-\alpha_{1}(s_{2}\dots s_{N-2}s_{N-1}(\eta(h_{\gamma_{j}})))$
$\displaystyle=\begin{cases}2&\mbox{if $j=N-1$}\\\ -1&\mbox{if $j=N-2$}\\\
0&\mbox{else}\end{cases}$ $\displaystyle=\gamma_{N-1}(h_{\gamma_{j}}).$

Inserting the above Equation into (4.11) we obtain the second equation in
(4.7).

To verify Equation (4.9) we consider $j=1,\dots,N-2$ and calculate

(4.12) $\displaystyle
K_{\eta(h_{\gamma_{j}})}E_{N}K_{\eta(h_{\gamma_{j}})}^{-1}=q^{\alpha_{N}(\eta(h_{\gamma_{j}}))}E_{N}=q^{\alpha_{N}(h_{\alpha_{j+1}})}E_{i}=q^{-\delta_{j,N-3}}E_{N},$
(4.13) $\displaystyle
K_{\eta(h_{\gamma_{N-1}})}E_{N}K_{\eta(h_{\gamma_{N-1}})}^{-1}=q^{\alpha_{N}(\eta(h_{\gamma_{N-1}}))}E_{N}=q^{\alpha_{N}(h_{\alpha_{N}}-h_{\alpha_{N-1}})}E_{N}=q^{2}E_{N}.$

On the other hand, as
$\gamma_{N-2}+2\gamma_{N-1}=\epsilon_{N-2}+\epsilon_{N-1}$ we obtain

$\displaystyle(\gamma_{N-2}+2\gamma_{N-1})(h_{\gamma_{j}})=\begin{cases}-1&\mbox{if
$j=N-3$,}\\\ 2&\mbox{if $j=N-1$,}\\\ 0&\mbox{else.}\end{cases}$

Comparison of the above with (4.12) and (4.13) proves Equation (4.9). Equation
(4.10) now follows from (3.2). ∎

###  4.3. The PBW-basis for ${\mathcal{B}}_{\mathbf{c}}$

Recall the definition (4.3) of the PBW-monomials $F_{\mathcal{J}}\in U^{-}$
for ${\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}$. Similarly, we use the root
vectors (4.5) to define PBW-monomials for ${\mathcal{B}}_{\mathbf{c}}$ by

(4.14) $\displaystyle B_{\mathcal{J}}=B_{\beta_{N(N-1)}}^{j_{N(N-1)}}\dots
B_{\beta_{2}}^{j_{2}}B_{\beta_{1}}^{j_{1}}\qquad\mbox{ for
${\mathcal{J}}=(j_{1},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{N(N-1)}$.}$

Recall the algebra isomorphism
$\varphi:{\mathcal{A}}\rightarrow{\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})$
from Section 3.4. Lemma 3.5 and Equations (4.4), (4.5) imply that
$\varphi(F_{\beta_{j}})=B_{\beta_{j}}$. This in turn implies that for any
${\mathcal{J}}=(j_{1},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{N(N-1)}$ we have

(4.15)
$\displaystyle\varphi({\mathcal{F}}_{\mathcal{J}})=B_{\mathcal{J}}\in{\mathcal{F}}_{j}({\mathcal{B}}_{\mathbf{c}})/{\mathcal{F}}_{j-1}({\mathcal{B}}_{\mathbf{c}})\qquad\mbox{with
$j=\sum_{\ell=1}^{2N-2}j_{\ell}$.}$

This observation is the main ingredient in the proof of the following
Proposition.

######  Proposition 4.5.

The set $\\{B_{\mathcal{J}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}\\}$
is a basis of $B_{\mathbf{c}}$ as a right (or left)
$U^{0}_{X}{\mathcal{M}}_{X}^{+}$-module.

###### Proof.

By the PBW Theorem for $U^{-}$ and the triangular decomposition (3.17), the
algebra ${\mathcal{A}}$ is a free right
$U^{0}_{X}{\mathcal{M}}_{X}^{+}$-module with basis
$\\{F_{\mathcal{J}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}\\}$ given by
(4.3). Lemma 3.5 and Equation (4.15) imply that the set

$\displaystyle\\{B_{\mathcal{J}}\,|\,{\mathcal{J}}=(j_{1},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{N(N-1)},\sum_{\ell=1}^{2N-2}j_{\ell}\leq
j\\}$

forms a basis of ${\mathcal{F}}_{j}({\mathcal{B}}_{\mathbf{c}})$ as a right
$U^{0}_{X}{\mathcal{M}}_{X}^{+}$-module. This proves the proposition
considering ${\mathcal{B}}_{\mathbf{c}}$ as a right
$U^{0}_{X}{\mathcal{M}}_{X}^{+}$-module. The corresponding statement about
${\mathcal{B}}_{\mathbf{c}}$ as a left $U^{0}_{X}{\mathcal{M}}_{X}^{+}$-module
is obtained by swapping tensor factors in the triangular decomposition (3.17).
∎

In analogy to (4.2) we define positive root vectors $E_{\beta_{j}}\in U^{+}$
by

$\displaystyle E_{\beta_{j}}=T_{i_{1}}\dots
T_{i_{j-1}}(E_{i_{j}})\qquad\mbox{for $j=1,\dots,N(N-1)$.}$

For
${\mathcal{I}}=(i_{2N-1},\dots,i_{N(N-1)})\in{\mathbb{N}}_{0}^{(N-1)(N-2)}$
define

$\displaystyle E_{{\mathcal{I}}}=E_{\beta_{N(N-1)}}^{i_{N(N-1)}}\dots
E_{\beta_{2N}}^{i_{2N}}E_{\beta_{2N-1}}^{i_{2N-1}}.$

By the PBW-Theorem for ${\mathcal{M}}^{+}_{X}$ the ordered monomials
$E_{\mathcal{I}}$ for ${\mathcal{I}}\in{\mathbb{N}}_{0}^{(N-1)(N-2)}$ form a
$k$-vector space basis of ${\mathcal{M}}_{X}^{+}$. Moreover, the set
$\\{K_{\beta}\,|\,\beta\in Q_{X}\\}$ is a $k$-vector space basis of
$U^{0}_{X}$. Combining these facts with Proposition 4.5 we obtain the desired
PBW-Theorem for ${\mathcal{B}}_{\mathbf{c}}$.

######  Theorem 4.6.

The set of ordered monomials

$\displaystyle\\{B_{\mathcal{J}}K_{\beta}E_{\mathcal{I}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)},\beta\in
Q_{X},{\mathcal{I}}\in{\mathbb{N}}_{0}^{(N-1)(N-2)}\\}$

is a $k$-vector space basis of ${\mathcal{B}}_{\mathbf{c}}$.

By Lemma 4.4 and Lemma 4.2 the root vectors
$B_{\beta_{1}},\dots,B_{\beta_{N-1}}$ correspond to positive roots of
${\mathfrak{so}}(2N-1)$ while the root vectors
$B_{\beta_{N}},\dots,B_{\beta_{N(N-1)}}$ correspond to negative roots. Hence
it is reasonable to define

$\displaystyle{\mathcal{B}}_{\mathbf{c}}^{-}$
$\displaystyle=\mathrm{span}_{k}\\{B_{\beta_{N(N-1)}}^{j_{N(N-1)}}\dots
B_{\beta_{N}}^{j_{N}}\\}\,|\,(j_{N},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{(N-1)^{2}}\\},$
$\displaystyle{\mathcal{B}}_{\mathbf{c}}^{+}$
$\displaystyle=\mathrm{span}_{k}\\{B_{\beta_{N-1}}^{j_{N-1}}\dots
B_{\beta_{1}}^{j_{1}}E_{{\mathcal{I}}}\\}\,|\,(j_{1},\dots,j_{N-1})\in{\mathbb{N}}_{0}^{N-1},{\mathcal{I}}\in{\mathbb{N}}_{0}^{(N-1)(N-2)}\\}$

and we obtain a triangular decomposition of ${\mathcal{B}}_{\mathbf{c}}$.

######  Corollary 4.7.

The multiplication map ${\mathcal{B}}_{\mathbf{c}}^{-}\otimes
U^{0}_{X}\otimes{\mathcal{B}}_{\mathbf{c}}^{+}\rightarrow{\mathcal{B}}_{\mathbf{c}}$
is a linear isomorphism.

######  Remark 4.8.

Note that ${\mathcal{B}}_{\mathbf{c}}^{+}$ and
${\mathcal{B}}_{\mathbf{c}}^{-}$ are not subalgebras of
${\mathcal{B}}_{\mathbf{c}}$. Indeed, $B_{\beta_{1}}=B_{1}$ and
$B_{\beta_{2}}=T_{2}^{-1}(B_{1})$ belong to ${\mathcal{B}}_{\mathbf{c}}^{+}$.
However, the deformed $q$-Serre relation (3.12) implies that
$[B_{\beta_{2}},B_{\beta_{1}}]_{q^{-1}}\notin{\mathcal{B}}_{\mathbf{c}}^{+}$
because of the term $F_{2}{\mathcal{Z}}_{1}$.

######  Remark 4.9.

The definitions of ${\mathcal{B}}_{\mathbf{c}}^{+}$ and
${\mathcal{B}}_{\mathbf{c}}^{-}$ are symmetric as the ordered monomials
$B_{\beta_{N(N-1)}}^{j_{N(N-1)}}\dots B_{\beta_{2N-1}}^{j_{2N-1}}$ form a PBW-
basis of ${\mathcal{M}}_{X}^{-}$.

###  4.4. Commutation of root vectors

For later use we investigate some $q$-commutators of root vectors. Let
${\mathcal{M}}_{X,+}^{+}$ be the augmentation ideal of
${\mathcal{M}}_{X}^{+}$, that is, the ideal generated by the set
$\\{E_{i}\,|\,i\in X\\}$.

######  Lemma 4.10.

For $1\leq\ell<m\leq N-1$ the relation
$[B_{\beta_{\ell}},B_{\beta_{m}}]_{q}\in{\mathcal{M}}_{X}{\mathcal{M}}_{X,+}^{+}$
holds.

###### Proof.

To avoid notational confusion we assume $\ell\geq 2$, but the argument for
$\ell=1$ is similar. By definition of the root vectors we have

$\displaystyle[B_{\beta_{\ell}},B_{\beta_{m}}]_{q}$
$\displaystyle=[T_{\ell}^{-1}\dots T_{2}^{-1}(B_{1}),T_{m}^{-1}\dots
T_{2}^{-1}(B_{1})]_{q}$ (4.16) $\displaystyle=T_{\ell}^{-1}\dots
T_{2}^{-1}T_{m}^{-1}\dots T_{3}^{-1}([B_{1},T_{2}^{-1}(B_{1})]_{q}).$

By (3.15) and (3.12) we have

$\displaystyle[B_{1},T_{2}^{-1}(B_{1})]_{q}\in
kF_{2}{\mathcal{Z}}_{1}+kK_{2}r_{2}({\mathcal{Z}}_{1})+kK_{2}^{-1}{}_{2}r({\mathcal{Z}}_{1})$

where
${\mathcal{Z}}_{1}\in({\mathcal{M}}_{X}^{+})_{w_{X}(\alpha_{1})-\alpha_{1}}$
and
$r_{2}({\mathcal{Z}}_{1}),{}_{2}r({\mathcal{Z}}_{1})\in({\mathcal{M}}_{X}^{+})_{w_{X}(\alpha_{1})-\alpha_{1}-\alpha_{2}}$.
Combining the above formula with Equation (4.16) shows that

$\displaystyle[B_{\beta_{\ell}},B_{\beta_{m}}]_{q}\in{\mathcal{M}}_{X}({\mathcal{M}}_{X})_{\mu}+{\mathcal{M}}_{X}({\mathcal{M}}_{X})_{\nu}$

where

$\displaystyle\mu$ $\displaystyle=s_{\ell}\dots s_{2}s_{m}\dots
s_{3}(w_{X}(\alpha_{1})-\alpha_{1})$ $\displaystyle=w_{X}(s_{\ell}\dots
s_{2}(\alpha_{1}))-s_{\ell}\dots s_{2}(\alpha_{1})$
$\displaystyle=2(\alpha_{\ell+1}+\dots+\alpha_{N-2})+\alpha_{N-1}+\alpha_{N}$
$\displaystyle\nu$ $\displaystyle=s_{\ell}\dots s_{2}s_{m}\dots
s_{3}(w_{X}(\alpha_{1})-\alpha_{1}-\alpha_{2})$ $\displaystyle=\mu-
s_{\ell}\dots s_{2}s_{m}\dots s_{3}(\alpha_{2})$
$\displaystyle=\mu-(\alpha_{\ell+1}+\dots+\alpha_{m})$

As $0\neq\mu,\nu\in Q_{X}^{+}$, the triangular decomposition of
${\mathcal{M}}_{X}$ implies that $({\mathcal{M}}_{X})_{\mu}$ and
$({\mathcal{M}}_{X})_{\nu}$ are contained in
${\mathcal{M}}_{X}{\mathcal{M}}_{X,+}^{+}$. This completes the proof of the
Lemma. ∎

As an immediate consequence of the above Lemma we obtain the following result.

######  Corollary 4.11.

For each $m\in\\{N-1,N\\}$ we have
$[B_{\beta_{N-2}},B_{\beta_{m}}]_{q}\in{\mathcal{M}}_{X}{\mathcal{M}}_{X,+}^{+}$.

###### Proof.

The case $m=N-1$ is a special case of the above Lemma. The case $m=N$ holds by
symmetry between $N$ and $N-1$. ∎

The next Lemma follows from Equation (4.6) and the defining relations of
${\mathcal{B}}_{\mathbf{c}}$ given in Section 3.3.

######  Lemma 4.12.

For $i\in X$ and $m=1,\dots,N-1$ we have

$\displaystyle[E_{i},B_{\beta_{m}}]=\begin{cases}B_{\beta_{m-1}}K_{m}^{-1}&\mbox{if
$i=m$,}\\\ 0&\mbox{if $i\neq m$.}\end{cases}$

Similarly, we need commutation relations between the root vectors
$B_{\beta_{m}}$ and the generators $F_{i}$ for $i\in X$.

######  Lemma 4.13.

The following relations hold in ${\mathcal{B}}_{\mathbf{c}}$:

(4.17) $\displaystyle[B_{\beta_{m}},F_{m}]_{q}$
$\displaystyle=0,\qquad\qquad\mbox{for $m=1,\dots,N$}$ (4.18)
$\displaystyle[F_{m+1},B_{\beta_{m}}]_{q}$
$\displaystyle=B_{\beta_{m+1}}\qquad\,\,\mbox{for $m=1,\dots,N-2$,}$

Moreover, for $m=1,\dots,N$, $i\in X\setminus\\{m,m+1\\}$ with ($m\neq N-2$ if
$i=N$) and ($m\neq N$ if $i=N-1$) the relation

(4.19) $\displaystyle[F_{i},B_{\beta_{m}}]$ $\displaystyle=0$

holds.

###### Proof.

By symmetry between $N-1$ and $N$ we may assume that $m\neq N$. To verify
Equation (4.17) we calculate

$\displaystyle[B_{\beta_{m}},F_{m}]_{q}$ $\displaystyle=T_{m}^{-1}\dots
T_{2}^{-1}([B_{1},T_{2}\dots T_{m}(F_{m})]_{q})$
$\displaystyle=-T_{m}^{-1}\dots T_{2}^{-1}([B_{1},K_{2}^{-1}\dots
K_{m}^{-1}T_{2}\dots T_{m-1}(E_{m})]_{q})$

which vanishes by the defining relations of ${\mathcal{B}}_{\mathbf{c}}$ in
Section 3.3. Relation (4.18) follows from Equation (4.6). For $i>m+1$ Equation
(4.19) follows again from the relations in Section 3.3. For $i<m$ we calculate

$\displaystyle[F_{i},B_{\beta_{m}}]$ $\displaystyle=T_{m}^{-1}\dots
T_{2}^{-1}([T_{2}\dots T_{i}T_{i+1}(F_{i}),B_{1}])$
$\displaystyle=T_{m}^{-1}\dots T_{2}^{-1}([F_{i+1},B_{1}])=0$

where we used the relation $T_{i}T_{i+1}(F_{i})=F_{i+1}$ which holds by
[Jan96, Remark 8.20]. ∎

By symmetry between $N-1$ and $N$, Formula (4.18) gives
$[F_{N},B_{\beta_{N-2}}]_{q}=B_{\beta_{N}}$. More generally, induction over
$m$ gives the following Lemma. Recall the $q$-numbers
$[m]_{q}=(q^{m}-q^{-m})/(q-q^{-1})$ for $m\in{\mathbb{N}}_{0}$.

######  Lemma 4.14.

For all $m\in{\mathbb{N}}_{0}$ the relation

$\displaystyle
B_{\beta_{N-2}}F_{N}^{m}=q^{-m}F_{N}^{m}B_{\beta_{N-2}}-q^{-1}[m]_{q}F_{N}^{m-1}B_{\beta_{N}}$

holds in ${\mathcal{B}}_{\mathbf{c}}$.

##  5\. Finite-dimensional irreducible representations of
${\mathcal{B}}_{\mathbf{c}}$

The identification of root vectors for ${\mathcal{B}}_{\mathbf{c}}$ and the
triangular decomposition in Corollary 4.7 allow us to follow a Verma module
approach towards the classification of finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-modules. To this end we show in Section 5.4 that
$\\{B_{\beta_{N-1}},B_{\beta_{N}},(K_{N}K_{N-1}^{-1})^{\pm 1}\\}$ act as a
copy of $U_{q}({\mathfrak{sl}}(2))$ on suitable subspaces of representations.
We consider embeddings of Verma modules in Section 5.6 and complete the
classification of finite dimensional, simple ${\mathcal{B}}_{\mathbf{c}}$
modules in Section 5.8 using a filtered-graded argument. First, however, we
briefly discuss signs for the action of $U^{0}_{X}$ which can occur in
complete analogy to the quantum group case in [Jan96, 5.1].

###  5.1. Representations of type $\mathbf{1}$

Let $V$ be a finite-dimensional representation of the $k$-algebra
${\mathcal{B}}_{\mathbf{c}}$. By restriction, $V$ can be considered as a
module over the subalgebra ${\mathcal{M}}_{X}$. Hence, the action of
$U_{X}^{0}$ on $V$ is diagonalisable.

Assume that $\mathrm{char}(k)\neq 2$. Let
$P_{2N-1}={\mathrm{Hom}}(Q^{\vee}_{2N-1},{\mathbb{Z}})$ be the weight lattice
of ${\mathfrak{so}}(2N-1)$. For every group homomorphism
$\chi\in{\mathrm{Hom}}(Q^{\vee}_{2N-1},\\{\pm 1\\})$ and any $\lambda\in
P_{2N-1}$ define

(5.1) $\displaystyle V_{\lambda,\chi}=\\{v\in
V\,|\,K_{\eta(h)}v=\chi(h)q^{\lambda(h)}v\quad\mbox{for all $h\in
Q^{\vee}_{2N-1}$}\\}.$

Following [Jan96, 5.2] we have

(5.2) $\displaystyle V=\bigoplus_{\chi}V^{\chi}\qquad\mbox{where}\quad
V^{\chi}=\bigoplus_{\lambda\in P_{2N-1}}V_{\lambda,\chi}$

where the first sum is over all group homomorphisms
$\chi\in{\mathrm{Hom}}(Q^{\vee}_{2N-1},\\{\pm 1\\})$. As the generator $B_{1}$
of ${\mathcal{B}}_{\mathbf{c}}$ $q$-commutes with all $K_{i}$ for $i\in X$,
the ${\mathcal{M}}_{X}$-submodule $V^{\chi}$ is a
${\mathcal{B}}_{\mathbf{c}}$-submodule of $V$. We say that $V$ is of type
$\chi$ if $V=V^{\chi}$. Let ${\mathcal{B}}_{\mathbf{c}}\mbox{-mod}_{f,\chi}$
denote the category of finite dimensional ${\mathcal{B}}_{\mathbf{c}}$ modules
of type $\chi$.

For every $\chi\in{\mathrm{Hom}}(Q^{\vee}_{2N-1},\\{\pm 1\\})$ there exists a
one-dimensional $U$-module $k_{\chi}=k{\mathbbm{1}}_{\chi}$ defined by

$\displaystyle u{\mathbbm{1}}_{\chi}$
$\displaystyle=\varepsilon(u){\mathbbm{1}}_{\chi}\quad\mbox{for
$u=E_{i},F_{i}$ for $i\in I$, and for $u=K_{1}$,}$ $\displaystyle
K_{\eta(h)}{\mathbbm{1}}_{\chi}$
$\displaystyle=\chi(h){\mathbbm{1}}_{\chi}\quad\mbox{for all $h\in
Q^{\vee}_{2N-1}$.}$

As ${\mathcal{B}}_{\mathbf{c}}\subset U$ is a right coideal subalgebra, the
category of finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-modules is a right
module category over the category of finite dimensional $U$-modules. As in
[Jan96, 5.2] this allows us to formulate an equivalence of categories. Let
${\mathbf{1}}\in{\mathrm{Hom}}(Q^{\vee}_{2N-1},\\{\pm 1\\})$ be the trivial
group homomorphism defined by ${\mathbf{1}}(h)=1$ for all $h\in
Q^{\vee}_{2N-1}$.

######  Proposition 5.1.

Assume that $\mathrm{char}(k)\neq 2$ and let
$\chi\in{\mathrm{Hom}}(Q^{\vee}_{2N-1},\\{\pm 1\\})$. The functor

$\displaystyle{\mathcal{B}}_{\mathbf{c}}\mbox{{-mod}}_{f,{\mathbf{1}}}\rightarrow{\mathcal{B}}_{\mathbf{c}}\mbox{{-mod}}_{f,\chi},\qquad
V\mapsto V\otimes k_{\chi}$

is an equivalence of categories.

Recall the involutive algebra anti-automorphism $\sigma^{B}$ defined by
(3.16). Let $V$ be a finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-module.
We define a ${\mathcal{B}}_{\mathbf{c}}$-module structure on the linear dual
space $V^{\ast}$ by

(5.3) $\displaystyle(bf)(v)=f(\sigma^{B}(b)v)\qquad\mbox{for all $f\in
V^{\ast}$, $b\in{\mathcal{B}}_{\mathbf{c}}$, $v\in V$.}$

Note that for any $\chi\in{\mathrm{Hom}}(Q^{\vee}_{2N-1},\\{\pm 1\\})$ the
category ${\mathcal{B}}_{\mathbf{c}}\mbox{-mod}_{f,\chi}$ is closed under
taking duals.

Proposition 5.1 shows that we lose no information if we restrict to type
$\mathbf{1}$ ${\mathcal{B}}_{\mathbf{c}}$-modules. Moreover, this category is
closed under taking duals. We shall henceforth only consider finite
dimensional ${\mathcal{B}}_{\mathbf{c}}$-modules of type $\mathbf{1}$.

######  Remark 5.2.

If $\mathrm{char}(k)=2$ then all types coincide. In this case the category of
finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-modules of type $\mathbf{1}$
is just the category of all finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-modules.

###  5.2. Existence of a highest weight vector

Let $V$ be a finite dimensional ${\mathcal{B}}_{\mathbf{c}}$-module of type
$\mathbf{1}$. For any $\lambda\in P_{2N-1}$ we write
$V_{\lambda}=V_{\lambda,\mathbf{1}}$ and hence the second equation in (5.1)
becomes

$V=\bigoplus_{\lambda\in P_{2N-1}}V_{\lambda}.$

We call the elements of $V_{\lambda}$ weight vectors of weight $\lambda$. The
commutation relations in Lemma 4.4 imply the following result.

######  Lemma 5.3.

Let $V$ be a representation of ${\mathcal{B}}_{\mathbf{c}}$. For
$i=2,\dots,N-1$ and $\lambda\in P_{2N-1}$ the following relations hold:

$\displaystyle E_{i}V_{\lambda}$ $\displaystyle\subseteq
V_{\lambda+\gamma_{i-1}},$ $\displaystyle F_{i}V_{\lambda}$
$\displaystyle\subseteq V_{\lambda-\gamma_{i-1}},$ $\displaystyle
B_{\beta_{N-1}}V_{\lambda}$ $\displaystyle\subseteq V_{\lambda+\gamma_{N-1}},$
$\displaystyle B_{\beta_{N}}V_{\lambda}$ $\displaystyle\subseteq
V_{\lambda-\gamma_{N-1}},$ $\displaystyle E_{N}V_{\lambda}$
$\displaystyle\subseteq V_{\lambda+\gamma_{N-2}+2\gamma_{N-1}},$
$\displaystyle F_{N}V_{\lambda}$ $\displaystyle\subseteq
V_{\lambda-\gamma_{N-2}-2\gamma_{N-1}}.$

The above Lemma implies the existence of a highest weight vector.

######  Lemma 5.4.

Let $V\neq\\{0\\}$ be a finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-module. There exists a weight vector $v\in
V\setminus\\{0\\}$ such that

$\displaystyle E_{i}v$ $\displaystyle=0\qquad\mbox{for $i=2,\dots,N$},$
$\displaystyle B_{\beta_{N-1}}v$ $\displaystyle=0.$

Moreover, the weight $\lambda\in P_{2N-1}$ of $v$ satisfies
$\lambda(h_{\gamma_{j}})\geq 0$ for $j=1,\dots,N-2$.

###### Proof.

The second statement follows from the fact that
$K_{\eta(h_{\gamma_{j}})}=K_{j+1}$ for $j=1,\dots,N-2$ and that
$\\{E_{j+1},F_{j+1},K^{\pm 1}_{j+1}\\}$ generates a copy of
$U_{q}({\mathfrak{sl}}(2))$. ∎

###  5.3. The commutator $[B_{\beta_{N-1}},B_{\beta_{N}}]$

Lemma 5.4 shows that the weight $\lambda$ of a highest weight vector $v$ in a
finite-dimensional representation $V$ of ${\mathcal{B}}_{\mathbf{c}}$ is
dominant with respect to $\\{h_{\gamma_{1}},\dots,h_{\gamma_{N-2}}\\}$. To
show that $\lambda$ is also dominant with respect to $h_{\gamma_{N-1}}$, we
need to investigate the commutator

(5.4) $\displaystyle\Omega:=[B_{\beta_{N-1}},B_{\beta_{N}}].$

To simplify notation, we define $\Omega_{1}=[F_{1},E_{1}]$ and

(5.5) $\displaystyle\Omega_{j}$ $\displaystyle=[T_{j}^{-1}\dots
T_{2}^{-1}(F_{1}),T_{j}\dots T_{2}(E_{1})]\qquad\mbox{for $j=2,\dots,N-1$,}$
(5.6) $\displaystyle\Omega_{N}$ $\displaystyle=[T_{N}^{-1}T_{N-2}^{-1}\dots
T_{2}^{-1}(F_{1}),T_{N}T_{N-2}\dots T_{2}(E_{1})].$

The commutator $\Omega$ can be expressed in terms of $\Omega_{N-1}$ and
$\Omega_{N}$.

######  Lemma 5.5.

The commutator $\Omega$ given by (5.4) satisfies the relation

$\displaystyle\Omega=c_{1}\big{(}\Omega_{N}K_{N-1}^{-1}-\Omega_{N-1}K_{N}^{-1}\big{)}K_{1}^{-1}\dots
K_{N-2}^{-1}.$

In particular, we have $\Omega\in U_{0}$.

###### Proof.

Set $A=T_{N-2}^{-1}\dots T_{2}^{-1}(F_{1})$, $C=T_{N-2}\dots T_{2}(E_{1})$ and
$K=K_{1}K_{2}\dots K_{N-2}$. By (3.3) for $j\in\\{N-1,N\\}$ we have

$\displaystyle T_{j}^{-1}(A)$ $\displaystyle=T_{1}T_{2}\dots T_{N-2}(F_{j}),$
$\displaystyle T_{j}(C)$ $\displaystyle=T_{1}^{-1}T_{2}^{-1}\dots
T_{N-2}^{-1}(E_{j})$

and hence

$\displaystyle[T_{N}^{-1}(A),T_{N-1}^{-1}(A)]$ $\displaystyle=0,$
$\displaystyle[T_{N}^{-1}(C),T_{N-1}^{-1}(C)]$ $\displaystyle=0.$

Moreover, $[T_{N-1}^{-1}(A),KK_{N}]=0=[T_{N}(C),KK_{N-1}]$ for weight reasons.
With these preparations we calculate

$\displaystyle\Omega$ $\displaystyle=[B_{\beta_{N-1}},B_{\beta_{N}}]$
$\displaystyle=[T_{N-1}^{-1}(A)-c_{1}T_{N}(C)K^{-1}K_{N-1}^{-1},T_{N}^{-1}(A)-c_{1}T_{N-1}(C)K^{-1}K_{N}^{-1}]$
$\displaystyle=-c_{1}[T_{N-1}^{-1}(A),T_{N-1}(C)]K^{-1}K_{N}^{-1}-c_{1}[T_{N}(C),T_{N}(A)]K^{-1}K_{N-1}^{-1}$
$\displaystyle=-c_{1}\Omega_{N-1}K_{N}^{-1}K^{-1}+c_{1}\Omega_{N}K_{N-1}^{-1}K^{-1}$

which proves the desired formula. ∎

By Equations (5.5), (5.6) and Lemma 5.5 we have $\Omega\in
U_{0}\cap{\mathcal{B}}_{\mathbf{c}}$. By Proposition 4.5 or [Kol14,
Proposition 6.2] we have
$U_{0}\cap{\mathcal{B}}_{\mathbf{c}}={\mathcal{M}}_{X,0}$. Hence, $\Omega$ is
an element of weight $0$ in ${\mathcal{M}}_{X}$, that is,

(5.7) $\displaystyle\Omega\in{\mathcal{M}}_{X,0}.$

The triangular decomposition (3.1) of $U$ gives a direct sum decomposition

$\displaystyle
U_{0}=U^{0}\oplus\bigoplus_{\nu>0}U^{-}_{-\nu}U^{0}U^{+}_{\nu}.$

Let $\pi:U\rightarrow U^{0}$ be the projection with respect to this
decomposition, see [Jan96, 6.2]. For our purposes, it suffices to know
$\pi(\Omega)$. As a first step we determine $\pi(\Omega_{j})$ for the elements
$\Omega_{j}$ given by (5.5), (5.6). For any $j\in I$ there exists a well-
defined algebra automorphism $\Psi_{j}:U^{0}\rightarrow U^{0}$ such that

$\displaystyle\Psi_{j}(K_{i})=q^{\delta_{i,j}}K_{i}.$

######  Lemma 5.6.

For all $j\in\\{1,\dots,N-2\\}$ we have the iterative formula

$\displaystyle\pi(\Omega_{j+1})=\pi(\Omega_{j})\frac{K_{j+1}-(2-q^{-2})K_{j+1}^{-1}}{q-q^{-1}}-q\Psi_{j}(\pi(\Omega_{j}))\frac{K_{j+1}-K_{j+1}^{-1}}{q-q^{-1}}.$

###### Proof.

For $j\in\\{1,\dots,N-2\\}$ define $A_{j}=T_{j}^{-1}\dots T_{2}^{-1}(F_{1})$
and $C_{j}=T_{j}\dots T_{2}(E_{1})$, and in particular $A_{1}=F_{1}$ and
$C_{1}=E_{1}$. We have $\Omega_{j}=[A_{j},C_{j}]$ and
$\Omega_{j+1}=[[F_{j+1},A_{j}]_{q},[E_{j+1},C_{j}]_{q^{-1}}]$. Using the
relations $[F_{j+1},C_{j}]=0=[E_{j+1},A_{j}]$ we obtain

$\displaystyle\pi$
$\displaystyle(\Omega_{j+1})=-\pi\big{(}[E_{j+1},C_{j}]_{q^{-1}}\,[F_{j+1},A_{j}]_{q}\big{)}$
$\displaystyle=-\pi\big{(}E_{j+1}F_{j+1}C_{j}A_{j}{-}q^{-1}C_{j}E_{j+1}F_{j+1}A_{j}{-}qE_{j+1}C_{j}A_{j}F_{j+1}{+}C_{j}A_{j}E_{j+1}F_{j+1}\big{)}$
$\displaystyle=-\frac{K_{j+1}-K_{j+1}^{-1}}{q-q^{-1}}\pi([C_{j},A_{j}])+q^{-1}\pi([C_{j},A_{j}])\frac{qK_{j+1}-q^{-1}K_{j+1}^{-1}}{q-q^{-1}}$
$\displaystyle\qquad+q\pi\big{(}E_{j+1}\pi([C_{j},A_{j}])F_{j+1}\big{)}-\pi([C_{j},A_{j}])\frac{K_{j+1}-K_{j+1}^{-1}}{q-q^{-1}}$
$\displaystyle=2\pi(\Omega_{j})\frac{K_{j+1}-K_{j+1}^{-1}}{q-q^{-1}}-\pi(\Omega_{j})\frac{K_{j+1}-q^{-2}K_{j+1}^{-1}}{q-q^{-1}}-q\Psi_{j}(\pi(\Omega_{j}))\frac{K_{j+1}-K_{j+1}^{-1}}{q-q^{-1}}$

which gives the desired formula. ∎

The above lemma allows us to compare $\pi(\Omega_{j})$ and
$\Psi_{j}(\pi(\Omega_{j}))$.

######  Lemma 5.7.

For all $j\in\\{1,\dots,N-1\\}$ we have

(5.8)
$\displaystyle\pi(\Omega_{j})-q\Psi_{j}(\pi(\Omega_{j}))=(-1)^{j-1}q^{j}K_{1}\dots
K_{j}.$

###### Proof.

We perform induction on $j$. For $j=1$ we have
$\Omega_{1}=\frac{K_{1}^{-1}-K_{1}}{q-q^{-1}}$ and Equation (5.8) holds. For
$j=1,\dots,N-2$ we now use Lemma 5.6 and the fact that $\pi(\Omega_{j})$ is
invariant under $\Psi_{j+1}$ to calculate

$\displaystyle\pi(\Omega_{j+1})-$ $\displaystyle
q\Psi_{j+1}(\pi(\Omega_{j+1}))$ $\displaystyle=$
$\displaystyle\pi(\Omega_{j})\left[\frac{K_{j+1}-(2-q^{-1})K_{j+1}^{-1}}{q-q^{-1}}-q\frac{qK_{j+1}-(2-q^{-1})q^{-1}K_{j+1}^{-1}}{q-q^{-1}}\right]$
$\displaystyle-q\Psi_{j}(\pi(\Omega_{j}))\left[\frac{K_{j+1}-K_{j+1}^{-1}}{q-q^{-1}}-q\frac{qK_{j+1}-q^{-1}K_{j+1}^{-1}}{q-q^{-1}}\right]$
$\displaystyle=$
$\displaystyle\big{(}\pi(\Omega_{j})-q\Psi_{j}(\pi(\Omega_{j}))\big{)}(-q)K_{j+1}.$

This formula provides the induction step. ∎

Using the three lemmas above, we obtain the desired expression for
$\pi(\Omega)$.

######  Proposition 5.8.

The commutator $\Omega$ given by (5.4) satisfies the relation

$\displaystyle\pi(\Omega)=c_{1}(-1)^{N-1}q^{N-2}\frac{K_{N}K_{N-1}^{-1}-K_{N-1}K_{N}^{-1}}{q-q^{-1}}.$

###### Proof.

Using Lemmas 5.5 and 5.6 we get

$\displaystyle\pi(\Omega)=$ $\displaystyle
c_{1}\big{(}\pi(\Omega_{N})K_{N-1}^{-1}-\pi(\Omega_{N-1})K_{N}^{-1}\big{)}K_{1}^{-1}\dots
K_{N-2}^{-1}$ $\displaystyle=$ $\displaystyle
c_{1}\Big{[}\pi(\Omega_{N-2})\frac{K_{N}{-}(2{-}q^{-2})K_{N}^{-1}}{q-q^{-1}}$
$\displaystyle\qquad\qquad-q\Psi_{N-2}(\pi(\Omega_{N-2}))\frac{K_{N}{-}K_{N}^{-1}}{q-q^{-1}}\Big{]}K_{1}^{-1}\dots
K_{N-2}^{-1}K_{N-1}^{-1}$ $\displaystyle-
c_{1}\Big{[}\pi(\Omega_{N-2})\frac{K_{N-1}{-}(2{-}q^{-2})K_{N-1}^{-1}}{q-q^{-1}}$
$\displaystyle\qquad\qquad-q\Psi_{N-2}(\pi(\Omega_{N-2}))\frac{K_{N-1}{-}K_{N-1}^{-1}}{q-q^{-1}}\Big{]}K_{1}^{-1}\dots
K_{N-2}^{-1}K_{N}^{-1}$ $\displaystyle=$ $\displaystyle
c_{1}\big{(}\pi(\Omega_{N-2})-q\Psi_{N-2}(\pi(\Omega_{N-2}))\big{)}\frac{K_{N}K_{N-1}^{-1}-K_{N-1}K_{N}^{-1}}{q-q^{-1}}K_{1}^{-1}\dots
K_{N-2}^{-1}.$

Now the statement of the proposition follows from Lemma 5.7. ∎

###  5.4. The missing ${\mathfrak{sl}}_{2}$-triple

Let $V$ be a representation of ${\mathcal{B}}_{\mathbf{c}}$. Proposition 5.8
suggests that the triple $B_{\beta_{N-1}},B_{\beta_{N}},K_{N}K_{N-1}^{-1}$
provides a representation of $U_{q}({\mathfrak{sl}}_{2})$ on a suitable
subspace of $V$. We define

(5.9) $\displaystyle H(V)=\\{v\in
V\,|\,B_{\beta_{N-2}}v=E_{i}v=0\quad\mbox{for all $i\in X$}\\}.$

Recall from Lemma 4.12 that
$B_{\beta_{N-2}}=[E_{N-1},B_{\beta_{N-1}}]K_{N-1}$. Hence, Lemma 5.4 implies
that $H(V)\neq\\{0\\}$ if $0<\dim(V)<\infty$.

######  Lemma 5.9.

Let $V$ be a ${\mathcal{B}}_{\mathbf{c}}$-module. The subspace $H(V)$ is
invariant under the action of the root vectors $B_{\beta_{N-1}}$ and
$B_{\beta_{N}}$.

###### Proof.

Fix $v\in H(V)$. Recall the root vectors $B_{\beta_{j}}$ defined by (4.5). For
$\ell\in\\{N-1,N\\}$ it follows from (4.6) that

(5.10) $\displaystyle B_{\beta_{\ell}}=[F_{\ell},B_{\beta_{N-2}}]_{q}.$

Hence we have $B_{\beta_{\ell}}v=-qB_{\beta_{N-2}}F_{\ell}v$ by definition of
$H(V)$. To prove the Lemma, it remains to verify the relations

(5.11) $\displaystyle E_{i}B_{\beta_{N-2}}F_{\ell}v$
$\displaystyle=0\qquad\mbox{for all $i\in X$,}$ (5.12) $\displaystyle
B^{2}_{\beta_{N-2}}F_{\ell}v$ $\displaystyle=0.$

Again by (4.6) we have

$\displaystyle[E_{i},B_{\beta_{N-2}}]=\begin{cases}qK_{N-2}^{-1}B_{\beta_{N-3}}&\mbox{if
$i=N-2$,}\\\ 0&\mbox{if $i\in X\setminus\\{N-2\\}$.}\end{cases}$

The above relation implies Equation (5.11). Indeed, for $i=N-2$ we obtain

$\displaystyle
E_{N-2}B_{\beta_{N-2}}F_{\ell}v=B_{\beta_{N-2}}E_{N-2}F_{\ell}v+qK_{N-2}^{-1}B_{\beta_{N-3}}F_{\ell}v=0$

as $[E_{N-2},F_{\ell}]=[B_{\beta_{N-3}},F_{\ell}]=0$. And for $i=\ell$ we have

$\displaystyle
E_{\ell}B_{\beta_{N-2}}F_{\ell}v=B_{\beta_{N-2}}\frac{K_{\ell}-K_{\ell}^{-1}}{q-q^{-1}}v=\frac{qK_{\ell}-q^{-1}K_{\ell}^{-1}}{q-q^{-1}}B_{\beta_{N-2}}v=0.$

To verify Equation (5.12), observe that Equation (5.10) and the definition of
$H(V)$ imply that

$\displaystyle
B^{2}_{\beta_{N-2}}F_{\ell}v=-q^{-1}B_{\beta_{N-2}}B_{\beta_{\ell}}v=-q^{-1}[B_{\beta_{N-2}},B_{\beta_{\ell}}]_{q}v.$

This expression vanishes by Corollary 4.11 and by the definition of $H(V)$. ∎

Lemma 5.9 and Proposition 5.8 provide us with the desired representation of
$U_{q}({\mathfrak{sl}}_{2})$ on the subspace $H(V)$. Observe that the subspace
$H(V)$ is also invariant under $U^{0}_{X}$.

######  Proposition 5.10.

Let $V$ be a ${\mathcal{B}}_{\mathbf{c}}$-module and $a_{N-1},a_{N}\in k$
satisfying the relation $a_{N-1}a_{N}=(-1)^{N-1}c_{1}^{-1}q^{2-N}$. There
exists an algebra homomorphism
$U_{q}({\mathfrak{sl}}_{2})\rightarrow\mathrm{End}(H(V))$ such that

$\displaystyle E\mapsto a_{N-1}B_{\beta_{N-1}},\quad F\mapsto
a_{N}B_{\beta_{N}},\quad K\mapsto K_{N}K_{N-1}^{-1}.$

###### Proof.

By Lemma 5.9 the subspace $H(V)$ is invariant under $B_{\beta_{N-1}}$ and
$B_{\beta_{N}}$. By (5.7) the commutator $\Omega$ defined by (5.4) satisfies

$\displaystyle\Omega v=\pi(\Omega)v\qquad\mbox{for all $v\in H(V)$.}$

Now the statement of the Proposition follows from Proposition 5.8. ∎

Proposition 5.10 implies that the weight $\lambda\in P_{2N-1}$ in Lemma 5.4 is
dominant.

######  Corollary 5.11.

Let $V$ be a finite-dimensional representation of ${\mathcal{B}}_{\mathbf{c}}$
and $v\in V$ a weight vector of weight $\lambda\in P_{2N-1}$ such that
$B_{\beta_{N-1}}v=E_{i}v=0$ for all $i\in X$. Then
$\lambda(h_{\gamma_{j}})\in{\mathbb{N}}_{0}$ for all $j=1,\dots,N-1$.

###  5.5. Verma modules for ${\mathcal{B}}_{\mathbf{c}}$

For any $\lambda\in P_{2N-1}$ consider the left ideal

$\displaystyle
J_{\lambda}=\sum_{i=2}^{N}\bigg{(}{\mathcal{B}}_{\mathbf{c}}E_{i}+{\mathcal{B}}_{\mathbf{c}}(K_{i}-q^{\lambda(\eta^{-1}(h_{\alpha_{i}}))})\bigg{)}+{\mathcal{B}}_{\mathbf{c}}B_{\beta_{N-1}}\subseteq{\mathcal{B}}_{\mathbf{c}}$

and define a corresponding Verma module $M(\lambda)$ for
${\mathcal{B}}_{\mathbf{c}}$ by

$\displaystyle M(\lambda)={\mathcal{B}}_{\mathbf{c}}/J_{\lambda}.$

We write $v_{\lambda}\in M(\lambda)$ to denote the coset $1+J_{\lambda}$.

######  Remark 5.12.

By Lemma (4.4) the root vectors in ${\mathcal{B}}_{\mathbf{c}}$ corresponding
to positive, simple roots are given by $E_{2},\dots,E_{N-1},B_{\beta_{N-1}}$.
Hence, it would be natural to consider the left ideal

$\displaystyle
I_{\lambda}=\sum_{i=2}^{N-1}\bigg{(}{\mathcal{B}}_{\mathbf{c}}E_{i}+{\mathcal{B}}_{\mathbf{c}}(K_{i}{-}q^{\lambda(h_{\gamma_{i-1}})})\bigg{)}+{\mathcal{B}}_{\mathbf{c}}B_{\beta_{N-1}}+{\mathcal{B}}_{\mathbf{c}}(K_{N}K_{N-1}^{-1}{-}q^{\lambda(h_{\gamma_{N-1}})}).$

It is not obvious from the relations of ${\mathcal{B}}_{\mathbf{c}}$ that
$E_{N}\in I_{\lambda}$ and hence it is convenient to work with $J_{\lambda}$
instead. At $q=1$ we have $I_{\lambda}=J_{\lambda}$.

The PBW-Theorem for ${\mathcal{B}}_{\mathbf{c}}$ allows us to prove a PBW-
Theorem for the Verma module $M(\lambda)$. Recall the definition
$\eqref{eq:BJ-def}$ of the PBW-monomial $B_{\mathcal{J}}$ for
${\mathcal{J}}=(j_{1},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{N(N-1)}$. By
Lemmma 4.12 the root vectors $B_{\beta_{j}}$ satisfy the relation

$\displaystyle B_{\beta_{j-1}}K_{j}^{-1}=[E_{j},B_{\beta_{j}}]\qquad\mbox{for
$j=2,\dots,N-1$.}$

Hence $B_{\mathcal{J}}v_{\lambda}=0$ unless $j_{1}=\cdots=j_{N-1}=0$. By the
following theorem, the remaining vectors $B_{\mathcal{J}}v_{\lambda}$ form a
basis of the Verma module $M(\lambda)$.

######  Theorem 5.13.

For any $\lambda\in P_{2N-1}$ the family of vectors

$\displaystyle\\{B_{\mathcal{J}}v_{\lambda}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)},j_{1}=\dots=j_{N-1}=0\\}$

is a basis of the Verma module $M(\lambda)$.

###### Proof.

Let $U_{X}^{0}(\lambda)$ denote the maximal ideal of $U_{X}^{0}$ generated by
the elements $K_{i}-q^{\lambda(\eta^{-1}(h_{\alpha_{i}}))}$ for $i\in X$.
Define linear subspaces of ${\mathcal{B}}_{\mathbf{c}}$ by

$\displaystyle M$
$\displaystyle=\mathrm{span}_{k}\\{B_{\mathcal{J}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)},j_{1}=\dots=j_{N-1}=0\\},$
$\displaystyle M^{\prime}$
$\displaystyle=\mathrm{span}_{k}\\{B_{\mathcal{J}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)},\exists
i\in\\{1,\dots,N-1\\}\mbox{ such that }j_{i}\neq 0\\},$ $\displaystyle N$
$\displaystyle=\mathrm{span}_{k}\\{B_{\mathcal{J}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}\\}U_{X}^{0}(\lambda),$
$\displaystyle P$
$\displaystyle={\mathcal{B}}_{\mathbf{c}}{\mathcal{M}}_{X,+}^{+}$

where ${\mathcal{M}}_{X,+}^{+}$ denotes the augmentation ideal of
${\mathcal{M}}_{X}^{+}$ defined at the beginning of Section 4.4. By Theorem
4.6 we have a direct sum decomposition

$\displaystyle{\mathcal{B}}_{\mathbf{c}}=M\oplus M^{\prime}\oplus N\oplus P.$

By definition of $J_{\lambda}$ we have $N,P\subset J_{\lambda}$. By Lemma 4.12
we know that $B_{\beta_{k}}\in J_{\lambda}$ for $k=1,\dots,N-1$. Hence
$M^{\prime}\subset J_{\lambda}$ and thus $M^{\prime}\oplus N\oplus P\subset
J_{\lambda}$.

It remains to verify the opposite inclusion. To this end, it suffices to show
that $bB_{\beta_{N-1}}\in M^{\prime}\oplus N\oplus P$ for all
$b\in{\mathcal{B}}_{\mathbf{c}}$. More generally, we will show by induction on
$j$ that

(5.13) $\displaystyle bB_{\beta_{j}}\in M^{\prime}\oplus N\oplus
P\quad\mbox{for $j=1,\dots,N-1$}$

for any $b\in{\mathcal{B}}_{\mathbf{c}}$. To prove (5.13) we may assume
without loss of generality that $b=B_{\mathcal{J}}K_{\beta}E_{\mathcal{I}}$
for some ${\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}$, $\beta\in Q_{X}$ and
${\mathcal{I}}\in{\mathbb{N}}_{0}^{(N-1)(N-2)}$, see Theorem 4.6. We fist
consider the case $j=1$. In this case we may assume
${\mathcal{I}}=(0,\dots,0)$, because otherwise $bB_{\beta_{1}}=bB_{1}\in P$.
Moreover, we may assume $K_{\beta}=1$ because $K_{\beta}$ and $B_{1}$ always
$q$-commute. As $B_{\mathcal{J}}B_{\beta_{1}}\in M^{\prime}$ by definition of
the PBW-basis, we have established (5.13) for $j=1$.

Now assume that (5.13) holds for $j=1,\dots,\ell-1<N-1$ and consider
$bB_{\beta_{\ell}}$. By Lemma 4.12 we may again assume that
${\mathcal{I}}=(0,\dots,0)$, and again we may $q$-commute the factor
$K_{\beta}$ to the right hand side of $B_{\beta_{\ell}}$ and replace it by a
power of $q$. Hence we may assume that $b=B_{\mathcal{J}}$ for some
${\mathcal{J}}\in{\mathbb{N}}^{N(N-1)}$. The relation
$B_{{\mathcal{J}}}B_{\beta_{\ell}}\in M^{\prime}\oplus N\oplus P$ now follows
from Lemma 4.10 and the induction hypothesis. This completes the proof of the
theorem. ∎

###  5.6. Proper submodules of $M(\lambda)$

By Lemma 5.4 and Corollary 5.11, every finite-dimensional, simple
${\mathcal{B}}_{\mathbf{c}}$-module (of type ${\mathbf{1}}$) is given as a
quotient of $M(\lambda)$ by a maximal proper submodule $N(\lambda)\subset
M(\lambda)$ for some dominant, integral weight $\lambda\in P_{2N-1}^{+}$. For
the classification of finite-dimensional, simple
${\mathcal{B}}_{\mathbf{c}}$-modules, it remains to show that
$\dim(M(\lambda)/N(\lambda))<\infty$ for all $\lambda\in P_{2N-1}^{+}$. To
gain some understanding of the submodule $N(\lambda)$ we first construct
homomorphism of Verma modules, in analogy to [Jan96, Lemma 5.6]. Recall that
$\eta(h_{\gamma_{i}})=h_{\alpha_{i+1}}$ for $1\leq i\leq N-2$ and that hence,
in this case, $F_{i+1}\in{\mathcal{B}}_{\mathbf{c}}$ is a negative root vector
corresponding to the negative simple root $-\gamma_{i}$. Moreover, from (4.5)
or (4.6), recall the definition of the root vector $B_{\beta_{N}}$
corresponding to the negative simple root $-\gamma_{N-1}$.

######  Proposition 5.14.

Let $\lambda\in P_{2N-1}$ and $i\in\\{1,2,\dots,N-1\\}$ with
$n_{i}:=\lambda(h_{\gamma_{i}})\geq 0$. There is a homomorphism of
${\mathcal{B}}_{\mathbf{c}}$-modules
$\varphi_{i}:M(\lambda-(n_{i}+1)\gamma_{i})\rightarrow M(\lambda)$ with

$\displaystyle\varphi_{i}(v_{\lambda-(n_{i}+1)\gamma_{i}})=\begin{cases}F_{i+1}^{n_{i}+1}v_{\lambda}&\mbox{if
$i\leq N-2$,}\\\ B_{\beta_{N}}^{n_{N-1}+1}v_{\lambda}&\mbox{if
$i=N-1$.}\end{cases}$

###### Proof.

First consider the case $i\leq N-2$. By the universal property of
$M(\lambda-(n_{i}+1)\gamma_{i})$ we need to verify the following two relations

(5.14) $\displaystyle E_{j}F_{i+1}^{n_{i}+1}v_{\lambda}$
$\displaystyle=0\qquad\mbox{for $j\in X$,}$ (5.15) $\displaystyle
B_{\beta_{N-1}}F_{i+1}^{n_{i}+1}v_{\lambda}$ $\displaystyle=0.$

The proof of Equation (5.14) is identical to the proof of [Jan96, Lemma 5.6].
Equation (5.15) follows from the commutation relations
$[B_{\beta_{N-1}},F_{N-1}]_{q}=0$ and $[B_{\beta_{N-1}},F_{i}]=0$ which hold
by Lemma 4.13.

It remains to consider the case $i=N-1$. In this case, again by the universal
property of $M(\lambda-(n_{N-1}+1)\gamma_{N-1})$, we need to verify the
following two relations

(5.16) $\displaystyle E_{j}B_{\beta_{N}}^{n_{N-1}+1}v_{\lambda}$
$\displaystyle=0\qquad\mbox{for $j\in X$,}$ (5.17) $\displaystyle
B_{\beta_{N-1}}B_{\beta_{N}}^{n_{N-1}+1}v_{\lambda}$ $\displaystyle=0.$

By symmetry between $N-1$ and $N$, Lemma 4.12 implies that

(5.18)
$\displaystyle[E_{j},B_{\beta_{N}}]=\begin{cases}B_{\beta_{N-2}}K_{N}^{-1}&\mbox{if
$j=N$},\\\ 0&\mbox{if $j\neq N$.}\end{cases}$

This implies Equation (5.16) in the case $j\neq N$. To address the case $j=N$
we prove by induction that $E_{N}B_{\beta_{N}}^{a}v_{\lambda}=0$ for all
$a\in{\mathbb{N}}_{0}$. In the case $a=0$ this holds by definition of
$M(\lambda)$. Assuming now that $E_{N}B_{\beta_{N}}^{d}v_{\lambda}=0$ for all
$d\leq a$ we calculate

$\displaystyle
E_{N}B_{\beta_{N}}^{a+1}v_{\lambda}=[E_{N},B_{\beta_{N}}]B_{\beta_{N}}^{a}v_{\lambda}\stackrel{{\scriptstyle\eqref{eq:EBBEN}}}{{=}}qK_{N}^{-1}B_{\beta_{N-2}}B_{\beta_{N}}^{a}v_{\lambda}.$

This expression vanishes by Corollary 4.11 and induction hypothesis, and hence
Equation (5.16) also holds for $j=N$.

To verify Equation (5.17), we can calculate inside the subspace
$H(M(\lambda))$ defined by (5.9). By Proposition (5.10), the root vectors
$B_{\beta_{N-1}}$ and $B_{\beta_{N}}$ give rise to a representation of
$U_{q}({\mathfrak{sl}}_{2})$ on $H(M(\lambda))$ up to scaling. Hence Equation
(5.17) can again be verified by the calculation in the proof of [Jan96, Lemma
5.6] ∎

By the above proposition, the submodule
$\widetilde{N}(\lambda):=\sum_{i=1}^{N-1}{\mbox{Im}}(\varphi_{i})$ of
$M(\lambda)$ is proper. Ideally, we would like to adapt the arguments in
[Jan96, 5.7-5.9] to show that
$\widetilde{L}(\lambda)=M(\lambda)/\widetilde{N}(\lambda)$ is finite-
dimensional. In attempting to do so, however, we encounter some problems. In
particular, the triple $B_{\beta_{N-1}},B_{\beta_{N}},K_{N}K_{N-1}^{-1}$
provides a copy of $U_{q}({\mathfrak{sl}}_{2})$ only on the subspace
$H(\widetilde{L}(\lambda))$ and not on all of $\widetilde{L}(\lambda)$. Hence
it is unclear how to adapt the arguments in [Jan96, 5.7-5.9] to show that the
set of weights of $\widetilde{L}(\lambda)$ is stable under the Weyl group.

For this reason we take a different approach. We will consider a quotient of
$M(\lambda)$ by a submodule which at first sight seems larger than
$\widetilde{N}(\lambda)$. The following proposition will provide the main step
in our argument.

######  Proposition 5.15.

Let $\lambda\in P_{2N-1}$ with
$n_{N-1}:=\lambda(h_{\gamma_{N-1}})\in{\mathbb{N}}_{0}$ and
$n_{N-2}:=\lambda(h_{\gamma_{N-2}})\in{\mathbb{N}}_{0}$ and set
$n_{N}=n_{N-1}+n_{N-2}$. The submodule
${\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda}\subset M(\lambda)$ is
contained in $\sum_{\mu<\lambda}M(\lambda)_{\mu}$.

We will prove Proposition 5.15 in Section 5.7. By the above proposition the
submodule

$\displaystyle\widetilde{N}(\lambda)+{\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda}$

of $M(\lambda)$ is contained in $\sum_{\mu<\lambda}M(\lambda)_{\mu}$ and hence
proper. We will show in Section 5.8 using filtered-graded arguments that the
quotient
$M(\lambda)/\big{(}\widetilde{N}(\lambda)+{\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda}\big{)}$
is finite-dimensional.

######  Remark 5.16.

The element $F_{N}^{m+1}v_{\lambda}\in M(\lambda)$ is not annihilated by the
positive simple root vector $B_{\beta_{N-1}}$ for any $m\in{\mathbb{N}}_{0}$.
Indeed, by Equation (4.6) and the quantum Serre relations, we have

$\displaystyle[B_{\beta_{N+1}},F_{N}]_{q}$
$\displaystyle=[[F_{N},[F_{N-1},B_{\beta_{N-2}}]_{q}]_{q},F_{N}]_{q}=[F_{N-1},[[F_{N},B_{\beta_{N-2}}]_{q},F_{N}]_{q}]_{q}=0$

because $F_{N-1}$ and $F_{N}$ commute. Using the relation
$[F_{N},B_{\beta_{N-1}}]_{q}=B_{\beta_{N+1}}$ which also holds by (4.6), we
hence obtain for any $m\in{\mathbb{N}}_{0}$ the relation

$\displaystyle B_{\beta_{N-1}}F_{N}^{m+1}v_{\lambda}$
$\displaystyle=q^{-1}F_{N}B_{\beta_{N-1}}F_{N}^{m}v_{\lambda}-q^{-1}B_{\beta_{n+1}}F_{N}^{m}v_{\lambda}$
$\displaystyle=-q^{-1}\sum_{k=0}^{m}(q^{-1})^{k}F_{N}^{k}B_{\beta_{N+1}}F_{N}^{m-k}v_{\lambda}$
$\displaystyle=-q^{-1}\sum_{k=0}^{m}q^{m-2k}F_{N}^{m}B_{\beta_{N+1}}v_{\lambda}$
$\displaystyle=-q^{-1}[m+1]_{q}F_{N}^{m}B_{\beta_{N+1}}v_{\lambda}.$

By the PBW-Theorem for $M(\lambda)$, Theorem 5.13, the above expression is not
zero in $M(\lambda)$.

###  5.7. Proof of Proposition 5.15

We begin with a preparatory Lemma. Recall that by convention $[0]^{!}_{q}=1$
and that $[m]^{!}_{q}=[m]_{q}[m-1]^{!}_{q}$ for all $m\in{\mathbb{N}}$.

######  Lemma 5.17.

Let $\lambda\in P_{2N-1}$. For all $\ell,m\in{\mathbb{N}}_{0}$ with $\ell\leq
m$ the relation

(5.19) $\displaystyle
B_{\beta_{N-2}}^{\ell}F_{N}^{m}\,v_{\lambda}=(-q^{-1})^{\ell}\frac{[m]^{!}_{q}}{[m-\ell]^{!}_{q}}F_{N}^{m-\ell}B_{\beta_{N}}^{\ell}\,v_{\lambda}$

holds in the Verma module $M(\lambda)$.

###### Proof.

We proceed by induction on $\ell$. For $\ell=0$ there is nothing to show.
Assume that Equation (5.19) holds for some $\ell<m$. Using Lemma 4.14 we
calculate

$\displaystyle B_{\beta_{N-2}}^{\ell+1}F_{N}^{m}\,v_{\lambda}$
$\displaystyle=(-q^{-1})^{\ell}\frac{[m]^{!}_{q}}{[m-\ell]^{!}_{q}}B_{\beta_{N-2}}F_{N}^{m-\ell}B_{\beta_{N}}^{\ell}\,v_{\lambda}$
(5.20)
$\displaystyle=(-q^{-1})^{\ell}\frac{[m]^{!}_{q}}{[m-\ell]^{!}_{q}}\bigg{(}q^{-(m-\ell)}F_{N}^{m-\ell}B_{\beta_{N-2}}B_{\beta_{N}}^{\ell}\,v_{\lambda}$
$\displaystyle\qquad\qquad\qquad-q^{-1}[m-\ell]_{q}F_{N}^{m-\ell-1}B_{\beta_{N}}^{\ell+1}\,v_{\lambda}\bigg{)}$

Recall from Lemma 4.4 that $B_{\beta_{N}}$ is a root vector of weight
$-\gamma_{N-1}$ and that
$B_{\beta_{N-2}}=[E_{N-1},B_{\beta_{N-1}}]_{q}K_{N-1}$ is a root vector of
weight $\gamma_{N-2}+\gamma_{N-1}$. Hence

$\displaystyle B_{\beta_{N-2}}B_{\beta_{N}}^{\ell}\,v_{\lambda}\in
M(\lambda)_{\lambda-(\ell-1)\gamma_{N-1}+\gamma_{N-2}}=\\{0\\}$

as $\lambda-(\ell-1)\gamma_{N-1}+\gamma_{N-2}\nleq\lambda$. This implies that
the first term in the bracket in Equation (5.20) vanishes. Hence we have

$\displaystyle B_{\beta_{N-2}}^{\ell+1}F_{N}^{m}\,v_{\lambda}$
$\displaystyle=(-q^{-1})^{\ell+1}\frac{[m]^{!}_{q}}{[m-(\ell+1)]^{!}_{q}}F_{N}^{m-(\ell+1)}B_{\beta_{N}}^{\ell+1}\,v_{\lambda}$

as desired. ∎

Recall the homomorphism of Verma module
$\varphi_{N-1}:M(\lambda-(n_{N-1}+1)\gamma_{N-1})\rightarrow M(\lambda)$ from
Proposition 5.14. Lemma 5.17 has the following consequence.

######  Corollary 5.18.

Let $\lambda\in P_{2N-1}$ with
$n_{N-1}=\lambda(h_{\gamma_{N-1}})\in{\mathbb{N}}_{0}$ and
$n_{N-2}=\lambda(h_{\gamma_{N-2}})\in{\mathbb{N}}_{0}$ and set
$n_{N}=n_{N-1}+n_{N-2}$. The following relation holds

$\displaystyle
B_{\beta_{N-2}}^{n_{N}+1}F_{N}^{n_{N}+1}\,v_{\lambda}\in\mathrm{Im}(\varphi_{N-1}).$

###### Proof.

By definition, $\mathrm{Im}(\varphi_{N-1})$ is the
${\mathcal{B}}_{\mathbf{c}}$-submodule of $M(\lambda)$ generated by the
element $B_{\beta_{N}}^{n_{N-1}+1}\,v_{\lambda}$. By Lemma 5.17 we have

(5.21) $\displaystyle
B_{\beta_{N-2}}^{n_{N}+1}F_{N}^{n_{N}+1}\,v_{\lambda}=(-q^{-1})^{n_{N}+1}[n_{N}+1]^{!}_{q}B_{\beta_{N}}^{n_{N}+1}\,v_{\lambda}\in\mathrm{Im}(\varphi_{N-1})$

as $n_{N}\geq n_{N-1}$. ∎

###### Proof of Proposition 5.15.

For all $i\in X$ one obtains $E_{i}F_{N}^{n_{N}+1}v_{\lambda}=0$, again by the
argument in [Jan96, Lemma 5.6]. Recall from Lemma 4.4 that $F_{N}$ is a root
vector for ${\mathcal{B}}_{\mathbf{c}}$ of weight
$-\gamma_{N-2}-2\gamma_{N-1}$. Similarly, $B_{\beta_{j}}$ for $j=1,\dots,N-1$
is a root vector for ${\mathcal{B}}_{\mathbf{c}}$ of weight
$\sum_{\ell=j}^{N-1}\gamma_{k}$. Using the PBW-Theorem for
${\mathcal{B}}_{\mathbf{c}}$, Theorem 4.6, we conclude that

$\displaystyle({\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda})_{\lambda}$
$\displaystyle=\mathrm{span}_{k}\Big{\\{}B_{\beta_{N-1}}^{j_{N-1}}\dots
B_{\beta_{1}}^{j_{1}}F_{N}^{n_{N}+1}v_{\lambda}\,\big{|}$
$\displaystyle\qquad\qquad\qquad\qquad\sum_{i=1}^{N-1}j_{i}\sum_{\ell=i}^{N-1}\gamma_{\ell}=(n_{N}+1)(\gamma_{N-2}+2\gamma_{N-1})\Big{\\}}$
$\displaystyle=kB_{\beta_{N-1}}^{n_{N}+1}B_{\beta_{N-2}}^{n_{N}+1}F_{N}^{n_{N}+1}v_{\lambda}.$

By Corollary 5.18 we know that
$B_{\beta_{N-1}}^{n_{N}+1}B_{\beta_{N-2}}^{n_{N}+1}F_{N}^{n_{N}+1}v_{\lambda}\in\mathrm{Im}(\varphi_{N-1})_{\lambda}=\\{0\\}$.
Hence $v_{\lambda}\notin{\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda}$.
∎

###  5.8. Classification of finite-dimensional, irreducible
${\mathcal{B}}_{\mathbf{c}}$-modules

Fix $\lambda\in P_{2N-1}^{+}$. Recall that we use the notation
$n_{i}=\lambda(\gamma_{i})$ for $i=1,\dots,N-1$ and
$n_{N}=\lambda(h_{\gamma_{N-2}}+h_{\gamma_{N-1}})$. Define a submodule
$\accentset{\approx}{N}(\lambda)$ of the Verma module $M(\lambda)$ by

$\displaystyle\accentset{\approx}{N}(\lambda)=\sum_{i=1}^{N-2}{\mathcal{B}}_{\mathbf{c}}F_{i+1}^{n_{i}+1}v_{\lambda}+{\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda}.$

######  Remark 5.19.

Equation (5.21) implies that
${\mbox{Im}}(\varphi_{N-1})\subset{\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}v_{\lambda}$.
Indeed, by an ${\mathfrak{sl}}(2)$-argument and Equation (5.21) we see that
$B_{\beta_{N-1}}^{n_{N-2}}B_{\beta_{N-2}}^{n_{N}+1}F_{N}^{n_{N}+1}v_{\lambda}$
is a nonzero scalar multiple of $B_{\beta_{N}}^{n_{N-1}+1}v_{\lambda}$. For
this reason, we do not need to include a summand
${\mathcal{B}}_{\mathbf{c}}B_{\beta_{N}}^{n_{N-1}+1}v_{\lambda}$ in the
definition of $\accentset{\approx}{N}(\lambda)$.

By Propositions 5.14 and 5.15 we have $\accentset{\approx}{N}(\lambda)\neq
M(\lambda)$. Consider the factor module

$\displaystyle\accentset{\approx}{L}(\lambda)=M(\lambda)/\accentset{\approx}{N}(\lambda).$

We claim that $\accentset{\approx}{L}(\lambda)$ is finite-dimensional.

######  Proposition 5.20.

For all $\lambda\in P^{+}_{2N-1}$ the relation
$0<\dim(\accentset{\approx}{L}(\lambda))<\infty$ holds.

###### Proof.

The left-hand inequality holds because
$v_{\lambda}\notin\accentset{\approx}{N}(\lambda)$. To verify the right-hand
inequality define a left ideal
$\accentset{\approx}{J}_{\lambda}\subset{\mathcal{B}}_{\mathbf{c}}$ by

$\displaystyle\accentset{\approx}{J}_{\lambda}=J_{\lambda}+\sum_{i=1}^{N-2}{\mathcal{B}}_{\mathbf{c}}F_{i+1}^{n_{i}+1}+{\mathcal{B}}_{\mathbf{c}}F_{N}^{n_{N}+1}.$

By construction we have

(5.22)
$\displaystyle\accentset{\approx}{L}(\lambda)={\mathcal{B}}_{\mathbf{c}}/\accentset{\approx}{J}_{\lambda}.$

The standard filtration ${\mathcal{F}}_{\ast}$ on ${\mathcal{B}}_{\mathbf{c}}$
from Section 3.4 induces a filtration ${\mathcal{G}}_{\ast}$ on
$\accentset{\approx}{L}(\lambda)$ defined by

$\displaystyle{\mathcal{G}}_{n}(\accentset{\approx}{L}(\lambda))={\mathcal{F}}_{n}({\mathcal{B}}_{\mathbf{c}})/\accentset{\approx}{J}_{\lambda}\cap{\mathcal{F}}_{n}({\mathcal{B}}_{\mathbf{c}}).$

In this way $\accentset{\approx}{L}(\lambda)$ is a filtered left module over
the filtered algebra ${\mathcal{B}}_{\mathbf{c}}$, see e.g. [KL00, Chapter 6].
In particular, the associated graded space
${\mathrm{gr}}(\accentset{\approx}{L}(\lambda))$ is a graded module over the
graded algebra ${\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})\cong{\mathcal{A}}$.
By the isomorphism theorems we have

$\displaystyle{\mathrm{gr}}_{n}(\accentset{\approx}{L}(\lambda))=\frac{{\mathcal{F}}_{n}({\mathcal{B}}_{\mathbf{c}})/\accentset{\approx}{J}_{\lambda}\cap{\mathcal{F}}_{n}({\mathcal{B}}_{\mathbf{c}})}{{\mathcal{F}}_{n-1}({\mathcal{B}}_{\mathbf{c}})/\accentset{\approx}{J}_{\lambda}\cap{\mathcal{F}}_{n-1}({\mathcal{B}}_{\mathbf{c}})}\cong{\mathcal{A}}_{n}\big{/}{\mathrm{gr}}_{n}(\accentset{\approx}{J}_{\lambda}).$

Hence it suffices to show that
${\mathcal{A}}/{\mathrm{gr}}(\accentset{\approx}{J}_{\lambda})$ is a finite-
dimensional vector space. As $\accentset{\approx}{J}_{\lambda}$ contains the
elements

$\displaystyle
K_{i}-q^{\lambda(\eta^{-1}(h_{\alpha_{i}}))},E_{i},B_{1},F_{j}^{n_{j{-}1}+1},F_{N}^{n_{N}+1}\qquad\mbox{for
$i\in X$, $j\in X\setminus\\{N\\}$},$

the left ideal
${\mathrm{gr}}(\accentset{\approx}{J}_{\lambda})\subset{\mathrm{gr}}({\mathcal{B}}_{\mathbf{c}})\cong{\mathcal{A}}$
contains the elements

(5.23) $\displaystyle
K_{i}-q^{\lambda(\eta^{-1}(h_{\alpha_{i}}))},E_{i},F_{1},F_{j}^{n_{j{-}1}+1},F_{N}^{n_{N}+1}\qquad\mbox{for
$i\in X$, $j\in X\setminus\\{N\\}$}.$

Let $J^{\prime}_{\lambda}\subset{\mathcal{A}}$ be the left ideal generated by
the elements in (5.23). As
$J^{\prime}_{\lambda}\subset{\mathrm{gr}}(\accentset{\approx}{J}_{\lambda})$,
we see that ${\mathrm{gr}}(\accentset{\approx}{L}(\lambda))$ is a quotient of
${\mathcal{A}}/J^{\prime}_{\lambda}$. By the triangular decomposition (3.17)
of ${\mathcal{A}}$ we have

$\displaystyle{\mathcal{A}}/J^{\prime}_{\lambda}\cong
U^{-}\Big{/}\Big{(}U^{-}F_{1}+\sum_{j=2}^{N-1}U^{-}F_{j}^{n_{j{-}1}+1}+U^{-}F_{N}^{n_{N}+1}\Big{)}.$

Hence the quotient ${\mathcal{A}}/J^{\prime}_{\lambda}$ is finite-dimensional
by [Jan96, Lemma 5.9]. This implies that
${\mathrm{gr}}(\accentset{\approx}{L}(\lambda))$ and hence
$\accentset{\approx}{L}(\lambda)$ is finite-dimensional. ∎

######  Remark 5.21.

In general, the inclusion
$J^{\prime}_{\lambda}\subset{\mathrm{gr}}(\accentset{\approx}{J}(\lambda))$ is
proper. Indeed, the ideal $\accentset{\approx}{J}_{\lambda}$ contains the root
vector $B_{\beta_{N-1}}$ and hence
$F_{\beta_{N-1}}\in{\mathrm{gr}}(\accentset{\approx}{J}(\lambda))$ by (4.4),
(4.6). However, if $n_{N-1}\geq 1$ then $F_{\beta_{N-1}}\notin
J^{\prime}_{\lambda}$.

For any $\lambda\in P_{2N-1}$ the Verma module $M(\lambda)$ has a unique
simple quotient $L(\lambda)=M(\lambda)/N(\lambda)$, where $N(\lambda)\subset
M(\lambda)$ is the unique maximal proper submodule. By construction
$\accentset{\approx}{N}(\lambda)\subset N(\lambda)$ if $\lambda\in
P_{2N-1}^{+}$ and hence Proposition 5.20 implies that
$\dim(L(\lambda))<\infty$ in this case. We can now adapt [Jan96, Theorem 5.10]
to our setting.

######  Theorem 5.22.

Let $k$ be any field and $q\in k$ not a root of unity. For each $\lambda\in
P_{2N-1}^{+}$ the simple ${\mathcal{B}}_{\mathbf{c}}$-module $L(\lambda)$ has
finite dimension. Each finite-dimensional simple
${\mathcal{B}}_{\mathbf{c}}$-module (of type ${\bf 1}$) is isomorphic to
exactly one $L(\lambda)$ with $\lambda\in P_{2N-1}^{+}$.

###### Proof.

The first statement has been explained above. The second statement follows
from Lemma 5.4, Corollary 5.11 and the universal property of the Verma module
$M(\lambda)$. ∎

##  6\. Non-restricted specialisation

In this section, unless stated otherwise, we assume that $k={\mathbb{K}}(q)$
is the field of rational functions in a variable $q$ where ${\mathbb{K}}$ is a
field of characteristic zero. Let $\mathbf{A}={\mathbb{K}}[q]_{(q-1)}$ denote
the localisation of the polynomial ring ${\mathbb{K}}[q]$ with respect to the
prime ideal generated by $q-1$. For any $i\in I$ define

$\displaystyle(K_{i};0)_{q}=\frac{K_{i}-1}{q-1}.$

The $\mathbf{A}$-form $U_{\mathbf{A}}$ of $U$ is the $\mathbf{A}$-subalgebra
of $U$ generated by the elements $E_{i},F_{i},K_{i}^{\pm 1}$ and
$(K_{i};0)_{q}$ for all $i\in I$. As usual, we consider ${\mathbb{K}}$ as an
$\mathbf{A}$-module via the map sending $q$ to $1\in{\mathbb{K}}$. The
${\mathbb{K}}$-algebra $U_{1}={\mathbb{K}}\otimes_{\mathbf{A}}U_{\mathbf{A}}$
is called the specialisation of $U={U_{q}(\mathfrak{g})}$ at $q=1$. For any
$x\in U_{\mathbf{A}}$ we denote its image in $U_{1}$ by $\overline{x}$. It is
well known [CK90, Proposition 1.5], [HK02, Theorem 3.4.9] that there is an
isomorphism of algebras $U_{1}\rightarrow U({\mathfrak{g}})$ given by
$\overline{E_{i}}\mapsto e_{\alpha_{1}}$, $\overline{F}_{i}\mapsto
f_{\alpha_{i}}$, $\overline{(K_{i};0)_{q}}\mapsto h_{\alpha_{i}}$. Here
${\mathfrak{g}}={\mathfrak{gl}}_{S}(n)_{\mathbb{K}}$ is defined over
${\mathbb{K}}$ as in Remark 2.1.

###  6.1. $\mathbf{A}$-forms of triangular decompositions

For any subspace $W\subseteq U$ we define $W_{\mathbf{A}}=W\cap
U_{\mathbf{A}}$ and
$\overline{W}={\mathbb{K}}\otimes_{\mathbf{A}}W_{\mathbf{A}}\subset U_{1}$. By
[HK02, Proposition 3.3.3] the multiplication map yields an isomorphism

(6.1) $\displaystyle
U^{-}_{\mathbf{A}}\otimes_{\mathbf{A}}U^{0}_{\mathbf{A}}\otimes_{\mathbf{A}}U^{+}_{\mathbf{A}}\cong
U_{\mathbf{A}}.$

We aim to establish a similar $\mathbf{A}$-form of the triangular
decomposition of ${\mathcal{B}}_{\mathbf{c}}$ in Corollary 4.7. To this end,
the main ingredient is provided by the following Lemma, which is well known,
but we include a proof for the convenience of the reader. Recall the
definition of the PBW-monomials $F_{\mathcal{J}}$ for
${\mathcal{J}}=(j_{1},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{N(N-1)}$ from
(4.3).

######  Lemma 6.1.

The algebra $U^{-}_{\mathbf{A}}$ is a free $\mathbf{A}$-module with basis
$\\{F_{\mathcal{J}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}\\}$.

###### Proof.

Let
$V^{-}_{\mathbf{A}}=\oplus_{{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}}\mathbf{A}F_{\mathcal{J}}$
denote the free $\mathbf{A}$-submodule of $U^{-}$ with basis
$\\{F_{\mathcal{J}}\,|\,{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}\\}$. As the
Lusztig operators $T_{i}$ for $i\in I$ leave $U_{\mathbf{A}}$ invariant, we
have $V^{-}_{\mathbf{A}}\subseteq U^{-}_{\mathbf{A}}$. Conversely,
$U^{-}_{\mathbf{A}}=U^{-}\cap U_{\mathbf{A}}$ is the $\mathbf{A}$-subalgebra
of $U_{\mathbf{A}}$ generated by all $F_{i}$ for $i\in I$. As $F_{i}\in
V_{\mathbf{A}}^{-}$ it remains to show that $V^{-}_{\mathbf{A}}$ is closed
under multiplication. Levedorskii and Soibelman showed that for all $1\leq
i<j\leq N(N-1)$ we have

$\displaystyle
F_{\beta_{i}}F_{\beta_{j}}-q^{(\beta_{i},\beta_{j})}F_{\beta_{j}}F_{\beta_{i}}=\sum_{{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}}\mathbf{z}_{\mathcal{J}}F_{\mathcal{J}}$

where $\mathbf{z}_{\mathcal{J}}\in\mathbb{Q}[q,q^{-1}]$ and
$\mathbf{z}_{\mathcal{J}}=0$ unless $j_{r}=0$ for $r\leq i$ and $r\geq j$; see
[CP93, Theorem 9.3] for a proof and [BG02, I.6.10] for a discussion of the
literature. An inductive argument using Levendorskii and Soibelman’s result
shows that $F_{\mathcal{J}}F_{{\mathcal{J}}^{\prime}}\in V^{-}_{\mathbf{A}}$
for all ${\mathcal{J}},{\mathcal{J}}^{\prime}\in{\mathbb{N}}_{0}^{N(N-1)}$. ∎

Recall the definition of the monomials $B_{\mathcal{J}}$ for
${\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}$ from (4.14). Define

(6.2)
$\displaystyle{\mathcal{B}}_{\mathbf{c},{\mathrm{rt}}}=\bigoplus_{{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}}kB_{\mathcal{J}}.$

By Proposition 4.5 the multiplication map
${\mathcal{B}}_{\mathbf{c},\mathrm{rt}}\otimes
U^{0}_{X}\otimes{\mathcal{M}}_{X}^{+}\rightarrow{\mathcal{B}}_{\mathbf{c}}$ is
a $k$-linear isomorphism. The following Lemma is a version of [Kol14, Theorem
10.7.(4)] where we replaced monomials in the generators $B_{i}$ by monomials
in the root vectors $B_{\beta_{i}}$.

######  Theorem 6.2.

Let $c_{1}\in\mathbf{A}\setminus\\{0\\}$. The multiplication map

(6.3)
$\displaystyle({\mathcal{B}}_{\mathbf{c},\mathrm{rt}})_{\mathbf{A}}\otimes_{\mathbf{A}}(U^{0}_{X})_{\mathbf{A}}\otimes_{\mathbf{A}}({\mathcal{M}}_{X}^{+})_{\mathbf{A}}\rightarrow({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}$

is an isomorphism of $\mathbf{A}$-modules. Moreover,
$\displaystyle({\mathcal{B}}_{\mathbf{c},\mathrm{rt}})_{\mathbf{A}}=\bigoplus_{{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}}\mathbf{A}B_{{\mathcal{J}}}$.

###### Proof.

By Proposition 4.5 and [Kol14, Lemma 10.6], to prove the first statement, it
suffices to show that the map (6.3) is surjective. Let
$b\in{\mathcal{B}}_{\mathbf{c}}$. By Proposition 4.5 we have
$b=\sum_{{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}}B_{\mathcal{J}}a_{\mathcal{J}}$
for some $a_{\mathcal{J}}\in U_{X}^{0}{\mathcal{M}}_{X}^{+}$. For any
$\mu=\sum_{i\in I}n_{i}\alpha_{i}\in Q^{+}$ define $|\mu|=\sum_{i\in I}n_{i}$
and for
${\mathcal{J}}=(j_{1},j_{2},\dots,j_{N(N-1)})\in{\mathbb{N}}_{0}^{N(N-1)}$
define $|{\mathcal{J}}|=|\sum_{i=1}^{N(N-1)}j_{i}\beta_{i}|$. Let
$m\in{\mathbb{N}}_{0}$ be maximal such that $a_{\mathcal{J}}\neq 0$ for some
${\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}$ with $|{\mathcal{J}}|=m$. By the
triangular decomposition (6.1) we can write

$\displaystyle
b=\sum_{{\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)},|{\mathcal{J}}|\leq
m}F_{\mathcal{J}}b_{\mathcal{J}}$

for some $b_{\mathcal{J}}\in(U^{0}U^{+})_{\mathbf{A}}$. As
$B_{1}-F_{1}\in(U^{0}U^{+})_{\mathbf{A}}$ one obtains
$a_{\mathcal{J}}=b_{\mathcal{J}}$ if $|{\mathcal{J}}|=m$. Hence
$a_{\mathcal{J}}\in(U^{0}_{X}{\mathcal{M}}_{X}^{+})_{\mathbf{A}}$ for all
${\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}$ with $|{\mathcal{J}}|=m$. By
induction on $m$ one obtains
$a_{\mathcal{J}}\in(U^{0}_{X}{\mathcal{M}}_{X}^{+})_{\mathbf{A}}$ for all
${\mathcal{J}}\in{\mathbb{N}}_{0}^{N(N-1)}$. The second statement of the
proposition also follows from the above argument. ∎

Theorem 6.2 can be reformulated in terms of the triangular decomposition of
${\mathcal{B}}_{\mathbf{c}}$ from Corollary 4.7.

######  Corollary 6.3.

Assume that $c_{1}\in\mathbf{A}$. The multiplication map

(6.4)
$\displaystyle({\mathcal{B}}_{\mathbf{c}}^{-})_{\mathbf{A}}\otimes_{\mathbf{A}}(U^{0}_{X})_{\mathbf{A}}\otimes_{\mathbf{A}}({\mathcal{B}}_{\mathbf{c}}^{+})_{\mathbf{A}}\rightarrow({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}$

is an isomorphism of $\mathbf{A}$-modules.

###  6.2. Specialisation of ${\mathcal{B}}_{\mathbf{c}}$

Recall from Equation (3.5) that

$\displaystyle T_{w_{X}}$ $\displaystyle(E_{1})=T_{2}\hbox
to10.00002pt{.\hss.\hss.}T_{N-2}T_{N-1}T_{N}T_{N-2}\dots T_{2}(E_{1})$
$\displaystyle=\big{[}[E_{2},\dots[E_{N-2},E_{N}]_{q^{-1}}\hbox
to10.00002pt{.\hss.\hss.}]_{q^{-1}},[E_{N-1},[E_{N-2},\hbox
to10.00002pt{.\hss.\hss.}[E_{2},E_{1}]_{q^{-1}}\hbox
to10.00002pt{.\hss.\hss.}]_{q^{-1}}]_{q^{-1}}\big{]}_{q^{-1}}.$

On the other hand Equation (2.24) gives us

$\displaystyle\theta(f_{1})=(-1)^{N+1}\big{[}[e_{2},[e_{3},\hbox
to10.00002pt{.\hss.\hss.}[e_{N-2},e_{N}]\hbox
to10.00002pt{.\hss.\hss.}]],[e_{N-1},[e_{N-2},\hbox
to10.00002pt{.\hss.\hss.}[e_{2},e_{1}]\hbox
to10.00002pt{.\hss.\hss.}]]\big{]}.$

Hence, the element $B_{1}=F_{1}-c_{1}T_{w_{X}}(E_{1})K_{1}^{-1}$ for
$c_{1}\in{\mathcal{A}}$ satisfies $\overline{B_{1}}=f_{1}+\theta(f_{1})$ if
and only if $c_{1}(1)=(-1)^{N}$. We call an element $c\in k={\mathbb{K}}(q)$
specialisable if $c\in\mathbf{A}$ and $c(1)=(-1)^{N}$.

######  Remark 6.4.

In [Kol14, Section 10.1] we called $c_{1}\in\mathbf{A}$ specialisable if
$c_{1}(1)=1$. However, in that paper the definition of the generators $B_{i}$
involved an additional coefficient $s(\alpha_{\tau(i)})$ which can be chosen
to be in $\\{\pm 1\\}$, see [BK15, Remark 3.1], and which we have subsumed in
the coefficient $c_{1}$ in the present paper.

In the following we let
${\mathfrak{k}}={\mathfrak{so}}(2N)^{\theta}\cong{\mathfrak{so}}(2N-1)$ be the
subalgebra of ${\mathfrak{g}}={\mathfrak{so}}(2N)$ considered in Section 2.3.
Following Remark 2.1 we consider ${\mathfrak{g}}$ and ${\mathfrak{k}}$ defined
over the field ${\mathbb{K}}$.

######  Theorem 6.5 ([Let99, Theorem 4.8], see also [Kol14, Theorem 10.8]).

If $c_{1}\in\mathbf{A}$ is specialisable then
$\overline{{\mathcal{B}}_{\mathbf{c}}}=U({\mathfrak{k}})$.

We will now discuss the specialisation of the root vectors $B_{\beta_{j}}$ for
$1\leq j\leq 2N-2$. Recall the definition (2.8) of the elements
$b_{\ell},d_{\ell}\in{\mathfrak{so}}(2N-1)$ for $\ell=1,\dots,N-1$.

######  Proposition 6.6.

Let $c_{1}\in\mathbf{A}$ be specialisable and $1\leq\ell\leq N-1$. Then we
have
$B_{\beta_{\ell}},B_{\beta_{N-1+\ell}}\in({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}$
and

$\displaystyle\overline{B_{\beta_{\ell}}}$ $\displaystyle=\eta(b_{\ell}),$
(6.5) $\displaystyle\overline{B_{\beta_{N-1+\ell}}}$
$\displaystyle=(-1)^{\ell}2\eta(d_{N-\ell}).$

###### Proof.

For specialisable $c_{1}$ we have seen in Remark 4.3 that
$\overline{B_{1}}=\eta(b_{1})$ and more generally
$\overline{B_{\beta_{\ell}}}=\eta(b_{\ell})$ for $\ell=1,\dots,N-1$. Moreover,
we have seen that
$\overline{B_{\beta_{N}}}=-\eta(f_{\gamma_{N-1}})=-\eta(2d_{N-1})$. By
Equation (4.6) and the third equation of (2.9) we have

$\displaystyle\overline{B_{\beta_{N+1}}}=-\eta([f_{\gamma_{N-2}},f_{\gamma_{N-1}}])=\eta(2d_{N-2}).$

Finally, for $2\leq k\leq N-2$ the element $T_{N}\dots T_{k+1}(F_{k})$
specialises to

$\displaystyle[[[\dots[f_{\alpha_{k}},f_{\alpha_{k+1}}],$
$\displaystyle\dots],f_{\alpha_{N-1}}],f_{\alpha_{N}}]=(-1)^{N-k+1}[E_{N,k}-E_{2N-k+1,N+1},f_{\alpha_{N}}]$
$\displaystyle=(-1)^{N-k+1}(E_{N+2,k}-E_{2N-k+1,N-1})$

in ${\mathfrak{so}}(2N)$. Hence, for $N+2\leq j\leq 2(N-1)$ we have

$\displaystyle\overline{B_{\beta_{j}}}$
$\displaystyle=(-1)^{N-k+1}\big{(}[E_{N+2,k}-E_{2N-k+1,N-1},\eta(b_{N-2})]\big{)}$
$\displaystyle=(-1)^{N-k+1}\eta\big{(}[E_{N+2,k-1}-E_{2N-k+1,N-2},E_{N-2,N}-E_{N,N+2}]\big{)}$
$\displaystyle=(-1)^{N-k+1}\eta(2d_{k-1})$

where $k=2N-j$. This translates into (6.5) for $j=(N-1)+\ell$ and
$k=N+1-\ell$. ∎

###  6.3. Specialisation of highest weight
${\mathcal{B}}_{\mathbf{c}}$-modules

Let $V$ be a highest weight ${\mathcal{B}}_{\mathbf{c}}$-module of highest
weight $\lambda\in P_{2N-1}$ with highest weight vector $v_{\lambda}\in V$. We
define

$\displaystyle
V_{\mathbf{A}}=({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}v_{\lambda}.$

The triangular decomposition (6.4) of
$({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}$ implies that
$V_{\mathbf{A}}=({\mathcal{B}}_{\mathbf{c}}^{-})_{\mathbf{A}}v_{\lambda}$. The
negative part $({\mathcal{B}}_{\mathbf{c}}^{-})_{\mathbf{A}}$ of
$({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}$ is a direct sum of
$P_{2N-1}$-weight spaces under the adjoint action of $U^{0}_{X}$. Moreover,
the weight spaces of $({\mathcal{B}}_{\mathbf{c}}^{-})_{\mathbf{A}}$ are
finite-dimensional, free $\mathbf{A}$-modules. Hence

$\displaystyle V_{\mathbf{A}}=\bigoplus_{\mu\leq\lambda}V_{\mathbf{A},\mu}$

is a direct sum of $P_{2N-1}$-weight spaces. The weight spaces
$V_{\mathbf{A},\mu}$ are finitely generated, torsion-free and hence free
$\mathbf{A}$-modules. By [Kol14, Lemma 10.6] we have

(6.6)
$\displaystyle\mathrm{rank}_{\mathbf{A}}(V_{\mathbf{A},\mu})=\dim_{k}(V_{\mu}).$

The tensor product $V^{1}:={\mathbb{K}}\otimes_{\mathbf{A}}V_{\mathbf{A}}$ is
called the specialisation of $V$ at $q=1$. For any $v\in V_{\mathbf{A}}$ we
write ${\overline{v}}=1\otimes v\in V^{1}$. The specialisation $V^{1}$ is a
module over the specialisation
${\mathbb{K}}\otimes_{\mathbf{A}}({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}=\overline{{\mathcal{B}}_{\mathbf{c}}}=U({\mathfrak{k}})$.
By construction we have $\overline{bv}={\overline{b}}\,{\overline{v}}$ for all
$b\in({\mathcal{B}}_{\mathbf{c}})_{\mathbf{A}}$ and $v\in V_{\mathbf{A}}$. For
any weight vector $v_{\mu}\in V_{\mathbf{A},\mu}$ and any $h\in
Q^{\vee}_{2N-1}$ we have

$\displaystyle\eta(h)\overline{v_{\mu}}=\overline{(K_{\eta(h)};0)_{q}}\,\overline{v_{\mu}}=\overline{\frac{q^{\mu(h)}-1}{q-1}v_{\mu}}=\mu(h)\overline{v_{\mu}}.$

and hence $V^{1}$ is a ${\mathfrak{k}}$-weight module. Proposition 6.6 implies
that $V^{1}$ is a highest weight module. Moreover, by (6.6) we have
$\dim_{k}(V_{\mu})=\dim_{\mathbb{K}}(V^{1}_{\mu})$ for all $\mu\in P_{2N-1}$.
We summarise the discussion in the following proposition.

######  Proposition 6.7.

Assume that $k={\mathbb{K}}(q)$ where ${\mathbb{K}}$ is a field of
characteristic $0$. Let $V$ be a highest weight
${\mathcal{B}}_{\mathbf{c}}$-module of highest weight $\lambda\in P_{2N-1}$.
Then the specialisation $V^{1}$ of $V$ at $q=1$ is a highest weight
${\mathfrak{k}}$-module of highest weight $\lambda$ and

$\displaystyle\dim_{k}(V_{\mu})=\dim_{\mathbb{K}}(V^{1}_{\mu})\qquad\mbox{for
all $\mu\in P_{2N-1}$.}$

We can apply Proposition 6.7 in particular to the simple
${\mathcal{B}}_{\mathbf{c}}$-module $L(\lambda)$ from Theorem 5.22. Moreover,
as in [Jan96, Theorem 5.15] we can work over any field $k$ of characteristic
$0$ with $q\in k$ transcendental over $\mathbb{Q}$. Recall the definition of
the submodules $\widetilde{N}(\lambda)$ and $\accentset{\approx}{N}(\lambda)$
over the Verma module $M(\lambda)$ for $\lambda\in P^{+}_{2N-1}$, and the
corresponding quotients $\widetilde{L}(\lambda)$ and
$\accentset{\approx}{L}(\lambda)$, respectively.

######  Corollary 6.8.

Assume that $k$ is a field of characteristic $0$ and $q\in k$ is
transcendental over $\mathbb{Q}$. Let $\lambda\in P^{+}_{2N-1}$. Then
$\widetilde{L}(\lambda)=\accentset{\approx}{L}(\lambda)=L(\lambda)$ and the
dimension of the weight spaces $L(\lambda)_{\mu}$ for $\mu\in P_{2N-1}$ are
given by Weyl’s character formula.

###### Proof.

Assume first that $k=\mathbb{Q}(q)$. Each of $\accentset{\approx}{L}(\lambda)$
and $L(\lambda)$ are quotients of
$\widetilde{L}(\lambda)=M(\lambda)/\widetilde{N}(\lambda)$. Hence the highest
weight vector $v_{\lambda}$ in each of $\widetilde{L}(\lambda)$,
$\accentset{\approx}{L}(\lambda)$ and $L(\lambda)$ satisfies
$B_{\beta_{N}}^{n_{N-1}+1}v_{\lambda}=0=F_{i+1}^{n_{i}+1}v_{\lambda}$ for
$i=1,\dots,N-2$ where $n_{i}=\lambda(h_{\gamma_{i}})$ and
$n_{N-1}=\lambda(h_{\gamma_{N-1}})$. This means that
$f_{\gamma_{i}}^{n_{i}+1}\overline{v_{\lambda}}=0$ for $i=1,\dots,N-1$ in each
of the three cases. Hence, each of $\widetilde{L}(\lambda)^{1}$,
$\accentset{\approx}{L}(\lambda)^{1}$ and $L(\lambda)^{1}$ are isomorphic to
the unique simple ${\mathfrak{k}}$-module of highest weight $\lambda$. By
Proposition 6.7 the surjective ${\mathcal{B}}_{\mathbf{c}}$-module
homomorphisms
$\widetilde{L}(\lambda)\twoheadrightarrow\accentset{\approx}{L}(\lambda)\twoheadrightarrow
L(\lambda)$ are isomorphisms. The result extends from $\mathbb{Q}(q)$ to $k$
as in [Jan96, Theorem 5.15]. ∎

We can also use Proposition 6.7 to show that every finite dimensional highest
weight ${\mathcal{B}}_{\mathbf{c}}$-module is simple.

######  Corollary 6.9.

Assume that $k$ is a field of characteristic $0$ and $q\in k$ is
transcendental over $\mathbb{Q}$. Let $V$ be a finite dimensional highest
weight ${\mathcal{B}}_{\mathbf{c}}$-module of highest weight $\lambda\in
P^{+}_{2N-1}$. Then $V\cong L(\lambda)$.

###### Proof.

Again we first consider the case $k={\mathbb{K}}(q)$. In this case, by
Proposition 6.7, the specialisation $V^{1}$ is a finite dimensional highest
weight ${\mathfrak{k}}$-module of highest weight $\lambda$. Hence $V^{1}\cong
L(\lambda)^{1}$. Hence Proposition 6.7 implies that
$\dim_{k}(V)=\dim_{k}(L(\lambda))$ and as $L(\lambda)$ is a quotient of $V$ we
get $V\cong L(\lambda)$. Again we can extend from $\mathbb{Q}(q)$ to a general
$k$ of characteristic zero. ∎

######  Remark 6.10.

Recall the definition of the dual $V^{\ast}$ of a finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-module $V$ from (5.3). As
${\mathfrak{k}}\cong{\mathfrak{so}}(2N-1)$ we have
$\dim(L(\lambda)_{\mu}^{1})=\dim(L(\lambda)_{-\mu})$ for all $\mu\in
P_{2N-1}$. Hence we have $L(\lambda)\cong L(\lambda)^{\ast}$ for all
$\lambda\in P^{+}_{2N-1}$ as they are simple
${\mathcal{B}}_{\mathbf{c}}$-modules with weight spaces of the same dimension.

With Corollary 6.9 and Remark 6.10 we can show that every finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-module is semisimple. This can be proved by a
word by word translation of the proof of [Jan96, Theorem 5.17].

######  Theorem 6.11.

Assume that $k$ is a field of characteristic $0$ and $q\in k$ is
transcendental over $\mathbb{Q}$. Then every finite dimensional
${\mathcal{B}}_{\mathbf{c}}$-module is semisimple.

######  Remark 6.12.

We compare Theorem 5.22 and the results of the present section with Watanabe’s
more general approach in [Wat21]. By Proposition 3.2 we may assume that
$c_{1}\in\mathbf{A}$ is specialisable. Watanabe works over the algebraic
closure $k=\overline{{\mathbb{C}}(q)}$ in order to guarantee that Letzter’s
Cartan subalgebra acts diagonalisably, see [Wat21, Section 3.1]. In the
specific case of the QSP-coideal subalgebra of type $DII$, the Cartan
subalgebra is generated by the elements $K_{i}^{\pm 1}$ for $i\in X$ which act
diagonalisably on finite-dimensional ${\mathcal{B}}_{\mathbf{c}}$-modules over
any field. Hence, in type $DII$, Watanabe’s approach also works over
$k={\mathbb{K}}(q)$ for any field ${\mathbb{K}}$ of characteristic zero. Set

$\displaystyle{\mathcal{X}}$
$\displaystyle=\\{B_{\beta_{\ell}},E_{\beta_{j}}\,|\,j=2N-1,\dots,N(N-1),\,\ell=1,\dots,N-1\\},$
$\displaystyle{\mathcal{Y}}$
$\displaystyle=\\{B_{\beta_{\ell}},\,|\,\ell=N,\dots,N(N-1)\\},$
$\displaystyle{\mathcal{W}}$ $\displaystyle=\\{(K_{i};0)_{q},K_{i}^{\pm
1}\,|\,i\in X\\}$

and let ${\mathcal{B}}_{{\mathcal{X}},\mathbf{A}}$,
${\mathcal{B}}_{{\mathcal{Y}},\mathbf{A}}$ and
${\mathcal{B}}_{{\mathcal{W}},\mathbf{A}}$ be the ${\mathcal{A}}$-subalgebras
of ${\mathcal{B}}_{\mathbf{c},\mathbf{A}}$ generated by ${\mathcal{X}}$,