# Spectrum of the $\overline{\partial}$-Laplace operator on zero forms for the
quantum quadric ${\mathcal{O}}_{q}(\textbf{Q}_{N})$

Fredy Díaz García
Mathematical Institute of Charles University, Prague, Czech Republic
E-mail address<EMAIL_ADDRESS>
###### Abstract

We study the Laplacian operator $\Delta_{\overline{\partial}}$ associated to a
Kähler structure $(\Omega^{(\bullet,\bullet)},\kappa)$ for the
Heckenberger–Kolb differential calculus of the quantum quadrics
${\mathcal{O}}_{q}(\textbf{Q}_{N})$, which is to say, the irreducible quantum
flag manifolds of types $B_{n}$ and $D_{n}$. We show that the eigenvalues of
$\Delta_{\overline{\partial}}$ on zero forms tend to infinity and have finite
multiplicity.

##  1 Introduction

Since the discovery of quantum groups in the 1980’s mathematicians have been
trying to fit them into Connes’ framework of noncommutative spin geometry. In
fact, the construction of spectral triples for the Drinfeld–Jimbo quantum
groups has been a very difficult problem in noncommutative geometry. One of
the best-studied spectral triples is that of the Podleś sphere [6] which
satisfies most of the Connes conditions for a noncommutative spin geometry, up
to some slight modifications. Other constructions have been made in this
direction, for example the equivariant isospectral Dirac operators for all the
Podleś spheres [4], for the quantum projective spaces [2], for the quantum
group ${\mathcal{O}}_{q}(SU(2))$ [5] and for all the compact quantum groups
[23]. In [7] a Dirac operator associated to Kähler structures is constructed
for the quantum projective spaces ${\mathcal{O}}_{q}(\mathbb{CP}^{n})$
generalizing the work of S. Majid in [20].

In recent years it has been shown that the quantum flag manifolds provide a
large family of well behaved quantum homogeneous spaces whose noncommutative
geometry is quite similar to the classical situation. It was shown by I.
Heckenberger and S. Kolb that the subclass of irreducible quantum flag
manifolds ${\mathcal{O}}_{q}(G/L_{S})$ admit a unique differential calculus
which is a $q$-deformation of their classical de Rham complex [12], [13]. It
is important to note that the existence of this differential calculus is an
extremely especial property of these quantum spaces and it is not the case for
all quantum spaces. Moreover the existence of this canonical deformation
allows us to construct Dirac operators and investigate their spectrum.

The problem of the spectrum of the Dirac operator has been attacked for some
cases. For example in [3] the spectrum for
${\mathcal{O}}_{q}(\mathbb{CP}^{2})$ was calculated. In [2] the authors showed
that the spectrum of the Dolbeault–Dirac operator for the quantum projective
spaces has exponential growth by relating a Casimir operator. In [18] the
authors gave a new approach for the construction of Dolbeault–Dirac operator
and quantum Clifford algebras for the irreducible quantum flag manifolds.
Later, it was shown in [22] that this Dolbeault–Dirac operator gives a
spectral triple for the quantum Lagrangian Grassmannian of rank 2 by giving a
suitable version of the Parthasarathy formula.

Recently, a different approach to constructing Dolbeault–Dirac operators has
appeared in [24] where the notion of noncommutative Kähler structure for a
differential calculus was introduced. In the same paper the author showed the
existence of a covariant Kähler structure for the Heckenberger–Kolb calculus
for the quantum projective spaces. Later this result was generalized in [21]
for all the irreducible quantum flag manifolds, for all but a finite number of
values $q$. The existence of such a structure allows us to construct metrics,
Hilbert space completions and to verify all the axioms of a spectral triple
except the compact resolvent condition, see [8] for a more detailed
discussion. This last axiom has turned out to be the most challenging one.
This problem was solved in [7] for the quantum projective spaces
${\mathcal{O}}_{q}(\mathbb{CP}^{n})$ by using the existence of the Kähler
structure and the multiplicity-free property of the space of anti-holomorphic
forms. Also the spectrum was calculated for the quantum quadric
${\mathcal{O}}_{q}(\textbf{Q}_{5})$ in [10] by using the quantum version of
the Gelfand–Bernstein–Gelfand resolution for quantized irreducible flag
manifolds [14].

In this paper we investigate the asymptotic behaviour of the Laplace operator
$\Delta_{\overline{\partial}}$ on the space of zero forms $\Omega^{(0,0)}$ for
the irreducible quantum flag manifolds of quadric type. Besides of the
existence of the Kähler form we make use of the FRT description of the quantum
group ${\mathcal{O}}_{q}(SO(N))$ as well as the fact that $\Omega^{(0,0)}$ is
multiplicity-free as a $U_{q}(\mathfrak{g})$-module to make explicit
calculations of the eigenvalues. It has been conjectured in [7] that the
Dolbeault–Dirac operator $D_{\overline{\partial}}$ allows us to construct a
spectral triple for all the irreducible quantum flag manifolds, here we take a
small but significant step towards proving this conjecture.

The paper is organized as follows: in §2 we recall some necessary definitions
and properties of the quantized enveloping algebras, differential calculi,
Kähler structures as well as irreducible quantum flag manifolds and Takeuchi’s
equivalence. We also recall the FRT description of the algebras
${\mathcal{O}}_{q}(SO(N))$.

In §3 we give the most important properties of the Heckenberger–Kolb calculus
for the irreducible quantum flag manifolds and a proof of the existence of a
Kähler structure for the case of quantum quadrics
${\mathcal{O}}_{q}(\textbf{Q}_{N})$ by using tools of representation theory.

In §4 we present the main result of the paper. It is shown by explicit
calculation that the eigenvalues of the Laplace operator
$\Delta_{\overline{\partial}}$ on zero forms tend to infinity, supporting the
conjecture that the asymptotic behaviour of the Dolbeault–Dirac operator
$D_{\overline{\partial}}$ satisfies the compact resolvent condition for all
the irreducible quantum flag manifolds.

##  2 Preliminaries

In this section we recall the necessary definitions of quantum enveloping
algebras, covariant differential calculi as well as complex and Kähler
structures. For a more detailed presentation see [17], [24], and [25]. We also
introduce the definition of quantum flag manifolds, Takeuchi’s equivalence and
the FRT presentation of the quantum coordinate algebra
${\mathcal{O}}_{q}(SO(N))$.

###  2.1 Drinfeld–Jimbo quantum groups

Let $\mathfrak{g}$ a finite dimensional semisimple Lie algebra of rank $r$.
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra of
rank $r$. We fix a Cartan subalgebra $\mathfrak{h}$ and choose a set of simple
roots $\Pi=\\{\alpha_{1},\dots,\alpha_{r}\\}$ for the corresponding root
system in the linear dual of $\mathfrak{g}$. We denote by $(\cdot,\cdot)$ the
symmetric bilinear form induced on $\mathfrak{h}^{*}$ by the Killing form of
$\mathfrak{g}$, normalised so that any shortest simple root $\alpha_{i}$
satisfies $(\alpha_{i},\alpha_{i})=2$. The Cartan matrix $(a_{ij})$ of
$\mathfrak{g}$ is defined by
$a_{ij}:=\big{(}\alpha_{i}^{\vee},\alpha_{j}\big{)}$, where
$\alpha_{i}^{\vee}:=2\alpha_{i}/(\alpha_{i},\alpha_{i})$ is the _coroot_ of
$\alpha_{i}$.

Throughout the paper $q$ stands for a real number in the interval
$(1,+\infty)$ and denote $q_{i}:=q^{(\alpha_{i},\alpha_{i})/2}$. The
Drinfeld–Jimbo quantised enveloping algebra $U_{q}(\mathfrak{g})$ is the
noncommutative associative algebra generated by the elements
$E_{i},F_{i},K_{i}$, and $K^{-1}_{i}$, for $i=1,\ldots,r$, with the relations

$\displaystyle K_{i}E_{j}=q_{i}^{a_{ij}}E_{j}K_{i},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
K_{i}F_{j}=q_{i}^{-a_{ij}}F_{j}K_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
K_{i}K_{j}=K_{j}K_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ K_{i}K_{i}^{-1}=K_{i}^{-1}K_{i}=1,$
$\displaystyle
E_{i}F_{j}-F_{j}E_{i}=\delta_{ij}\frac{K_{i}-K^{-1}_{i}}{q_{i}-q_{i}^{-1}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $

and the quantum Serre relations which we omit (see [17, §6.1.2] for details).

A Hopf algebra structure on $U_{q}(\mathfrak{g})$ is defined by

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

A Hopf $\ast$-algebra structure is given by

$\displaystyle K_{i}^{*}=K_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
E^{*}_{i}=K_{i}F_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
F^{*}_{i}=E_{i}K_{i}^{-1}.$

Let $\\{\varpi_{1},\ldots,\varpi_{r}\\}$ denote the corresponding set of
_fundamental weights_ of $\mathfrak{g}$ which is the dual basis of the simple
coroots $\\{\alpha_{i}^{\vee},\ldots,\alpha_{r}^{\vee}\\}$:

$\displaystyle(\alpha_{i}^{\vee},\varpi_{j})=\delta_{ij}.$

Let $\mathcal{P}$ be the weight lattice of $\mathfrak{g}$ and
$\mathcal{P}^{+}$ the set of _dominant integral weights_ which are the
$\mathbb{Z}$-span and $\mathbb{Z}_{\geq 0}$-span of the fundamental weights,
respectively. For each $\mu\in\mathcal{P}^{+}$ there exists an irreducible
finite dimensional $U_{q}(\mathfrak{g})$-module $V_{\mu}$ uniquely determined
by the existence of a vector $\nu_{\mu}$ which is called the _highest weight
vector_ , satisfying

$\displaystyle
K_{i}\triangleright\nu_{\mu}=q^{(\alpha_{i},\mu)}\nu_{\mu},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ E_{i}\triangleright\nu_{\mu}=0.$

Moreover $\nu_{\mu}$ is unique up to scalar multiple. A finite direct sum of
such representations is called a _type-1 representation_. A vector $\nu$ is
called a _weight vector_ of weight $\mathrm{wt}(\nu)\in\mathcal{P}$ if

$\displaystyle K_{i}\triangleright\nu=q^{(\alpha_{i},\mathrm{wt}(\nu))}\nu.$

In this paper we use the fact that $U_{q}(\mathfrak{g})$ has invertible
antipode and therefore we have an equivalence between
$\mbox{}_{U_{q}(\mathfrak{g})}$Mod and $\mathrm{Mod}_{U_{q}(\mathfrak{g})}$
induced by the antipode.

###  2.2 Differential calculi

A differential calculus
$\big{(}\Omega^{\bullet}\simeq\bigoplus_{k\in\mathbb{N}_{0}}\Omega^{k},\mbox{d}\big{)}$
is a differential graded algebra (dg-algebra) which is generated in degree $0$
as a dg-algebra, that is to say, it is generated as an algebra by the elements
$a,\mbox{d}b$, for $a,b\in\Omega^{0}$. We call an element
$\omega\in\Omega^{\bullet}$ a _form_ , and if $\omega\in\Omega^{k}$, for some
$k\in\mathbb{N}_{0}$, then $\omega$ is said to be _homogeneous_ of degree
$|\omega|:=k$. The product of two forms $\omega,\nu\in\Omega^{\bullet}$ is
usually denoted by $\omega\wedge\nu$, unless one of the forms is of degree
$0$, where we just denote the product by juxtaposition.

For a given algebra $B$, a _differential calculus over_ $B$ is a differential
calculus such that $\Omega^{0}=B$. Note that for a differential calculus over
$B$, each $\Omega^{k}$ is a $B$-bimodule. A differential calculus is said to
have _total degree_ $m\in\mathbb{N}_{0}$, if $\Omega^{m}\neq 0$, and
$\Omega^{k}=0$, for every $k>m$.

A differential $*$-calculus over a $*$-algebra $B$ is a differential calculus
over $B$ such that the $*$-map of $B$ extends to a (necessarily unique)
conjugate-linear involutive map $*:\Omega^{\bullet}\to\Omega^{\bullet}$
satisfying $\mbox{d}(\omega^{*})=(\mbox{d}\omega)^{*}$, and

$\displaystyle\big{(}\omega\wedge\nu\big{)}^{*}=(-1)^{kl}\nu^{*}\wedge\omega^{*},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ for all }\ \
\omega\in\Omega^{k},\,\nu\in\Omega^{l}.$

We say that $\omega\in\Omega^{\bullet}$ is _closed_ if $\mbox{d}\omega=0$, and
_real_ if $\omega^{*}=\omega$.

We now recall the definition of a complex structure as introduced in [25].
This abstracts the properties of the de Rham complex for a classical complex
manifold [16].

######  Definition 2.1.

An almost complex structure $\Omega^{(\bullet,\bullet)}$ for a differential
$*$-calculus $(\Omega^{\bullet},\mbox{d})$ is an $\mathbb{N}^{2}_{0}$-algebra
grading $\bigoplus_{(a,b)\in\mathbb{N}^{2}_{0}}\Omega^{(a,b)}$ for
$\Omega^{\bullet}$ such that, for all $(a,b)\in\mathbb{N}^{2}_{0}$ the
following hold:

  1. (i)

$\Omega^{k}=\bigoplus_{a+b=k}\Omega^{(a,b)}$,

  2. (ii)

$\big{(}\Omega^{(a,b)}\big{)}^{*}=\Omega^{(b,a)}$.

We call an element of $\Omega^{(a,b)}$ an $(a,b)$-form. For the associated
pair of projections
$\mathrm{proj}_{\Omega^{(a+1,b)}}:\Omega^{a+b+1}\to\Omega^{(a+1,b)}$ and
$\mathrm{proj}_{\Omega^{(a,b+1)}}:\Omega^{a+b+1}\to\Omega^{(a,b+1)}$, we
denote

$\displaystyle\partial|_{\Omega^{(a,b)}}:=\mathrm{proj}_{\Omega^{(a+1,b)}}\circ\mbox{d},$
$\displaystyle\overline{\partial}|_{\Omega^{(a,b)}}:=\mathrm{proj}_{\Omega^{(a,b+1)}}\circ\mbox{d}.$

A _complex structure_ is an almost complex structure which satisfies

$\displaystyle\mbox{d}\Omega^{(a,b)}\subseteq\Omega^{(a+1,b)}\oplus\Omega^{(a,b+1)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ for all }\
(a,b)\in\mathbb{N}^{2}_{0}.$ (1)

For a complex structure, (1) implies the identities

$\displaystyle\mbox{d}=\partial+\overline{\partial},$
$\displaystyle\overline{\partial}\circ\partial=-\,\partial\circ\overline{\partial},$
$\displaystyle\partial^{2}=\overline{\partial}^{2}=0.$

In particular,
$\big{(}\bigoplus_{(a,b)\in\mathbb{N}^{2}_{0}}\Omega^{(a,b)},\partial,\overline{\partial}\big{)}$
is a double complex, which we call the Dolbeault double complex of
$\Omega^{(\bullet,\bullet)}$. Moreover, it is easily seen that both $\partial$
and $\overline{\partial}$ satisfy the graded Leibniz rule, and

$\displaystyle\partial(\omega^{*})=\big{(}\overline{\partial}\omega\big{)}^{*},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{\partial}(\omega^{*})=\big{(}\partial\omega\big{)}^{*},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \omega\in\Omega^{\bullet}.$ (2)

####  2.2.1 Hermitian and Kähler structures

We now present the definition of an Hermitian structure, as introduced in
[24], as well as a Kähler structure, introduced in §7 of the same paper.

######  Definition 2.2.

An Hermitian structure $(\Omega^{(\bullet,\bullet)},\sigma)$ for a
differential $*$-calculus $\Omega^{\bullet}$ over $B$ of even total degree
$2n$ is a pair consisting of a complex structure $\Omega^{(\bullet,\bullet)}$
and a central real $(1,1)$-form $\sigma$, called the Hermitian form, such
that, with respect to the Lefschetz operator

$\displaystyle L:\Omega^{\bullet}\to\Omega^{\bullet},$
$\displaystyle\omega\mapsto\sigma\wedge\omega,$

isomorphisms are given by

$\displaystyle L^{n-k}:\Omega^{k}\to\Omega^{2n-k},$ $\displaystyle\text{ for
all }\ \ k=0,\dots,n-1.$

For $L$ the Lefschetz operator of an Hermitian structure, we denote

$\displaystyle
P^{(a,b)}:=\begin{cases}\\{\alpha\in\Omega^{(a,b)}\,|\,L^{n-a-b+1}(\alpha)=0\\},&\text{
\leavevmode\nobreak\ if }a+b\leq n,\\\ 0,&\text{ \leavevmode\nobreak\ if
}a+b>n.\end{cases}$

Moreover, we denote $P^{k}:=\bigoplus_{a+b=k}P^{(a,b)}$, and
$P^{\bullet}:=\bigoplus_{k\in\mathbb{N}_{0}}P^{k}$. An element of
$P^{\bullet}$ is called a primitive form.

An important consequence of the existence of the Lefschetz operator is the
Lefschetz decomposition of the differential $*$-calculus, which we now recall
below. For a proof see [24, §4.1].

######  Proposition 2.3 (Lefschetz decomposition).

For $L$ the Lefschetz operator of an Hermitian structure on a differential
$*$-calculus $\Omega^{\bullet}$, a $B$-bimodule decomposition of $\Omega^{k}$,
for all $k\in\mathbb{N}_{0}$, is given by

$\displaystyle\Omega^{k}\simeq\bigoplus_{j\geq 0}L^{j}\big{(}P^{k-2j}\big{)}.$

In classical Hermitian geometry, the Hodge map of an Hermitian metric is
related to the associated Lefschetz decomposition through the well-known Weil
formula [16, §1.2]. In the noncommutative setting, we take the direct
generalisation of the Weil formula for our definition of the Hodge map, and
build upon this to define an Hermitian metric.

######  Definition 2.4.

The Hodge map associated to an Hermitian structure
$\big{(}\Omega^{(\bullet,\bullet)},\sigma\big{)}$ is the $B$-bimodule map
$\ast_{\sigma}:\Omega^{\bullet}\to\Omega^{\bullet}$ satisfying, for any
$j\in\mathbb{N}_{0}$,

$\displaystyle\ast_{\sigma}\big{(}L^{j}(\omega)\big{)}=(-1)^{\frac{k(k+1)}{2}}\mathbf{i}^{a-b}\frac{j!}{(n-j-k)!}L^{n-j-k}(\omega),$
$\displaystyle\omega\in P^{(a,b)}\subseteq P^{k=a+b},$

where $\mathbf{i}^{2}=-1$.

The metric associated to a Hermitian structure
$(\Omega^{(\bullet,\bullet)},\sigma)$ is the map
$g_{\sigma}:\Omega^{\bullet}\times\Omega^{\bullet}\rightarrow B$, for which
$g(\Omega^{k},\Omega^{l})=0$ for $k\neq l$ and

$g_{\sigma}(\omega,\nu):=*_{\sigma}(*_{\sigma}(\omega^{*})\wedge\nu),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{ if }\ \omega,\nu\in\Omega^{k}.$

######  Definition 2.5.

We say that an Hermitian structure $(\Omega^{(\bullet,\bullet)},\sigma)$ is
_positive definite_ if the associated metric $g_{\sigma}$ satisfies

$\displaystyle g_{\sigma}(\omega,\omega)\in
B_{>0}:=\Big{\\{}\sum_{i=1}^{m}\lambda_{i}b^{*}b_{i}:b_{i}\in B,\ \
\lambda_{i}\in\mathbb{R}_{>0},\ m\in\mathbb{N}\Big{\\}},\ \ \mbox{for all non-
zero}\ \ \omega\in\Omega^{\bullet}.$

In this case we say that $\sigma$ is a positive definite Hermitian form.

######  Definition 2.6.

A Kähler structure for a differential $*$-calculus is an Hermitian structure
$(\Omega^{(\bullet,\bullet)},\kappa)$ such that $\kappa$ is closed, which is
to say, $\mbox{d}\kappa=0$. We call such a form $\kappa$ a Kähler form.

###  2.3 Irreducible quantum flag manifolds

Let $V$ be a finite-dimensional $U_{q}(\mathfrak{g})$-module, $v\in V$, and
$f\in V^{*}$, the linear dual of $V$. Consider the function
$c^{\textrm{\tiny$V$}}_{f,v}:U_{q}(\mathfrak{g})\to\mathbb{C}$ defined by
$c^{\textrm{\tiny$V$}}_{f,v}(X):=f\big{(}X(v)\big{)}$. The coordinate ring of
$V$ is the subspace

$\displaystyle
C(V):=\text{span}_{\mathbb{C}}\\!\left\\{c^{\textrm{\tiny$V$}}_{f,v}\,|\,v\in
V,\,f\in V^{*}\right\\}\subseteq U_{q}(\mathfrak{g})^{*}.$

It is easy to check that $C(V)$ is contained in $U_{q}(\mathfrak{g})^{\circ}$,
the Hopf dual of $U_{q}(\mathfrak{g})$, and moreover that a Hopf subalgebra of
$U_{q}(\mathfrak{g})^{\circ}$ is given by

$\displaystyle\mathcal{O}_{q}(G):=\bigoplus_{\lambda\in\mathcal{P}^{+}}C(V_{\lambda}).$
(3)

We call ${\mathcal{O}}_{q}(G)$ the quantum coordinate algebra of $G$, where
$G$ is the compact, simply-connected, simple Lie group having $\mathfrak{g}$
as its complexified Lie algebra. Moreover, we call the decomposition of
${\mathcal{O}}_{q}(G)$ given in (3) the _Peter–Weyl decomposition_ of
${\mathcal{O}}_{q}(G)$.

For $S$ a subset of simple roots, consider the Hopf subalgebra of
$U_{q}(\mathfrak{g})$ given by

$\displaystyle
U_{q}(\mathfrak{l}_{S}):=\big{<}K_{i},E_{j},F_{j}\,|\,i=1,\ldots,r;j\in
S\big{>}.$

There are obvious notions of weight vectors, highest weight vectors as well as
type-1 representations. It can be shown that every irreducible finite
dimensional type-1 $U_{q}(\mathfrak{l}_{S})$-module admits a highest weight
vector which is unique up to scalar multiple. Moreover, the weight of this
highest weight vector determines the module up to isomorphism and the there is
a correspondence between irreducible type-1 $U_{q}(\mathfrak{l}_{S})$ modules
and weights in the sub-lattice

$\displaystyle\mathcal{P}^{+}_{S}+\mathcal{P}_{S^{c}}\subseteq\mathcal{P},$

where $S^{c}=\Pi\setminus S$ and

$\displaystyle\mathcal{P}^{+}_{S}$
$\displaystyle:=\Big{\\{}\lambda\in\mathcal{P}:\lambda=\sum_{j\in
S}\lambda_{j}\varpi_{j},\ \ \lambda_{j}\in\mathbb{Z}_{\geq 0}\Big{\\}},$
$\displaystyle\mathcal{P}_{S^{c}}$
$\displaystyle:=\Big{\\{}\lambda\in\mathcal{P}:\lambda=\sum_{j\in
S^{c}}\lambda_{j}\varpi_{j}\ \ \lambda_{j}\in\mathbb{Z}\Big{\\}}.$

Just as for ${\mathcal{O}}_{q}(G)$, we can construct the type-1 dual of
$U_{q}(\mathfrak{l}_{S})$ using matrix coefficients, this Hopf algebra will be
denoted by ${\mathcal{O}}_{q}(L_{S})$. Restriction of domains gives us a
surjective Hopf $\ast$-algebra map
$\pi_{S}:{\mathcal{O}}_{q}(G)\rightarrow{\mathcal{O}}(L_{S})$, dual to the
inclusion of $U_{q}(\mathfrak{l}_{S})$ in $U_{q}(\mathfrak{g})$. The quantum
flag manifold associated to S is the quantum homogeneous space associated to
$\pi_{S}$

$\displaystyle{\mathcal{O}}_{q}\big{(}G/L_{S}\big{)}$
$\displaystyle:={\mathcal{O}}_{q}(G)^{co({\mathcal{O}}_{q}(L_{S}))}$
$\displaystyle\ =\\{b\in{\mathcal{O}}_{q}(G):b_{(1)}\langle
X,b_{(2)}\rangle=\varepsilon(X)b\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for all}\ \ X\in
U_{q}(\mathfrak{l}_{S})\\}.$

We say that the quantum flag manifold is irreducible if
$S=\Pi\setminus\\{\alpha_{x}\\}$ where $\alpha_{x}$ has coefficient $1$ in the
expansion of the highest rood of $\mathfrak{g}$. In the table 1 we give a
diagrammatic presentation of the set of simple roots defining the irreducible
quantum flag manifold, where the node corresponding to $\alpha_{x}$ is the
coloured one.

Table 1: Irreducible Quantum Flag Manifolds with nodes numbered according to [15, §11.4]. $A_{n}$ |  | ${\mathcal{O}}_{q}(\text{Gr}_{s,n+1})$ | quantum Grassmannian
---|---|---|---
$B_{n}$ |  | ${\mathcal{O}}_{q}(\mathbf{Q}_{2n+1})$ | odd quantum quadric
$C_{n}$ |  | ${\mathcal{O}}_{q}(\mathbf{L}_{n})$ | quantum Lagrangian Grassmannian
$D_{n}$ |  | ${\mathcal{O}}_{q}(\mathbf{Q}_{2n})$ | even quantum quadric
$D_{n}$ |  | ${\mathcal{O}}_{q}(\textbf{S}_{n})$ | quantum spinor variety
$E_{6}$ |  | ${\mathcal{O}}_{q}(\mathbb{OP}^{2})$ | quantum Caley plane
$E_{7}$ |  | ${\mathcal{O}}_{q}(\textrm{F})$ | quantum Freudenthal variety

####  2.3.1 Takeuchi’s equivalence

In the following we denote $\mbox{}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}_{q}(G/L_{S})}\mathrm{Mod}_{0}$ the
category whose objects are relative Hopf modules $\mathcal{F}$ such that
$\mathcal{F}{\mathcal{O}}_{q}(G/L_{S})^{+}={\mathcal{O}}_{q}(G/L_{S})^{+}\mathcal{F}$
and morphisms are left ${\mathcal{O}}_{q}(G)$-comodule,
${\mathcal{O}}(G/L_{S})$-bimodule maps. Let
$\mbox{}^{{\mathcal{O}}(L_{S})}\mathrm{Mod}$ be the category whose objects are
${\mathcal{O}}(L_{S})$-comodules and morphisms are
${\mathcal{O}}_{q}(L_{S})$-comodule maps. By _Takeuchi’s equivalence_ , see
for example [26], [13, §2.2], an equivalence of the categories
$\mbox{}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}_{q}(G/L_{S})}\mathrm{Mod}_{0}$ and
$\mbox{}^{{\mathcal{O}}_{q}(L_{S})}\mathrm{Mod}$ is given by the functors

$\displaystyle\Phi:\mbox{}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}_{q}(G/L_{S})}\mathrm{Mod}_{0}\rightarrow\mbox{}^{{\mathcal{O}}_{q}(L_{S})}\mathrm{Mod},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\Phi(\Gamma)=\Gamma/{\mathcal{O}}_{q}(G/L_{S})^{+}\Gamma,$
$\displaystyle\Psi:\mbox{}^{{\mathcal{O}}_{q}(L_{S})}\mathrm{Mod}\rightarrow\mbox{}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}(G/L_{S})}\mathrm{Mod}_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Psi(V)={\mathcal{O}}_{q}(G)\square_{{\mathcal{O}}_{q}(L_{S})}V.$

Here for $\Gamma\in\mbox{}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}(G/L_{S})}\mathrm{Mod}_{0}$ the left
${\mathcal{O}}(L_{S})$-comodule structure on
$\Gamma/{\mathcal{O}}_{q}(G/L_{S})^{+}\Gamma$ is induced by the left
${\mathcal{O}}_{q}(G)$-comodule structure of $\Gamma$. The left
${\mathcal{O}}_{q}(G)$-comodule and ${\mathcal{O}}(G/L_{S})$-bimodule
structures on the cotensor
${\mathcal{O}}_{q}(G)\square_{{\mathcal{O}}_{q}(L_{S})}V$ are given by the
coproduct and ${\mathcal{O}}(G/L_{S})$-bimodule structure of
${\mathcal{O}}_{q}(G)$, respectively. Takeuchi’s equivalence states that
natural isomorphisms are given by

$\displaystyle\mathrm{C}:\Phi\circ\Psi(V)\rightarrow V,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\Big{[}\sum_{i}a^{i}\otimes
v^{i}\Big{]}\mapsto\sum_{i}\varepsilon(a^{i})v^{i},$
$\displaystyle\mathrm{U}:\mathcal{F}\rightarrow\Psi\circ\Phi(\mathcal{F}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ f\mapsto f_{(-1)}\otimes[f_{(0)}].$

###  2.4 The Hopf algebras ${\mathcal{O}}_{q}(SO(N))$

In this subsection we give a brief description of the FRT presentation of the
Hopf algebras ${\mathcal{O}}_{q}(SO(N))$, see [17] for details. We first give
a bit of notation. We denote $i^{\prime}=N+1-i$. Let $N=2n$ if $N$ is even and
$N=2n+1$ if $N$ is odd. We define

$\displaystyle\rho_{i}=\frac{N}{2}-i\ \ \text{if}\ \ i<i^{\prime},\ \
\rho_{i}^{\prime}=-\rho_{i}\ \ \text{if}\ \ i\leq i^{\prime}.$

Recall that the R-matrix associated to the quantum ${\mathcal{O}}_{q}(SO(N))$
is given by

$R^{ij}_{mn}=q^{\delta_{ij}-\delta_{ij^{\prime}}}\delta_{im}\delta_{jn}+(q-q^{-1})\theta(i-m)(\delta_{jm}\delta_{in}-\delta_{ji^{\prime}}\delta_{mn^{\prime}}q^{-\rho_{j}-\rho_{m}}).$
(4)

Let $U_{q}(\mathfrak{so}_{N})^{\circ}$ denote the dual Hopf algebra of
$U_{q}(\mathfrak{so}_{N})$. Then ${\mathcal{O}}_{q}(SO(N))\subset
U_{q}(\mathfrak{so}_{N})^{\circ}$ is the Hopf subalgebra generated by matrix
coefficients of the $N$-dimensional irreducible
$U_{q}(\mathfrak{so}_{N})$-representation of highest weight $\varpi_{1}$. We
denote the matrix coefficients by $u^{i}_{j}$, $i,j=1,\ldots,N$, they satisfy
the relations

$\displaystyle\sum_{k,l=1}^{N}R^{ij}_{kl}u^{k}_{m}u^{l}_{n}=\sum_{k,l=1}^{N}u^{j}_{k}u^{i}_{l}R^{lk}_{mn},\
\ i,j,m,n=1,\ldots,N,$ (5) $\displaystyle\mathcal{D}_{q}=1,\ \
\sum_{i,j,k=1}^{N}C^{i}_{j}(C^{-1})^{k}_{m}u^{n}_{i}u^{k}_{j}=\sum_{i,j,k=1}^{N}C^{n}_{i}(C^{-1})^{j}_{k}u^{j}_{i}u^{k}_{m}=\delta_{mn},\
\ m,n=1,\ldots,N,$ (6)

where the C-matrix coefficients and the quantum determinant are given as in
[17, §9.3]. The Hopf $\ast$-algebra structure on ${\mathcal{O}}_{q}(SO(N))$ is
determined on generators by

$\displaystyle\Delta(u^{i}_{j})=\sum_{k}u^{i}_{k}\otimes
u^{k}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\varepsilon(u^{i}_{j})=\delta_{ij},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
S(u^{i}_{j})=q^{\rho_{j}-\rho_{i}}u^{j^{\prime}}_{i^{\prime}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ u^{i*}_{j}=S(u^{j}_{i}).$

It follows from [17, §9.4] that there exists a dual pairing
$\langle\cdot,\cdot\rangle$ of the Hopf algebras $U_{q}(\mathfrak{so}_{N})$
and ${\mathcal{O}}_{q}(SO(N))$ which in turn induces a left action
$U_{q}(\mathfrak{so}_{N})\otimes{\mathcal{O}}_{q}(SO(N))\ni X\otimes a\mapsto
X\triangleright a\in{\mathcal{O}}_{q}(SO(N))$ and a right action
${\mathcal{O}}_{q}(SO(N))\otimes U_{q}(\mathfrak{so}_{N})\ni a\otimes X\mapsto
a\triangleleft X\in{\mathcal{O}}_{q}(SO(N))$ such that
${\mathcal{O}}_{q}(SO(N))$ is a left and a right
$U_{q}(\mathfrak{so}_{N})$-module $\ast$-algebra. The actions on generators
are given as follows:

$\displaystyle E_{j}\triangleright u^{i}_{j}=u^{i}_{j+1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ E_{j}\triangleright
u^{i}_{(j+1)^{\prime}}=-u^{i}_{j^{\prime}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ F_{j}\triangleright
u^{i}_{j+1}=u^{i}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ F_{j}\triangleright
u^{i}_{j^{\prime}}=-u^{i}_{(j+1)^{\prime}},\ \ j<n,$ (7) $\displaystyle
E_{n}\triangleright
u^{i}_{n}=[2]^{1/2}_{q_{2}}u^{i}_{n+1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ E_{n}\triangleright
u^{i}_{n+1}=-q_{2}[2]^{1/2}_{q_{2}}u^{i}_{n+2},$ (8) $\displaystyle
F_{n}\triangleright
u^{i}_{n+1}=[2]^{1/2}_{q_{2}}u^{i}_{n},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ F_{n}\triangleright
u^{i}_{n+2}=-q^{-1}_{2}[2]^{1/2}_{q_{2}}u^{i}_{n+1},$ (9)

for $i=1,\ldots,N$ and $N=2n+1$. For $N=2n$ we have

$\displaystyle E_{j}\triangleright u^{i}_{j}=u^{i}_{j+1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ E_{j}\triangleright
u^{i}_{(j+1)^{\prime}}=-u^{i}_{j^{\prime}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ F_{j}\triangleright
u^{i}_{j+1}=u^{i}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ F_{j}\triangleright
u^{i}_{j^{\prime}}=u^{i}_{(j+1)^{\prime}}\ \ i<n,$ (10) $\displaystyle
E_{n}\triangleright u^{i}_{n}=-u^{i}_{n+2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ E_{n}\triangleright
u^{i}_{n-1}=u^{i}_{n+1},$ (11) $\displaystyle F_{n}\triangleright
u^{i}_{n+2}=-u^{i}_{n},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ F_{n}\triangleright u^{i}_{n+1}=u^{i}_{n-1}.$ (12)

The corresponding right actions for $N=2n+1$ are given by

$\displaystyle u^{i}_{j}\triangleleft F_{i}=u^{i+1}_{j},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ u^{(i+1)^{\prime}}_{j}\triangleleft
F_{i}=-u^{i^{\prime}}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ u^{i+1}_{j}\triangleleft
E_{i}=u^{i}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
u^{i^{\prime}}_{j}\triangleleft E_{i}=-u^{(i+1)^{\prime}}_{j},\ \ i<n,$ (13)
$\displaystyle u^{n}_{j}\triangleleft
F_{n}=[2]^{1/2}_{q_{2}}u^{n+1}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ u^{n+1}_{j}\triangleleft
F_{n}=-q_{2}[2]^{1/2}_{q_{2}}u^{n+2}_{j},$ (14) $\displaystyle
u^{n+1}_{j}\triangleleft E_{n}=[2]^{1/2}_{q_{2}}u^{n}_{j},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ u^{n+2}_{j}\triangleleft
E_{n}=-q^{-1}_{2}[2]^{1/2}_{q_{2}}u^{n+1}_{j},$ (15)

and for even $N=2n$

$\displaystyle u^{i}_{j}\triangleleft F_{i}=u^{i+1}_{j},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ u^{(i+1)^{\prime}}_{j}\triangleleft
F_{i}=-u^{i^{\prime}}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ u^{i+1}_{j}\triangleleft
E_{i}=u^{i}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
u^{i^{\prime}}_{j}\triangleleft E_{i}=u^{(i+1)^{\prime}}_{j},\ \ i<n,$ (16)
$\displaystyle u^{n}_{j}\triangleleft F_{n}=-u^{n+2}_{j},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ u^{n-1}_{j}\triangleleft
F_{n}=u^{n+1}_{j},$ (17) $\displaystyle u^{n+2}_{j}\triangleleft
E_{n}=-u^{n}_{j},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ u^{n+1}_{j}\triangleleft E_{n}=u^{n-1}_{j},$ (18)

for $i=1,\ldots,N$. The values $E_{j}\triangleright u^{k}_{l}$,
$F_{j}\triangleright u^{k}_{l}$, $u^{k}_{l}\triangleleft E_{j}$ and
$u^{k}_{l}\triangleleft F_{j}$ for all other cases are zero.

For the convenience of the reader, the next lemma collects the commutation
relations of the generators of ${\mathcal{O}}_{q}(SO(N))$ which will be used
in this paper.

######  Lemma 2.7.

If $u^{i}_{j},i,j=1,...,N$ are the generators of $\mathcal{O}_{q}(SO(N))$ then
the following commutation relations hold

$\displaystyle u^{i}_{1}u^{i}_{N}=q^{2}u^{i}_{N}u^{i}_{1},\ i\neq
i^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
(19) $\displaystyle u^{i}_{l}u^{i}_{k}=qu^{i}_{k}u^{i}_{l},\ l<k,\ i\neq
i^{\prime},$ (20) $\displaystyle u^{j}_{l}u^{i}_{k}=u^{i}_{k}u^{j}_{l},\ l<k,\
i<j,\ l\neq k^{\prime},\ i\neq j^{\prime},$ (21) $\displaystyle
u^{j}_{1}u^{i}_{N}=qu^{i}_{N}u^{j}_{1},\ i<j,\ i\neq j^{\prime},$ (22)
$\displaystyle
u^{i}_{1}u^{j}_{N}=qu^{j}_{N}u^{i}_{1}+(q^{2}-1)u^{i}_{N}u^{j}_{1},\ i<j,\
i\neq j^{\prime},$ (23) $\displaystyle
u^{i}_{k}u^{j}_{l}=u^{j}_{l}u^{i}_{k}-(q-q^{-1})u^{j}_{k}u^{i}_{l},\ i<j,\
k<l,\ i\neq j^{\prime},\ k\neq l^{\prime},$ (24) $\displaystyle
u^{i}_{k}u^{j}_{k}=qu^{j}_{k}u^{i}_{k},\ k\neq k^{\prime},\ i<j,\ i\neq
j^{\prime},$ (25)
$\displaystyle(u^{1}_{1}u^{2}_{a}-qu^{2}_{1}u^{1}_{a})(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})=q^{2}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})(u^{1}_{1}u^{2}_{a}-qu^{2}_{1}u^{1}_{a}),$
(26)
$\displaystyle(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})=q^{2}(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N}),$
(27)

where $a=2,\ldots,N-1$.

###### Proof.

Equations (19)-(24) follow directly from the R-matrix relations (5). For
instance, if we take $m=N$, $n=1$ and $i>j$ with $i\neq j^{\prime}$ in
relation (5) we obtain

$\displaystyle
u^{i}_{N}u^{j}_{1}+(q-q^{-1})u^{j}_{N}u^{i}_{1}=q^{-1}u^{j}_{1}u^{i}_{N},$

which yields equation (23). Equation (25) follows by choosing $k=l+1$ in (21)
and acting on both sides by the $E_{i}$’s or $F_{i}$’s from the left by using
formulas (7)-(12). We only give the proof of (27) since the proof of (26) is
analogous. By using (19)-(25) it can be show that
$u^{1}_{N}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})=q^{-1}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})u^{1}_{N}$.
If we act on both sides by $E_{1}$ from the left we get

$\displaystyle
u^{1}_{N}(u^{1}_{2}u^{2}_{N}-qu^{2}_{2}u^{1}_{N})=q^{-1}(u^{1}_{2}u^{2}_{N}-qu^{2}_{2}u^{1}_{N})u^{1}_{N}.$

Then the relation
$u^{1}_{N}(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})=q^{-1}(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})u^{1}_{N}$
can be obtained by acting from the left by the $E_{i}$’s according to
identities (7)-(12). The commutation relations

$\displaystyle
u^{1}_{1}(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})=q^{2}(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})u^{1}_{1},\quad\
u^{2}_{N}(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})=(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})u^{2}_{N},$
$\displaystyle
u^{2}_{1}(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})=(u^{1}_{a}u^{2}_{N}-qu^{2}_{a}u^{1}_{N})u^{2}_{1}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $

can be easily deduced from (19)-(25). Combining this results we have that (27)
holds for $a=2,\ldots,N-1$. ∎

##  3 The Heckenberger–Kolb calculi and the Kähler structure

In this section we recall important facts about Heckenberger–Kolb calculi for
irreducible quantum flag manifolds as well as the existence of a Kähler
structure for them. For more details we refer the reader to the seminal papers
[13] and [14] for the Heckenberger–Kolb calculus and [21], [24] for Kähler
structures.

As mentioned before, the irreducible quantum flag manifolds are distinguished
by existence of a unique $q$-deformed de Rham complex. The following theorem
collects the main results of [12], [13], and [21].

######  Theorem 3.1.

For any irreducible quantum flag manifold ${\mathcal{O}}_{q}(G/L_{S})$, there
exists a unique finite dimensional left ${\mathcal{O}}(G)$-covariant
differential $\ast$-calculus

$\displaystyle\Omega^{\bullet}_{q}(G/L_{S})\in\mbox{}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}_{q}(G/L_{S})}\mathrm{Mod}_{0},$

of classical dimension

$\displaystyle\mathrm{dim}\,\Phi\big{(}\Omega^{k}_{q}(G/L_{S})\big{)}=\left(\begin{array}[]{c}\\!\\!2M\\!\\!\\\
\\!\\!k\\!\\!\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{for all}\leavevmode\nobreak\
\leavevmode\nobreak\ k=1,\ldots,2M,$

where $M$ is the complex dimension of the corresponding classical manifold.
Moreover

  1. (i)

$\Omega^{\bullet}_{q}(G/L_{S})$ admits a unique left
${\mathcal{O}}_{q}(G)$-covariant complex structure

$\displaystyle\Omega^{\bullet}_{q}(G/L_{S})\simeq\bigoplus_{(a,b)\in\mathbb{N}_{0}}\Omega^{(a,b)}=:\Omega^{(\bullet,\bullet)},$

  2. (ii)

$\Omega^{(1,0)}$ and $\Omega^{(0,1)}$ are irreducible objects in
$\mbox{}^{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}_{q}(G/L_{S})}\mathrm{Mod}_{0}$.

We recall that the $\ast$-calculus and the covariant complex structure were
given in [21]. The following theorem gives us the existence of the Kähler
structure for the Heckenberger–Kolb calculus.

######  Theorem 3.2.

([21] Theorem 5.10). Let $\Omega^{\bullet}_{q}(G/L_{S})$ be the
Heckenberger–Kolb calculus of the irreducible quantum flag manifold
${\mathcal{O}}_{q}(G/L_{S})$. Then there exists a form
$\kappa\in\Omega^{(1,1)}$ such that $(\Omega^{(\bullet,\bullet)},\kappa)$ is a
covariant Kähler structure for all $q\in\mathbb{R}_{>0}\setminus F$, where $F$
is a finite, possibly empty, subset of $\mathbb{R}_{>0}$. Moreover, any
element of $F$ is necessarily non-transcendental.

It has been shown in [9, §5.7] that if ${\mathcal{O}}_{q}(G)$ is a compact
quantum group algebra and $q$ belongs to a suitable open interval around $1$
then an inner product on $\Omega^{(\bullet,\bullet)}$ can be defined by

$\displaystyle\langle\cdot,\cdot\rangle:\Omega^{(\bullet,\bullet)}\times\Omega^{(\bullet,\bullet)}\rightarrow\mathbb{C},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (\omega,\nu)\mapsto\mbox{{h}}\circ
g_{\sigma}(\omega,\nu),$ (28)

where is the Haar state associated to ${\mathcal{O}}_{q}(G)$ [17, §11.3.2].
From now on we consider $q\in(1,\epsilon)$ so that the inner product is well
defined. With respect to this inner product we consider the adjoints
$\mathrm{d}^{\dagger}$, $\partial^{\dagger}$ and
$\overline{\partial}^{\dagger}$. It is shown in [24, §5.4] that

$\displaystyle\mathrm{d}^{\dagger}=-\ast_{\kappa}\circ\,\mathrm{d}\circ\ast_{\kappa},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\partial^{\dagger}=-\ast_{\kappa}\circ\,\overline{\partial}\circ\ast_{\kappa},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\overline{\partial}^{\dagger}=-\ast_{\kappa}\circ\,\partial\circ\ast_{\kappa}.$
(29)

Then Theorem 3.1 and the inner product (28) allow us to define the
$\overline{\partial}$-Dirac operator and the $\overline{\partial}$-Laplacian

$\displaystyle
D_{\overline{\partial}}:=\overline{\partial}+\overline{\partial}^{\dagger},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Delta_{\overline{\partial}}:=(\overline{\partial}+\overline{\partial}^{\dagger})^{2}.$

######  Lemma 3.3.

With respect to the notation in [13] we have

  1. (i)

$u^{i}_{j}=c^{\varpi_{1}}_{f_{i},v_{j}}$,
$S(u^{i}_{j})=c^{-w_{0}(\varpi_{1})}_{v_{j},f_{i}}$ and
$z_{ij}:=c^{\varpi_{1}}_{f_{i},v_{N}}c^{-w_{0}\varpi_{1}}_{v_{j},f_{N}}=u^{i}_{N}S(u^{N}_{j})$
are the generators of the algebra ${\mathcal{O}}_{q}(G/L_{S})$,

  2. (ii)

$\Phi(\Omega^{(0,1)})=\Omega^{(0,1)}/{\mathcal{O}}_{q}(\mathrm{Q}_{N})^{+}\Omega^{(0,1)}=\mathrm{Lin}_{\mathbb{C}}\big{\\{}[\overline{\partial}z_{Ni}]:i=2,\ldots,N-1\big{\\}}$
and $\Phi(\Omega^{(1,0)})=\mathrm{Lin}_{\mathbb{C}}\big{\\{}[\partial
z_{iN}]:i=2,\ldots,N-1\big{\\}}$

  3. (iii)

if we consider $\mathcal{O}_{q}(G/L_{S})$ as a left
$U_{q}(\mathfrak{g})$-module with action $X\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}z:=z\triangleleft S(X)$ then the
generators of $\mathcal{O}_{q}(\mathrm{Q}_{N})$ are given by
$z:=u^{1}_{1}u^{1}_{N}$, $y:=u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N}$ with
weights $2\varpi_{1}$, $\varpi_{2}$ respectively.

###### Proof.

(i) follows from the notation in [13]. To prove (ii) we know from [13, §3.2]
that $I_{(1)}=\\{i\in
I:(\varpi_{1},\varpi_{1}-\alpha_{1}-\mbox{wt}(v_{i}))=0\\}$. Since $v_{N}$ is
the highest weight vector
$(\varpi_{1},\varpi_{1}-\alpha_{1}-\mbox{wt}(v_{N}))=(\varpi_{1},\varpi_{1}-\alpha_{1}-\varpi_{1})=-(\varpi_{1},\alpha_{1})\neq
0$, thus $v_{N}\notin I_{(1)}$. Similarly, we can consider $v_{1}$ as the
lowest weight vector, then
$\mbox{wt}(v_{1})=w_{0}(\varpi_{1})=-\mbox{id}(\varpi_{1})=-\varpi_{1}$. Then
$(\varpi_{1},\varpi_{1}-\alpha_{1}-\mbox{wt}(v_{1}))=(\varpi_{1},\varpi_{1}-\alpha_{1}+\varpi_{1})=2(\varpi_{1},\varpi_{1})-(\varpi_{1},\alpha_{1})=(2r_{1}-1)(\varpi_{1},\alpha_{1})$
where $r_{1}$ is the coefficient of $\alpha_{1}$ in the expression of
$\varpi_{1}$ as linear combination of the $\alpha_{1},\ldots,\alpha_{n}$. From
the form of the Cartan matrix of type $B_{n}$ and $D_{n}$ it can be shown that
$r_{1}=1$ and therefore $v_{1}\notin I_{(1)}$. It also can be shown that
$M=N-2$ for both cases ${\mathcal{O}}_{q}(\mathbf{Q}_{2n+1})$ and
${\mathcal{O}}_{q}(\mathbf{Q}_{2n})$, see for example table 2 in [9].
Therefore $I_{(1)}=\\{2,\ldots,N-1\\}$.

For the proof of (iii) we first note that the left and right actions on matrix
coefficients are given by

$\displaystyle X\triangleright c^{\varpi_{1}}_{f_{i},v_{j}}\triangleleft
Y=c^{\varpi_{1}}_{f_{i}\triangleleft Y,X\triangleright vj}.$ (30)

By (30) we have $E_{i}\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}(u^{1}_{1}u^{1}_{N})=(u^{1}_{1}u^{1}_{N})\triangleleft
S(E_{i})=-(c^{\varpi_{1}}_{f_{1}\triangleleft
E_{i},v_{1}}c^{\varpi_{1}}_{f_{1}\triangleleft
K_{i},v_{1}}+c^{\varpi_{1}}_{f_{1},v_{1}}c^{\varpi_{1}}_{f_{1}\triangleleft
E_{i},v_{N}})\triangleleft K_{i}^{-1}=0$ since $f_{1}$ is the highest weight
of the dual representation. On the other hand

$\displaystyle K_{i}\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}(u^{1}_{1}u^{1}_{N})=(u^{1}_{1}u^{1}_{N})\triangleleft
K^{-1}_{i}=c^{\varpi_{1}}_{f_{1}\triangleleft
K_{i},v_{j}}c^{\varpi_{1}}_{f_{1}\triangleleft
K_{i},v_{N}}=q^{(2\varpi_{1},\alpha_{i})}u^{1}_{1}u^{1}_{N}.$

Similarly, since $f_{1}$ is a highest weight vector we have for $i\neq 1$

$\displaystyle(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})\triangleleft
E_{i}=c^{\varpi_{1}}_{f_{1},v_{1}}c^{\varpi_{1}}_{f_{2}\triangleleft
E_{i},v_{N}}-qc^{\varpi_{1}}_{f_{2}\triangleleft
E_{i},v_{1}}c^{\varpi_{1}}_{f_{1}\triangleleft K_{i},v_{N}}.$ (31)

However, since $v_{1}$ is the lowest weight vector of $V_{\varpi_{1}}$ with
weight $w_{0}\varpi_{1}=-\varpi_{1}$ then for $i\neq 1$ the vector $v_{1}$ is
the lowest weight with weight zero of the
$U_{q}(\mathfrak{sl}_{2})_{i}$-module $U_{q}(\mathfrak{sl}_{2})_{i}v_{N}$
where $U_{q}(\mathfrak{sl}_{2})_{i}:=\langle E_{i},F_{i},K_{i}\rangle$.
Therefore $E_{i}v_{1}=0$ for $i\neq 1$ and $E_{1}v_{1}=v_{2}$. This in turn
implies that $f_{2}\triangleleft E_{i}=0$ for $i\neq 1$. Then the right hand
side of equation (31) is zero. Since $f_{2}\triangleleft E_{1}=f_{1}$ and
$f_{1}\triangleleft K_{1}=f_{1}$ we have

$\displaystyle E_{1}\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})$
$\displaystyle=-(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})\triangleleft
E_{1}K_{1}^{-1}$
$\displaystyle=-(c^{\varpi_{1}}_{f_{1},v_{1}}c^{\varpi_{1}}_{f_{2}\triangleleft
E_{1},v_{N}}-qc^{\varpi_{1}}_{f_{2}\triangleleft
E_{1},v_{1}}c^{\varpi_{1}}_{f_{1}\triangleleft K_{1},v_{N}})\triangleleft
K_{i}^{-1}$ $\displaystyle=0.$

Finally, since
$K_{i}E_{1}v_{1}=q^{(\alpha_{i},\alpha_{1})}E_{1}K_{i}v_{1}=q^{(\alpha_{i},\alpha_{1})-(\varpi_{1},\alpha_{i})}E_{1}v_{1}$
we have

$\displaystyle K_{i}\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})$
$\displaystyle=(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})\triangleleft
K^{-1}_{i}$ $\displaystyle=c^{\varpi_{1}}_{f_{1}\triangleleft
K^{-1}_{i},v_{1}}c^{\varpi_{1}}_{f_{2}\triangleleft
K^{-1}_{i},v_{N}}-qc^{\varpi_{1}}_{f_{2}\triangleleft
K^{-1}_{i},v_{1}}c^{\varpi_{1}}_{f_{1}\triangleleft K^{-1}_{i},v_{N}}$
$\displaystyle=q^{(2\varpi_{1}-\alpha_{1},\alpha_{i})}(c^{\varpi_{1}}_{f_{1},v_{1}}c^{\varpi_{1}}_{f_{2},v_{N}}-qc^{\varpi_{1}}_{f_{2},v_{1}}c^{\varpi_{1}}_{f_{1},v_{N}})$
$\displaystyle=q^{(\varpi_{2},\alpha_{i})}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N}),$

where we used the fact that $\alpha_{i}=\sum_{j}a_{ji}\varpi_{j}$ to derive
the third equality. ∎

Part (iii) of the Lemma 3.3 says that the weights $2\varpi_{1},\varpi_{2}$
form a minimal set of generators for the weights of the highest weights
vectors of ${\mathcal{O}}_{q}(Q_{N})$ as a monoid under addition which are
called spherical weights. In table 2 below we present the spherical weights
for each irreducible quantum flag manifold which was presented by Krämer [19]
in the classical case. We note that for the case of
${\mathcal{O}}_{q}(\mathbf{S}_{2n})$, the weight $2\varpi_{2n-1}$ or
$2\varpi_{2n}$ appears according to the defining crossed node, as presented in
table 1.

Table 2: Spherical weights according to the numbering of the Dynkin nodes in [15, §11.4]. ${\mathcal{O}}_{q}(\text{Gr}_{r,s})$ | $\varpi_{1}+\varpi_{r+s-1},\varpi_{2}+\varpi_{r+s-2},\ldots,\varpi_{r}+\varpi_{s}$
---|---
${\mathcal{O}}_{q}(\mathbf{Q}_{2n+1})$ | $2\varpi_{1},\varpi_{2}$
${\mathcal{O}}_{q}(\mathbf{L}_{n})$ | $2\varpi_{1},2\varpi_{2},\ldots,2\varpi_{n}$
${\mathcal{O}}_{q}(\mathbf{Q}_{2n})$ | $2\varpi_{1},\varpi_{2}$
${\mathcal{O}}_{q}(\textbf{S}_{2n})$ |  $\varpi_{2},\varpi_{4},\ldots,\varpi_{2n-2},2\varpi_{2n-1}$ or $2\varpi_{2n}$
${\mathcal{O}}_{q}(\textbf{S}_{2n+1})$ | $\varpi_{2},\varpi_{4},\ldots,\varpi_{2n-2},\varpi_{2n}+2\varpi_{2n+1}$
${\mathcal{O}}_{q}(\mathbb{OP}^{2})$ | $\varpi_{1}+\varpi_{6},\varpi_{2}$
${\mathcal{O}}_{q}(\textrm{F})$ | $2\varpi_{7},\varpi_{1},\varpi_{6}$

######  Lemma 3.4.

Let $\partial$, $\overline{\partial}$, $z,y$ be as before. Then the following
holds

  1. (i)

$\overline{\partial}zz=q^{2}z\overline{\partial}z$,

  2. (ii)

$\overline{\partial}yy=q^{2}y\overline{\partial}y$,

  3. (iii)

$\partial zz=q^{-2}z\partial z$,

  4. (iv)

$\partial yy=q^{-2}y\partial y$.

###### Proof.

We only show (iii) and (iv), equations (i) and (ii) can be shown in a similar
way by using Lemma 2.7 and Lemma 3.3. By Takeuchi’s equivalence there exists
an isomorphism in the category ${}^{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
{\mathcal{O}}_{q}(G)}_{{\mathcal{O}}_{q}(G/L_{S})}\mathrm{Mod}_{0}$ given by

$\displaystyle\mathrm{U}:\Omega^{(0,1)}\rightarrow\mathcal{O}(SO_{q}(5))\square_{{\mathcal{O}}_{q}(L)}\Phi(\Omega^{(0,1)}),\qquad\omega\mapsto\omega_{(-1)}\otimes[\omega_{(0)}].$

Now we recall by Lemma 3.3 that
$\\{[\overline{\partial}z_{Nj}]:j=1,\ldots,N-1\\}$ and $\\{[\partial z_{jN}]$,
$j=2,\ldots,N-1\\}$ are bases of the vector spaces $\Phi(\Omega^{(0,1)})$ and
$\Phi(\Omega^{(1,0)})$, respectively. Moreover, it was shown in [13, §3.2]
that $[\overline{\partial}z_{ij}]=[\partial z_{ji}]=0$ if $i\neq N$ and
$j\in\\{1,N\\}$.

Then by definition of $\mathrm{U}$ and relations in Lemma 2.7 we have

$\displaystyle\mathrm{U}\big{(}(\overline{\partial}z_{1N})z_{1N}\big{)}$
$\displaystyle=\mathrm{U}(\overline{\partial}z_{1N})z_{1N}$
$\displaystyle=(z_{1N})_{(1)}z_{1N}\otimes[\overline{\partial}(z_{1N})_{(2)}]$
$\displaystyle=(u^{1}_{N}S(u^{N}_{N}))_{(1)}z_{1N}\otimes[\overline{\partial}(u^{1}_{N}S(u^{N}_{N}))_{(2)}]$
$\displaystyle=\sum_{j,k}u^{1}_{j}S(u^{k}_{N})z_{1N}\otimes[\overline{\partial}(u^{j}_{N}S(u^{N}_{k}))]$
$\displaystyle=\sum_{j,k}u^{1}_{j}S(u^{k}_{N})z_{1N}\otimes[\overline{\partial}z_{jk}]$
$\displaystyle=\sum_{k=2}^{N-1}u^{1}_{N}S(u^{k}_{N})z_{1N}\otimes[\overline{\partial}z_{Nk}]$
$\displaystyle=\sum_{k=2}^{N-1}u^{1}_{N}S(u^{k}_{N})u^{1}_{N}S(u^{N}_{N})\otimes[\overline{\partial}z_{Nk}]$
$\displaystyle=\sum_{k=2}^{N-1}q^{\rho_{N}-\rho_{k}}u^{1}_{N}u^{1}_{k^{\prime}}u^{1}_{N}u^{1}_{1}\otimes[\overline{\partial}z_{Nk}]$
$\displaystyle=q^{-2}\sum_{k=2}^{N-1}q^{\rho_{N}-\rho_{k}}u^{1}_{N}u^{1}_{1}u^{1}_{N}u^{1}_{k^{\prime}}\otimes[\overline{\partial}z_{Nk}]$
$\displaystyle=q^{-2}z_{1N}\mathrm{U}(\overline{\partial}z_{1N}).$

This yields equation (iii).

Since $y=q(z_{2,N}-q^{-2}z_{1,N-1})$ then analogously by using relations of
Lemma 2.7 we have

$\displaystyle\mathrm{U}\big{(}(\overline{\partial}y)y\big{)}$
$\displaystyle=\mathrm{U}(\overline{\partial}y)y$
$\displaystyle=\mathrm{U}(\overline{\partial}(qz_{2,N}-q^{-1}z_{1,N-1}))(qz_{2,N}-q^{-1}z_{1,N-1})$
$\displaystyle=(qz_{2,N}-q^{-1}z_{1,N-1})_{(1)}(qz_{2,N}-q^{-1}z_{1,N-1})\otimes[\overline{\partial}(qz_{2,N}-q^{-1}z_{1,N-1})_{(2)}]$
$\displaystyle=\sum_{l=2}^{N-1}(qu^{2}_{N}S(u^{l}_{N})-q^{-1}u^{1}_{N}S(u^{l}_{N-1}))(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})\otimes[\overline{\partial}(qz_{Nl}-q^{-1}z_{Nl})]$
$\displaystyle=\sum_{l=2}^{N-1}q^{\rho_{N}-\rho_{l}}(qu^{2}_{N}u^{1}_{l^{\prime}}-u^{1}_{N}u^{2}_{l^{\prime}})(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})\otimes[\overline{\partial}(qz_{Nl}-q^{-1}z_{Nl})]$
$\displaystyle=\sum_{l=2}^{N-1}q^{\rho_{N}-\rho_{l}+1}(u^{1}_{l^{\prime}}u^{2}_{N}-qu^{2}_{l^{\prime}}u^{1}_{N})(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})\otimes[\overline{\partial}(qz_{Nl}-q^{-1}z_{Nl})]$
$\displaystyle=q^{-2}\sum_{l=2}^{N-1}q^{\rho_{N}-\rho_{l}+1}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})(u^{1}_{l^{\prime}}u^{2}_{N}-qu^{2}_{l^{\prime}}u^{1}_{N})\otimes[\overline{\partial}(qz_{Nl}-q^{-1}z_{Nl})]$
$\displaystyle=q^{-2}(u^{1}_{1}u^{2}_{N}-qu^{2}_{1}u^{1}_{N})\sum_{l=2}^{N-1}q^{\rho_{N}-\rho_{l}+1}(u^{1}_{l^{\prime}}u^{2}_{N}-qu^{2}_{l^{\prime}}u^{1}_{N})\otimes[\overline{\partial}(qz_{Nl}-q^{-1}z_{Nl})]$
$\displaystyle=q^{-2}y\mathrm{U}(\overline{\partial}y).$

This proves equation (iv).

∎

In the following lemma we give the existence of a Kähler structure for the
Heckenberger–Kolb calculus over the quantum quadric
${\mathcal{O}}_{q}(\textbf{Q}_{N})$. Since the space of coinvariants
$\mbox{}^{co(L_{S})}\Phi(\Omega^{(1,1)})$ is a 1-dimensional vector space (see
for example [11, §3.3]) it can be shown that this Kähler form is up to scalar
equal to the Kähler form given in [21, §5].

######  Lemma 3.5.

A Kähler form is given by

$\displaystyle\kappa=\mathrm{i}\sum_{i,j=1}^{N}q^{-\rho_{i}}\partial
z_{ij}\wedge\overline{\partial}z_{ji}.$

For $i=1,\ldots,N-2$ we denote $e^{+}_{i}:=[\partial z_{i+1,N}]$ and
$e^{-}_{i}:=q^{-\rho_{i+1}}[\overline{\partial}z_{N,i+1}]$, then

$\displaystyle\mathrm{U}(\kappa)=\mathrm{i}\sum_{i=1}^{N-2}1\otimes
e^{+}_{i}\wedge e^{-}_{i}$ (32)

and there exist polynomials $f_{I,J}(q)\in\mathbb{C}[q,q^{-1}]$ such that

$\displaystyle\mathrm{U}(\kappa^{l})=\mathrm{i}^{l}\sum_{I,J\in\Theta(l)}f_{l,I,J}(q)\\!\cdot\\!1\otimes
e^{+}_{I}\wedge e^{-}_{J},\quad\ \ f_{l,I,I}(1)\neq 0,\ \ \text{and}\ \
f_{l,I,J}(1)=0\ \ \text{for}\ \ I\neq J,$ (33)

where $\Theta(l)$ denotes the set of all ordered subsets of
$\\{1,\ldots,N-2\\}$ with $l$ elements. Moreover,
sgn$(f_{l,I,I}(1))=(-1)^{\frac{l(l-1)}{2}}$ for all $I\in\Theta(l)$.

###### Proof.

That $\kappa$ is real follows from identities (2) and the fact that
$z_{ij}^{*}=z_{ji}$. Equation (32) follows from the calculation

$\displaystyle\mathrm{U}\Big{(}\sum_{i,j=1}^{N}q^{-2\rho_{i}}\partial
z_{ij}\wedge\overline{\partial}z_{ji}\Big{)}$
$\displaystyle\\!=\sum_{i,j=1}^{N}q^{-2\rho_{i}}(z_{ij})_{(1)}(z_{ji})_{(1)}\otimes[\partial((z_{ij})_{(2)})]\wedge[\overline{\partial}((z_{ji})_{(2)})]$
$\displaystyle\\!=\sum_{i,j=1}^{N}\sum_{a,b,c,d}q^{-2\rho_{i}}u^{i}_{a}S(u^{b}_{j})u^{j}_{c}S(u^{d}_{i})\otimes[\partial(u^{a}_{N}S(u^{N}_{b}))]\wedge[\overline{\partial}(u^{c}_{N}S(u^{N}_{d}))]$
$\displaystyle\\!=\sum_{i=1}^{N}\sum_{a,b,d}q^{-2\rho_{i}}u^{i}_{a}S(u^{d}_{i})\otimes[\partial(u^{a}_{N}S(u^{N}_{b}))]\wedge[\overline{\partial}(u^{b}_{N}S(u^{N}_{d}))]$
$\displaystyle\\!=\sum_{i=1}^{N}\sum_{a,d}q^{-2\rho_{i}}u^{i}_{a}S(u^{d}_{i})\otimes[\partial
z_{aN}]\wedge[\overline{\partial}z_{Nd}]$
$\displaystyle\\!=\sum_{d=2}^{N-1}1\otimes[\partial z_{dN}]\wedge
q^{-\rho_{d}}[\overline{\partial}z_{Nd}],$

where the identity
$\sum_{i=1}^{N}q^{-2\rho_{i}}u^{i}_{a}S(u^{d}_{i})=q^{-2\rho_{d}}\delta_{da}\\!\cdot\\!1$
was used in the penultimate line. This implies the left
${\mathcal{O}}_{q}(G)$-coinvariance of $\kappa$ since

$\displaystyle\Big{[}\sum_{i,j=1}^{N}q^{-2\rho_{i}}\partial
z_{ij}\wedge\overline{\partial}z_{ji}\Big{]}=\sum_{i,j=1}^{N}q^{-2\rho_{i}}[\partial
z_{ij}]\wedge[\overline{\partial}z_{ji}]=\sum_{i=2}^{N-1}[\partial
z_{iN}]\wedge q^{-2\rho_{i}}[\overline{\partial}z_{Ni}].$

Now we show that $\kappa$ is closed by a Lie theoretic argument: we set
$V^{(a,b)}:=\Phi(\Omega^{(a,b)})$ and note that $\mathrm{d}\kappa=0$ if and
only if $\partial\kappa=\overline{\partial}\kappa=0$, and both are coinvariant
elements in $\Omega^{(2,1)}$ and $\Omega^{(1,2)}$, respectively. But
$\mbox{}^{co(L_{S})}V^{(1,2)}=\mbox{}^{co(L_{S})}(V^{(1,0)}\otimes
V^{(0,2)})$, then $\overline{\partial}\kappa=0$ if
$\mbox{}^{co(L_{S})}(V^{(1,0)}\otimes V^{(0,2)})=0$. Since $V^{(1,0)}\otimes
W$ contains the trivial representation if and only if
$W\simeq(V^{(1,0)})^{*}\simeq V^{(0,1)}$ it is enough to show that $V^{(0,2)}$
is not isomorphic to $V^{(1,0)}$. But $V^{(0,2)}$ is an irreducible component
of $V^{(0,1)}\otimes V^{(0,1)}$. The following formulas for the tensor product
decomposition can be found in Table 5 of [27]: for odd $N=2n+1$ we have

$\displaystyle V^{(0,1)}\otimes
V^{(0,1)}\simeq\left\\{\begin{array}[]{lc}V_{4\varpi_{1}}\oplus
V_{2\varpi_{1}}\oplus\mathbb{C},&n=2\ \ (V^{(0,1)}\simeq V_{2\varpi_{1}}\
\text{as}\ U_{q}(\mathfrak{sl}_{2})\text{-modules}),\\\ V_{2\varpi_{1}}\oplus
V_{2\varpi_{2}}\oplus\mathbb{C},&n=3\ \ (V^{(0,1)}\simeq V_{\varpi_{1}}\
\text{as}\ U_{q}(\mathfrak{so}_{5})\text{-modules}),\\\ V_{2\varpi_{1}}\oplus
V_{\varpi_{2}}\oplus\mathbb{C},&\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ n\geq 4\ \ (V^{(0,1)}\simeq
V_{\varpi_{1}}\ \text{as}\
U_{q}(\mathfrak{so}_{2n-1})\text{-modules}).\end{array}\right.$

For $n=2$, it can be shown that the $U_{q}(\mathfrak{l}^{s}_{S})$-invariant
element corresponding to $\mathbb{C}$ is not
$U_{q}(\mathfrak{l}_{S})$-invariant. For even $N=2n$ we have

$\displaystyle V^{(0,1)}\otimes
V^{(0,1)}\simeq\left\\{\begin{array}[]{lc}V_{2\varpi_{2}}\oplus
V_{\varpi_{3}+\varpi_{1}}\oplus\mathbb{C},&n=4\ \ (V^{(0,1)}\simeq
V_{\varpi_{2}}\ \text{as}\ \ U_{q}(\mathfrak{sl}_{4})\text{-modules}),\\\
V_{2\varpi_{1}}\oplus V_{\varpi_{2}}\oplus\mathbb{C},&\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ n\geq 5\ \ (V^{(0,1)}\simeq V_{\varpi_{1}}\ \text{as}\ \
U_{q}(\mathfrak{so}_{2n-2})\text{-modules}).\end{array}\right.$

This formulas show that $V^{(1,0)}$ does not appear as an irreducible
component of $V^{(0,1)}\otimes V^{(0,1)}$ and therefore
$\overline{\partial}\kappa=0$. A similar argument shows that $\partial k=0$.
By Lemma 4.6 in [24] we have that $\kappa$ is central. The bijectivity of the
Lefschetz operators $L^{n-k}$ is shown in [21] for $q$ belonging to a
sufficiently small open interval around $1$.

Now we proceed to calculate the powers of $\kappa$ by using the relations in
$\Phi(\Omega^{(\bullet,0)})$ and $\Phi(\Omega^{(0,\bullet)})$. It follows from
[13] that
$\Phi(\Omega^{(0,1)})=\mathrm{Lin}_{\mathbb{C}}\\{e^{-}_{1},\ldots,e^{-}_{N-2}\\}$
and the algebra $\Phi(\Omega^{(0,\bullet)})$ is isomorphic to the quantum
exterior algebra $\Lambda({\mathcal{O}}^{N-2}_{q})$ of the quantum Euclidean
space ${\mathcal{O}}^{N-2}_{q}$ in [17, §9.3]. Then the relations in [13]
imply that the relations in $\Phi(\Omega^{(0,\bullet)})$ for $N=2n+1$ are
given by

$\displaystyle e^{-}_{i}\wedge e^{-}_{i}=0,\ \ i\neq
i^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ e^{-}_{i}\wedge
e^{-}_{j}=-q^{-1}e^{-}_{j}\wedge e^{-}_{i},\ \ i<j,\ i\neq j^{\prime},$ (34)
$\displaystyle e^{-}_{i^{\prime}}\wedge e^{-}_{i}+e^{-}_{i}\wedge
e^{-}_{i^{\prime}}=(q-q^{-1})\sum_{1\leq
j<i}\lambda_{i}^{-1}\lambda_{j}q^{j-i+1}e^{-}_{j}\wedge e^{-}_{j^{\prime}},\
i<i^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ $ (35) $\displaystyle
e^{-}_{n}\wedge e^{-}_{n}=(q^{1/2}-q^{-1/2})\sum_{1\leq j\leq
n-1}\lambda_{n}^{-1}\lambda_{j}q^{j-(n-1)}e^{-}_{j}\wedge
e^{-}_{j^{\prime}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (36)

where $\lambda_{i}$ is a non-zero complex number. For even $N=2n$ the
relations are given by $\eqref{ii}$ and $\eqref{i'i}$ for possibly different
constants $\lambda_{i}$. The relations for $\Phi(\Omega^{(\bullet,0)})$ are
giving by applying the involution to the relations (34)-(36). If we apply the
involution to (35) we get

$\displaystyle e^{+}_{i}\wedge e^{+}_{i^{\prime}}+e^{+}_{i^{\prime}}\wedge
e^{+}_{i}=(q-q^{-1})\sum_{1\leq
j<i}\overline{\lambda}_{i}^{-1}\overline{\lambda}_{j}q^{j-i+1}e^{+}_{j^{\prime}}\wedge
e^{+}_{j},\ \ i<i^{\prime}.$ (37)

We claim that the relations in $\Phi(\Omega^{(\bullet,0)})$ are given by

$\displaystyle e^{+}_{i}\wedge e^{+}_{i}=0,\ \ i\neq
i^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ e^{+}_{i}\wedge
e^{+}_{j}=-qe^{+}_{j}\wedge e^{+}_{i},\ \ i<j,\ i\neq j^{\prime},$ (38)
$\displaystyle e^{+}_{i^{\prime}}\wedge e^{+}_{i}+e^{+}_{i}\wedge
e^{+}_{i^{\prime}}=-(q-q^{-1})\sum_{1\leq
j<i}\overline{\lambda}_{i}^{-1}\overline{\lambda}_{j}q^{-(j-i+1)}e^{+}_{j}\wedge
e^{+}_{j^{\prime}},\ i<i^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ $
(39) $\displaystyle e^{+}_{n}\wedge e^{+}_{n}=-(q^{1/2}-q^{-1/2})\sum_{1\leq
j\leq
n-1}\overline{\lambda}_{n}^{-1}\overline{\lambda}_{j}q^{-(j-(n-1))}e^{+}_{j}\wedge
e^{+}_{j^{\prime}}.\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (40)

Relations (38) are shown by applying the involution to relations (34). For the
rest of the paper we denote $\nu_{\tau}:=\tau-\tau^{-1}$ if $\tau>0$. We use
induction to prove (39):

$\displaystyle\overline{\lambda}_{i+1}\big{(}$ $\displaystyle
e^{+}_{i+1}\wedge e^{+}_{(i+1)^{\prime}}+e^{+}_{(i+1)^{\prime}}\wedge
e^{+}_{i+1}\big{)}=\nu_{q}\sum_{1\leq
j<i+1}\overline{\lambda}_{j}q^{j-(i+1)+1}e^{+}_{j^{\prime}}\wedge
e^{+}_{j}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{by}\ \
\eqref{invi'i}$ $\displaystyle=q^{-i}\nu_{q}\sum_{1\leq
j<i+1}\overline{\lambda}_{j}q^{j}\big{(}\\!-\nu_{q}\sum_{1\leq
k<j}\overline{\lambda}_{j}^{-1}\overline{\lambda}_{k}q^{-k+j-1}e^{+}_{k}\wedge
e^{+}_{k^{\prime}}-e^{+}_{j}\wedge
e^{+}_{j^{\prime}}\big{)}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (\text{ind. hyp.})$
$\displaystyle=q^{-i}\nu_{q}\Big{(}\\!\\!-\nu_{q}\sum_{1\leq j<i+1}\sum_{1\leq
k<j}\overline{\lambda}_{k}q^{2j-k-1}e^{+}_{k}\wedge
e^{+}_{k^{\prime}}-\sum_{1\leq
j<i+1}\overline{\lambda}_{j}q^{j}e^{+}_{j}\wedge e^{+}_{j^{\prime}}\Big{)}$
$\displaystyle=q^{-i}\nu_{q}\Big{(}\\!\\!-\nu_{q}\sum_{1\leq k<i}\sum_{k+1\leq
j\leq i}\overline{\lambda}_{k}q^{2j-k-1}e^{+}_{k}\wedge
e^{+}_{k^{\prime}}-\sum_{1\leq
j<i+1}\overline{\lambda}_{j}q^{j}e^{+}_{j}\wedge e^{+}_{j^{\prime}}\Big{)}$
$\displaystyle=q^{-i}\nu_{q}\Big{(}\\!\\!-\nu_{q}\sum_{1\leq
k<i}\overline{\lambda}_{k}q^{-k-1}\Big{(}\frac{q^{2(k+1)}-q^{2i+2}}{1-q^{2}}\Big{)}e^{+}_{k}\wedge
e^{+}_{k^{\prime}}-\sum_{1\leq
j<i+1}\overline{\lambda}_{j}q^{j}e^{+}_{j}\wedge e^{+}_{j^{\prime}}\Big{)}$
$\displaystyle=q^{-i}\nu_{q}\Big{(}\sum_{1\leq
k<i}\overline{\lambda}_{k}(q^{k}-q^{2i-k})e^{+}_{k}\wedge
e^{+}_{k^{\prime}}-\sum_{1\leq
j<i+1}\overline{\lambda}_{j}q^{j}e^{+}_{j}\wedge e^{+}_{j^{\prime}}\Big{)}$
$\displaystyle=q^{-i}\nu_{q}\Big{(}\sum_{1\leq
k<i}\overline{\lambda}_{k}q^{k}e^{+}_{k}\wedge e^{+}_{k^{\prime}}-\sum_{1\leq
k<i}\overline{\lambda}_{k}q^{2i-k}e^{+}_{k}\wedge
e^{+}_{k^{\prime}}-\sum_{1\leq
j<i+1}\overline{\lambda}_{j}q^{j}e^{+}_{j}\wedge e^{+}_{j^{\prime}}\Big{)}$
$\displaystyle=\nu_{q}\Big{(}-\sum_{1\leq
k<i}\overline{\lambda}_{k}q^{i-k}e^{+}_{k}\wedge
e^{+}_{k^{\prime}}-\overline{\lambda}_{i}e^{+}_{i}\wedge
e^{+}_{i^{\prime}}\Big{)}.$

Similar calculations show the identity (40) by using (39).

Now assuming that the equation (33) holds for $l$, we have

$\displaystyle\mathrm{U}(\kappa^{l+1})$
$\displaystyle=\mathrm{U}(\kappa)\wedge\mathrm{i}^{l}\sum_{I,J\in\Theta(l)}f_{I,J}(q)\\!\cdot\\!1\otimes
e^{+}_{I}\wedge e^{-}_{J}$
$\displaystyle=\mathrm{i}^{l+1}\sum_{I,J\in\Theta(l)}\sum_{i}f_{I,J}(q)\\!\cdot\\!1\otimes
e^{+}_{I}\wedge e^{+}_{i}\wedge e^{-}_{i}\wedge e^{-}_{J},$

where we use the fact that $\kappa$ is central. First we can focus on the case
$e^{+}_{I}\wedge e^{+}_{i}\wedge e^{-}_{i}\wedge e^{-}_{I}$ for
$I=\\{i_{1},\ldots,i_{l}\\}$ fixed. Then different cases arise:

Case $i\neq i^{\prime}$ and $i\in I$: $e^{+}_{I}\wedge e^{+}_{i}\wedge
e^{-}_{i}\wedge e^{-}_{I}=0$ since we can move $e^{-}_{i}$ to the right(resp.
left) by using (34)(resp. (38)) if $i<i^{\prime}$(resp. $i>i^{\prime}$).

Case $i,i^{\prime}\notin I$: in this case we can move $e^{+}_{i}$ to the left
and $e^{-}_{i}$ to the right by using (38) and (34), respectively and we get

$\displaystyle e^{+}_{I}\wedge e^{+}_{i}\wedge e^{-}_{i}\wedge
e^{-}_{I}=(-1)^{l}q^{l-2\alpha_{i}}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i}\wedge\cdots\wedge e^{+}_{l}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{i}\wedge\cdots\wedge e^{-}_{l},$

where $\alpha_{i}\in\mathbb{N}_{0}$.

Case there is a $k<l$ such that $i_{k}=i^{\prime}\geq i$ and $i\notin I$: in
this case we can move $e^{+}_{i}$ to the left by commuting with
$e^{+}_{i_{l}},\ldots,e^{+}_{i_{k+1}}$ by using relations (38) and we get

$\displaystyle e^{+}_{i_{1}}\wedge\cdots\wedge e^{+}_{i_{l}}$
$\displaystyle\wedge e^{+}_{i}\wedge e^{-}_{i}\wedge
e^{-}_{i_{1}}\wedge\cdots\wedge e^{-}_{i_{l}}$
$\displaystyle=\\!(-q)^{-(l-k)}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i_{k}}\wedge e^{+}_{i}\wedge\cdots\wedge e^{+}_{l}\wedge
e^{-}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge e^{-}_{i_{l}}.$

Now if $i_{k}=i^{\prime}>i$ then we can use (39) with
$\gamma_{ij}:=-\lambda^{-1}_{i}\lambda_{j}$ and we get

$\displaystyle e^{+}_{I}\wedge e^{+}_{i}\wedge e^{-}_{i}\wedge
e^{-}_{I}=\\!(-q)^{-(l-k)}e^{+}_{i_{1}}\wedge\cdots\wedge e^{+}_{i_{k}}\wedge
e^{+}_{i}\wedge\cdots\wedge e^{+}_{i_{l}}\wedge e^{-}_{i}\wedge
e^{-}_{i_{1}}\wedge\cdots\wedge e^{-}_{i_{l}}$
$\displaystyle=\\!(-q)^{-(l-k)}\nu_{q}\\!\sum_{1\leq
j<i}\overline{\gamma}_{ij}q^{-(j-i+1)}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i_{k-1}}\wedge e^{+}_{j}\wedge e^{+}_{j^{\prime}}\wedge\cdots\wedge
e^{+}_{i_{l}}\wedge e^{-}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{i_{l}}$
$\displaystyle\quad-(-q)^{-(l-k)}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i_{k-1}}\wedge e^{+}_{i}\wedge e^{+}_{i_{k}}\wedge\cdots\wedge
e^{+}_{i_{l}}\wedge e^{-}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{i_{l}}.$

Since $j^{\prime}\neq i_{r}$ and $i^{\prime}\neq i_{r}$ for $r\neq k$ then we
can move $e^{+}_{j}$ and $e^{+}_{i}$ to the left and $e^{+}_{j^{\prime}}$ and
$e^{-}_{i}$ to the right and we obtain

$\displaystyle e^{+}_{I}\wedge e^{+}_{i}\wedge e^{-}_{i}\wedge e^{-}_{I}$
$\displaystyle=\\!(-q)^{-(l-k)}\nu_{q}\\!\\!\sum_{1\leq
j_{s}<i}\\!\overline{\gamma}_{ij}q^{-(j_{s}-i+1)+\alpha_{is}}e^{+}_{i_{1}}\wedge\\!\cdots\\!\wedge
e^{+}_{j_{s}}\wedge\\!\cdots\\!\wedge
e^{+}_{j^{\prime}_{s}}\wedge\\!\cdots\\!\wedge e^{+}_{i_{l}}\wedge
e^{-}_{i_{1}}\wedge\\!\cdots\\!\wedge e^{-}_{i}\\!\wedge\\!\cdots\\!\wedge
e^{-}_{i_{l}}$
$\displaystyle\quad+(-1)^{l}q^{1-l+2\beta_{i}}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i}\wedge\cdots\wedge e^{+}_{i_{k}}\wedge\cdots\wedge
e^{+}_{i_{l}}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{i}\wedge\cdots\wedge e^{-}_{i_{k}}\wedge\cdots\wedge e^{-}_{i_{l}},$

where $\alpha_{is}\in\mathbb{Z}$ and $\beta_{i}\in\mathbb{N}_{0}$. Now if
$i=i^{\prime}=n$ and $n\in I$ then by (35) and (39) we have

$\displaystyle e^{+}_{I}\wedge e^{+}_{n}\wedge e^{-}_{n}\wedge
e^{-}_{I}=\\!(-1)^{l-1}q^{l-1-2\gamma_{n}}e^{+}_{i_{1}}\wedge\\!\cdots\\!\wedge
e^{+}_{n}\wedge e^{+}_{n}\wedge\\!\cdots\\!\wedge e^{+}_{i_{l}}\wedge
e^{-}_{i_{1}}\wedge\\!\cdots\\!\wedge e^{-}_{n}\wedge
e^{-}_{n}\wedge\\!\cdots\\!\wedge e^{-}_{i_{l}}$
$\displaystyle=\\!(-1)^{l}q^{l-1-2\gamma_{i}}\nu_{q^{1/2}}^{2}\\!\\!\sum_{1\leq
j,r<n}\\!\\!\overline{\gamma}_{nj}\gamma_{nr}q^{r-j}e^{+}_{i_{1}}\wedge\\!\cdots\\!\wedge
e^{+}_{j}\wedge e^{+}_{j^{\prime}}\wedge\\!\cdots\\!\wedge e^{+}_{i_{l}}\wedge
e^{-}_{i_{1}}\wedge\\!\cdots\\!\wedge e^{-}_{r}\wedge
e^{-}_{r^{\prime}}\wedge\\!\cdots\\!\wedge e^{-}_{i_{l}}$
$\displaystyle=\\!(-1)^{l}q^{l-1-2\gamma_{i}}\nu_{q^{1/2}}^{2}\\!\\!\sum_{1\leq
j<n}\\!\\!|\gamma_{nj}|^{2}e^{+}_{i_{1}}\wedge\cdots\wedge e^{+}_{j}\wedge
e^{+}_{j^{\prime}}\wedge\cdots\wedge e^{+}_{i_{l}}\wedge
e^{-}_{i_{1}}\wedge\cdots\wedge e^{-}_{j}\wedge
e^{-}_{j^{\prime}}\wedge\cdots\wedge e^{-}_{i_{l}}$
$\displaystyle+\\!(-1)^{l}q^{l-1-2\gamma_{i}}\nu_{q^{1/2}}^{2}\\!\\!\sum_{\begin{subarray}{c}1\leq
j,r<n\\\ j\neq
r\end{subarray}}\\!\\!\overline{\gamma}_{nj}\gamma_{nr}q^{r-j}e^{+}_{i_{1}}\\!\wedge\\!\cdots\\!\wedge
e^{+}_{j}\wedge e^{+}_{j^{\prime}}\wedge\\!\cdots\\!\wedge e^{+}_{i_{l}}\wedge
e^{-}_{i_{1}}\wedge\\!\cdots\\!\wedge e^{-}_{r}\wedge
e^{-}_{r^{\prime}}\wedge\\!\cdots\\!\wedge e^{-}_{i_{l}},$

where $\gamma_{n}\in\mathbb{Z}$. Analogously, we can move $e^{+}_{j}$,
$e^{-}_{j}$ to the left and $e^{+}_{j^{\prime}}$ and $e^{-}_{j^{\prime}}$ to
the right by using (34) and (38), respectively.

Case $i>i_{l}$ and $i=i^{\prime}_{k}$ for some $k$: then we can move
$e^{+}_{i}$ to the right by using (34) and we get

$\displaystyle e^{+}_{i_{1}}\wedge\cdots\wedge e^{+}_{i_{l}}\wedge
e^{+}_{i}\wedge e^{-}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge e^{-}_{i_{l}}$
$\displaystyle=(-1)^{k-1}q^{k-1}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i_{l}}\wedge e^{+}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{i}\wedge e^{-}_{i_{k}}\wedge\cdots\wedge e^{-}_{i_{l}}$
$\displaystyle=(-1)^{k-1}q^{k-1}\nu_{q}\sum_{j<{i_{k}}}\gamma_{i_{k}j}q^{j-i_{k}+1}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i_{l}}\wedge e^{+}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{j}\wedge e^{-}_{j^{\prime}}\wedge\cdots\wedge e^{-}_{i_{l}}$
$\displaystyle\quad-(-1)^{k-1}q^{k-1}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i_{l}}\wedge e^{+}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{i_{k}}\wedge e^{-}_{i}\wedge\cdots\wedge e^{-}_{i_{l}}$
$\displaystyle=(-1)^{k-1}q^{k-1}\nu_{q}\sum_{j_{s}<{i_{k}}}\gamma_{i_{k}j_{s}}q^{j_{s}-i_{k}+1+\theta_{is}}e^{+}_{i_{1}}\\!\wedge\\!\cdots\\!\wedge
e^{+}_{i_{l}}\wedge e^{+}_{i}\wedge e^{-}_{i_{1}}\wedge\\!\cdots\\!\wedge
e^{-}_{j_{s}}\wedge\\!\cdots\\!\wedge
e^{-}_{j^{\prime}_{s}}\wedge\\!\cdots\\!\wedge e^{-}_{i_{l}}$
$\displaystyle\quad+(-1)^{l}q^{l-1}e^{+}_{i_{1}}\wedge\cdots\wedge
e^{+}_{i_{l}}\wedge e^{+}_{i}\wedge e^{-}_{i_{1}}\wedge\cdots\wedge
e^{-}_{i_{k}}\wedge\cdots\wedge e^{-}_{i_{l}}\wedge e^{-}_{i}.$

where $\theta_{is}\in\mathbb{Z}$.

Case $I\neq J$: in the same way it can be show that $e^{+}_{I}\wedge
e^{+}_{i}\wedge e^{-}_{i}\wedge e^{-}_{J}$ is a linear combination of elements
of the form $e^{+}_{P}\wedge e^{-}_{P}$ and $e^{+}_{S}\wedge e^{-}_{T}$ for
$S\neq T$ where $P,S,T\in\Theta(l+1)$.

Finally, if $I^{\prime}\in\Theta(l+1)$ then from the previous cases we see
that the coefficient $f_{l+1,I^{\prime},I^{\prime}}(q)$ of
$e^{+}_{I^{\prime}}\wedge e^{-}_{I^{\prime}}$ is of the form

$\displaystyle(-1)^{l}\sum_{t}f_{l,I_{t},I_{t}}(q)q^{t}+(q-q^{-1})\xi(q)+(q^{1/2}-q^{-1/2})^{2}\zeta(q)+\sum_{r}\chi_{r}(q)f_{l,I_{r},J_{r}}(q),$

where $I_{t},I_{r},J_{r}\in\Theta(l)$ and
$\xi(q),\zeta(q),\chi_{r}(q)\in\mathbb{C}[q,q^{-1}]$. Since all the
$f_{l,I_{t},I_{t}}(1)$ have the same sign and $f_{l,I_{r},J_{r}}(1)=0$, then
$f_{l+1,I^{\prime},I^{\prime}}(1)=(-1)^{l}\sum_{t}f_{l,I_{t},I_{t}}(1)\neq 0$
and

$\displaystyle\mathrm{sgn}(f_{l+1,I^{\prime},I^{\prime}}(1))=(-1)^{l}\mathrm{sgn}\big{(}\sum_{t}f_{l,I_{t},I_{t}}(1)\big{)}=(-1)^{l}(-1)^{\frac{l(l-1)}{2}}=(-1)^{\frac{l(l+1)}{2}}.$

∎

##  4 Spectrum of the $\overline{\partial}$-Laplace operator on zero forms

In this last section we present the proof of the main result of the paper by
using previous results and the representation theory of the quantum group
$U_{q}(\mathfrak{g})$.

######  Lemma 4.1.

There exist constants $\theta$, $\theta^{\prime}$, $\theta^{\prime\prime}$ and
$\theta^{\prime\prime\prime}$ such that

  1. (i)

-$\ast_{\kappa}(\partial y\wedge\ast_{\kappa}(\overline{\partial}y))=\theta y^{2}$,

  2. (ii)

-$\ast_{\kappa}(\ast_{\kappa}(\overline{\partial}y)\wedge\partial z)=\theta^{\prime}yz$,

  3. (iii)

-$\ast_{\kappa}(\partial y\wedge\ast_{\kappa}(\overline{\partial}z))=\theta^{\prime\prime}yz$,

  4. (iv)

-$\ast_{\kappa}(\ast_{\kappa}(\overline{\partial}z)\wedge\partial z)=\theta^{\prime\prime\prime}z^{2}$,

where the constants $\theta$, $\theta^{\prime}$ and
$\theta^{\prime\prime\prime}$ are non-zero.

###### Proof.

We first note that both sides of the equations (i)-(iv) are highest weight
vectors with the same weight. Since the decomposition

$\displaystyle\Omega^{(0,0)}={\mathcal{O}}_{q}(\mathbf{Q}_{N})=\bigoplus_{k,l\in\mathbb{N}_{0}}U_{q}(\mathfrak{g})\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}z^{k}y^{l}$

is multiplicity-free we have that there are constants
$\theta,\theta^{\prime},\theta^{\prime\prime},\theta^{\prime\prime\prime}$
such that equations (i)-(iv) hold.

To show that $\theta$ is non-zero, we note first that $\partial
y\wedge\ast_{\kappa}(\overline{\partial}y)\neq 0$ if and only if $\partial
y\wedge\overline{\partial}y$ is not primitive. This follows from the fact that

$\displaystyle\ast_{\kappa}(\overline{\partial}y)\wedge\partial
y=-i^{-1}\frac{1}{(n-1)!}L^{n-1}(\overline{\partial}y)\wedge\partial
y=-i^{-1}\frac{1}{(n-1)!}L^{n-2+1}(\overline{\partial}y\wedge\partial y).$

A similar argument shows that
$\ast_{\kappa}(\overline{\partial}y)\wedge\partial z)$, $\partial
y\wedge\ast_{\kappa}(\overline{\partial}z)$ and
$\ast_{\kappa}(\overline{\partial}z)\wedge\partial z$ are non-zero if and only
if $\overline{\partial}y\wedge\partial z$, $\partial
y\wedge\overline{\partial}z$ and $\overline{\partial}z\wedge\partial z$ are
not primitive, respectively.

Since the Lefschetz map is a right $U_{q}(\mathfrak{g})$-module map, if
$\overline{\partial}z\wedge\partial z$ is primitive then
$X\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}(\overline{\partial}z\wedge\partial
z)$ is primitive for any $X\in U_{q}(\mathfrak{g})$. We claim that there
exists $X\in U_{q}(\mathfrak{g})$ such that

$\displaystyle[X\raisebox{0.86108pt}{
\scriptsize{\mbox{$\blacktriangleright$}}}(\overline{\partial}y\wedge\partial
y)]=\gamma[\overline{\partial}z_{Ni}]\wedge[\partial
z_{jN}]\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for some}\ \
i,j\in\\{2,\ldots,N-1\\}\ \ \text{and}\ \ \gamma\neq 0.$

or equivalently there exists $X^{\prime}\in U_{q}(\mathfrak{g})$ and
$\gamma^{\prime}\neq 0$ such that

$\displaystyle[(\overline{\partial}y\wedge\partial)\triangleleft
X^{\prime}]=\gamma^{\prime}[\overline{\partial}z_{Ni}]\wedge[\partial
z_{iN}].$

We show this for the case $\mathfrak{so}_{2n+1}$ since for
$\mathfrak{so}_{2n}$ the proof is similar. We define the sequence

$\displaystyle
X_{1}:=F_{2},\,X_{2}:=F_{3},\ldots,X_{n-1}:=F_{n},\,X_{n}:=F_{n},\,X_{n+1}:=F_{n-1},\ldots,X_{2n-2}:=F_{2},$
$\displaystyle\leavevmode\nobreak\
X_{2n-1}:=F_{2},\ldots,X_{3n-3}:=F_{n},\,X_{3n-2}:=F_{n},\,X_{3n-1}:=F_{n-1},\ldots,X_{4n-4}:=F_{2},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
X_{4n-3}:=F_{1},\,X_{4n-2}:=F_{1}.$

If $1\leq i_{1}<\cdots<i_{r}\leq 4n-2$ then by using the formulas (13)-(15) it
can be shown that if $y\triangleleft X_{i_{1}}\cdots X_{i_{r}}$ is non-zero
then it is a multiple of one element of the following list

  1. (i’)

$z_{i,N}-q^{\gamma_{i}}z_{1,i^{\prime}}$ $2\leq i\leq N-1$,
$\gamma_{i}\in\mathbb{Z}$,

  2. (ii’)

$z_{N,N}-q^{\eta_{1}}z_{N-1,N-1}+q^{\eta_{2}}z_{2,2}-q^{\eta_{3}}z_{1,1}$,
$\eta_{i}\in\mathbb{Z}$,

  3. (iii’)

$z_{i,N-1}-q^{a_{i}}z_{2,i^{\prime}}$, $2\leq i\leq N-2$,
$a_{i}\in\mathbb{Z}$,

  4. (iv’)

$z_{N,N-1}-\mu z_{2,1}$, $\mu<0$,

and the only way to obtain $z_{N,N-1}-\mu z_{2,1}$ is by acting by
$F_{2}\cdots F^{2}_{n}\cdots F_{2}F^{2}_{1}$ on $y=q(z_{2,N}-q^{-2}z_{1,N-1})$
from the right. If we set $X:=X_{1}\cdots X_{4n-2}$ then by using the previous
list (i’)-(iv’) and Lemma 3.3 (ii), we obtain

$\displaystyle[(\partial(y)\wedge\overline{\partial}(y))\triangleleft X]$
$\displaystyle\\!=\sum_{\begin{subarray}{c}r=0\\\
p\in\mathcal{P}_{4n-2,r}\end{subarray}}^{4n-2}q^{\alpha_{r}}[\partial(y\triangleleft
X_{p(1)}\cdots X_{p(r)}]\wedge[\overline{\partial}(y\triangleleft
X_{p(r+1)}\cdots X_{p(4n-2)})]$
$\displaystyle=q^{\alpha}\partial(y\triangleleft F_{2}\cdots F_{n}^{2}\cdots
F_{2})]\wedge[\overline{\partial}(y\triangleleft F_{2}\cdots F_{n}^{2}\cdots
F_{2}F_{1}^{2})]$
$\displaystyle=\gamma[\partial(z_{N-1,N}\\!-\\!q^{\gamma_{N-1}}z_{1,2})]\wedge[\overline{\partial}(z_{N,N-1}\\!-\\!\mu
z_{2,1})]$ $\displaystyle=\gamma e^{+}_{N-2}\wedge e^{-}_{N-2},$ (41)

where $\gamma<0$ and $\mathcal{P}_{4n-2,r}$ is the set of
$(r,4n-2-r)$-shuffles, that is, the set of permutations
$p\in\mathcal{P}_{4n-2}$ such that $p(1)<\cdots<p(r)$ and
$p(r+1)<\cdots<p(4n-2)$. On the other hand, the form $e^{+}_{N-2}\wedge
e^{-}_{N-2}$ is not primitive since by Lemma 3.5 we have

$\displaystyle[\kappa^{N-3}]\wedge e^{+}_{N-2}\wedge e^{-}_{N-2}$
$\displaystyle=e^{+}_{N-2}\wedge[\kappa]^{N-3}\wedge e^{-}_{N-2}$
$\displaystyle=\sum_{I,J\in\Theta(N-3)}f_{N-3,I,J}(q)e^{+}_{N-2}\wedge
e^{+}_{I}\wedge e^{-}_{J}\wedge e^{-}_{N-2}$
$\displaystyle=\sum_{I\in\Theta(N-3)}f_{N-3,I,J}(q)e^{+}_{N-2}\wedge
e^{+}_{I}\wedge e^{-}_{I}\wedge e^{-}_{N-2}$ $\displaystyle\quad+\sum_{I\neq
J\in\Theta(N-3)}f_{N-3,I,J}(q)e^{+}_{N-2}\wedge e^{+}_{I}\wedge
e^{-}_{J}\wedge e^{-}_{N-2}$ $\displaystyle=f_{N-3,J,J}(q)e^{+}_{N-2}\wedge
e^{+}_{1}\wedge\cdots\wedge e^{+}_{N-3}\wedge e^{-}_{1}\wedge\cdots\wedge
e^{-}_{N-3}\wedge e^{-}_{N-2}$
$\displaystyle=(-1)^{N-3}q^{-(N-3)}f_{N-3,J,J}(q)e^{+}_{1}\wedge\cdots\wedge
e^{+}_{N-2}\wedge e^{-}_{1}\wedge\cdots\wedge e^{-}_{N-2},$

where $J=\\{1,\ldots,N-3\\}$ and $f_{N-3,J,J}(q)\neq 0$ if $q$ belongs to a
suitable interval around $1$. Since $e^{+}_{N-2}\wedge e^{-}_{N-2}$ is not
primitive then $(\partial(y)\wedge\overline{\partial}(y))\triangleleft
X_{1}\cdots X_{4n-2}$ is not primitive and therefore
$\partial(y)\wedge\overline{\partial}(y)$ is not primitive. The calculation
for $\overline{\partial}y\wedge\partial z$ is similar by using the fact that

$\displaystyle(\overline{\partial}y\wedge\partial z)\triangleleft
F_{1}=q\overline{\partial}(z_{2,N}-q^{-2}z_{1,N-1})\wedge\partial(z_{2,N}+qz_{1,N-1}),$

and we can proceed by acting with $X$ and obtain

$\displaystyle[(\overline{\partial}y\wedge\partial z)\triangleleft F_{1}X]$
$\displaystyle=[(\overline{\partial}y\wedge\partial z)\triangleleft
F_{1}(F_{2}\cdots F_{n}^{2}\cdots F_{2})^{2}F_{1}^{2}]$
$\displaystyle=q^{\beta}[\overline{\partial}(y\triangleleft F_{2}\cdots
F_{n}^{2}\cdots
F_{2}F_{1}^{2})]\wedge[\partial\big{(}(z_{2,N}+qz_{1,N-1})\triangleleft
F_{2}\cdots F_{n}^{2}\cdots F_{2}\big{)}]$
$\displaystyle=\gamma^{\prime}[\overline{\partial}(z_{N,N-1}-\mu
z_{2,1})]\wedge[\partial(z_{N-1,N}+q^{\gamma^{\prime}_{N-1}}z_{1,2})]$
$\displaystyle=\gamma^{\prime}e^{-}_{N-2}\wedge e^{+}_{N-2},$

where $\gamma^{\prime}\neq 0$. Similarly, it can be shown that
$e^{-}_{N-2}\wedge e^{+}_{N-2}$ is not primitive by using the fact that the
Kähler form $\kappa$ is non-zero multiple of

$\displaystyle\mathrm{i}\sum_{i,j=1}^{N}q^{2(N-i)}\overline{\partial}z_{ij}\wedge\partial
z_{ji}$

satisfying analogous results as those of Lemma 3.5, i.e., there exist
$g_{l,I,J}(q)\in\mathbb{C}[q,q^{-1}]$ such that

$\displaystyle[\kappa^{l}]=\mathrm{i}^{l}\sum_{I,J\in\Theta(l)}g_{l,I,J}(q)e^{-}_{I}\wedge
e^{+}_{J},$

where $\mathrm{sgn}(g_{l,I,I}(1))=(-1)^{\frac{l(l-1)}{2}}$ for all
$I\in\Theta(l)$ and $g_{l,I,J}(1)=0$, $I\neq J$.

Much in the same way is the calculation for
$\overline{\partial}(z)\wedge\partial z$: if we act with $F_{1}$ from the
right we obtain

$\displaystyle(\overline{\partial}z_{1,N}\wedge\partial z_{1,N})\triangleleft
F_{1}^{2}$
$\displaystyle=\alpha\overline{\partial}(z_{2,N}+qz_{1,N-1})\wedge\partial(z_{2,N}+qz_{1,N-1})+\beta\overline{\partial}(z_{1,N})\wedge\partial(z_{2,N-1})$
$\displaystyle\quad+\zeta\overline{\partial}(z_{2,N-1})\wedge\partial(z_{1,N}),$
(42)

where $\alpha,\beta,\zeta>0$. Now a routine calculation shows that

$\displaystyle\big{[}(\overline{\partial}(z_{1,N})\wedge\partial(z_{2,N-1}))\triangleleft(F_{2}\cdots
F_{n}^{2}\cdots F_{2})^{2}F_{1}^{2}\big{]}=0,$ (43)
$\displaystyle(\overline{\partial}(z_{2,N-1})\wedge\partial(z_{1,N}))\triangleleft(F_{2}\cdots
F_{n}^{2}\cdots
F_{2})^{2}F_{1}^{2}=\chi\big{(}-\overline{\partial}(z_{N,2}+q^{\alpha}z_{N-1,1})\wedge\partial(z_{2,N}+q^{\beta}z_{1,N-1})$
$\displaystyle+\vartheta_{1}\overline{\partial}(z_{N-1,2})\wedge\partial(z_{2,N-1})+\vartheta_{2}\overline{\partial}(z_{N,1})\wedge\partial(z_{1,N})\big{)},$
(44)

for a certain real $\chi>0$ and real $\vartheta_{1}$, $\vartheta_{2}$. On the
other hand, similar calculations as given in (4) show that

$\displaystyle\big{[}(\overline{\partial}(z_{2,N}+qz_{1,N-1})\wedge\partial(z_{2,N}+qz_{1,N-1}))\triangleleft(F_{2}\cdots
F_{n}^{2}\cdots F_{2}$
$\displaystyle)^{2}F_{1}^{2}\big{]}=-\gamma^{\prime}e^{-}_{N-2}\wedge
e^{+}_{N-2},$ (45)

for a real $\gamma^{\prime}>0$. Now combining equations (4)-(45) we have

$\displaystyle\big{[}(\overline{\partial}z_{1,N}\wedge\partial
z_{1,N})\triangleleft F_{1}^{2}(F_{2}\cdots F_{n}^{2}\cdots F_{2}$
$\displaystyle)^{2}F_{1}^{2}\big{]}=-\gamma^{\prime}e^{-}_{N-2}\wedge
e^{+}_{N-2}-\chi e^{-}_{1}\wedge e^{+}_{1}$

which is not primitive since $-\gamma^{\prime}$ and $-\chi$ are both negative
and we are assuming that $q$ belongs to a suitable open interval around $1$.
Finally, similar calculations work for the case $\mathfrak{so}_{2n}$ where the
element $X$ is given by $X:=(F_{2}\cdots F_{n}F_{n-2}\cdots
F_{2})^{2}F_{1}^{2}.$

∎

With this lemma and results of previous sections we are now ready to calculate
the eigenvalues of the Laplace operator $\Delta_{\overline{\partial}}$ on zero
forms.

######  Proposition 4.2.

The eigenvalue of the Dolbeault Laplace operator
$\Delta_{\overline{\partial}}$ on zero forms goes to infinity with finite
multiplicity and therefore the operator has compact resolvent.

###### Proof.

Combining Lemma 3.4, Lemma 4.1 and the identities (29) we have

$\displaystyle\Delta_{\overline{\partial}}(y^{k}z^{l})$
$\displaystyle=\overline{\partial}^{\dagger}\overline{\partial}(y^{k}z^{l})$
$\displaystyle=\overline{\partial}^{\dagger}\big{(}(k)_{q^{2}}y^{k-1}\overline{\partial}yz^{l}+(l)_{q^{-2}}y^{k}\overline{\partial}zz^{l-1}\big{)}$
$\displaystyle=-\ast_{\kappa}\circ\partial\big{(}(k)_{q^{2}}y^{k-1}\ast_{\kappa}(\overline{\partial}y)z^{l}+(l)_{q^{-2}}y^{k}\ast_{\kappa}(\overline{\partial}z)z^{l-1}\big{)}$
$\displaystyle=-\ast_{\kappa}\big{(}(k)_{q^{2}}y^{k-2}\partial
y\wedge\ast_{\kappa}(\overline{\partial}y)z^{l}+(k)_{q^{2}}y^{k-1}(\partial\circ\ast_{\kappa}\circ\overline{\partial}y)z^{l}$
$\displaystyle\quad+(k)_{q^{2}}(l)_{q^{2}}y^{k-1}(\ast_{\kappa}\circ\overline{\partial}y)\wedge\partial
zz^{l}+(l)_{q^{-2}}(k)_{q^{-2}}y^{k-1}\partial
y\wedge\ast_{\kappa}(\overline{\partial}z)z^{l-1}$
$\displaystyle\quad+(l)_{q^{-2}}y^{k}\partial\circ\ast_{\kappa}(\overline{\partial}z)z^{l-1}+(l)_{q^{-2}}(l-1)_{q^{2}}y^{k}\ast_{\kappa}(\overline{\partial}z)\wedge\partial
zz^{l-2}\big{)}$ $\displaystyle=-(k)_{q^{2}}y^{k-2}\ast_{\kappa}(\partial
y\wedge\ast_{\kappa}(\overline{\partial}y))z^{l}-(k)_{q^{2}}y^{k-1}\ast_{\kappa}(\partial\circ\ast_{\kappa}\circ\overline{\partial}y)z^{l}$
$\displaystyle\quad-(k)_{q^{2}}(l)_{q^{2}}y^{k-1}\ast_{\kappa}((\ast_{\kappa}\circ\overline{\partial}y)\wedge\partial
z)z^{l}-(l)_{q^{-2}}(k)_{q^{-2}}y^{k-1}\ast_{\kappa}(\partial
y\wedge\ast_{\kappa}(\overline{\partial}z))z^{l-1}$
$\displaystyle\quad-(l)_{q^{-2}}y^{k}\ast_{\kappa}(\partial\circ\ast_{\kappa}(\overline{\partial}z))z^{l-1}-(l)_{q^{-2}}(l-1)_{q^{2}}y^{k}\ast_{\kappa}(\ast_{\kappa}(\overline{\partial}z)\wedge\partial
z)z^{l-2}$
$\displaystyle=\big{[}\theta(k)_{q^{2}}(k-1)_{q^{-2}}+(k)_{q^{2}}\mu_{y}+(l)_{q^{2}}(k)_{q^{2}}\theta^{\prime}+(l)_{q^{-2}}(k)_{q^{-2}}\theta^{\prime\prime}+(l)_{q^{-2}}\mu_{z}$
$\displaystyle\quad+(l)_{q^{-2}}(l-1)_{q^{2}}\theta^{\prime\prime\prime}\big{]}y^{k}z^{l},$

where $\mu_{z}$ and $\mu_{y}$ are the eigenvalues of $z$ and $y$ respectively.
A differential calculus
$(\Omega^{(\bullet,\bullet)},\partial,\overline{\partial})$ is said to be
_connected_ if

$\displaystyle\text{Ker}(\partial:\Omega^{(0,0)}\rightarrow\Omega^{(1,0)})=\text{Ker}(\overline{\partial}:\Omega^{(0,0)}\rightarrow\Omega^{(0,1)})=\mathbb{C}1.$

Since the Heckengerger–Kolb calculus is connected [1] and
$\mathrm{ker}(\Delta_{\overline{\partial}})=\mathrm{ker}(\partial)\cap\mathrm{ker}(\overline{\partial})$
[24], then $\mu_{z}$ and $\mu_{y}$ are nonzero. Now we note that
$(l)_{q^{-2}}(k)_{q^{-2}}\theta^{\prime\prime}$ and $(l)_{q^{-2}}\mu_{z}$ have
finite limit as $k,l\rightarrow\infty$ and therefore they do not contribute in
the asymptotic behaviour of the eigenvalue. By Lemma 4.1 we have that
$\theta^{\prime}\neq 0$. It is not difficult to see that if
$\theta^{\prime}<0$ then the eigenvalue would tend to $-\infty$ if $k$ is
fixed and $l\rightarrow\infty$. Therefore $\theta^{\prime}>0$.

Now we can consider the sum

$\displaystyle\big{(}\theta\mu_{y}^{-1}(k-1)_{q^{-2}}+1\big{)}(k)_{q^{2}}\mu_{y}+(l)_{q^{-2}}(l-1)_{q^{2}}\theta^{\prime\prime\prime}.$
(46)

First we note that

$\displaystyle\theta<-(1-q^{-2})\mu_{y}\Rightarrow\big{(}1+(k-1)_{q^{-2}}\theta\mu_{y}^{-1}\big{)}<0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{as}\ \ k\rightarrow\infty,$

and this would imply that the eigenvalues of $\Delta_{\overline{\partial}}$ on
$y^{k}$ are negative. Therefore $\theta\geq-(1-q^{-2})\mu_{y}$. If
$\theta=-(1-q^{-2})\mu_{y}$ then
$(\theta\mu_{y}^{-1}(k-1)_{q^{-2}}+1)(k)_{q^{2}}\mu_{y}$ converges when
$k\rightarrow\infty$. On the other hand, if $\theta>-(1-q^{-2})\mu_{y}$ then
$(\theta\mu_{y}^{-1}(k-1)_{q^{-2}}+1)(k)_{q^{2}}\mu_{y}$ tends to infinity.

Now if $\theta^{\prime\prime\prime}<0$ then

$\displaystyle\Delta_{\overline{\partial}}(z^{l})=\theta^{\prime\prime\prime}(l)_{q^{-2}}(l-1)_{q^{2}}+(l)_{q^{-2}}\mu_{z}=(\mu_{z}^{-1}\theta^{\prime\prime\prime}(l-1)_{q^{2}}+1)\mu_{z}(l)_{q^{-2}}<0,$

as $l\rightarrow\infty$ which contradicts the fact the eigenvalues of $z^{l}$
are non-negatives. Then $\theta^{\prime\prime\prime}\geq 0$. By Lemma 4.1 we
have that $\theta^{\prime\prime\prime}>0$ and this implies that the second
term of (46) tends to infinity as $l\rightarrow\infty$. Combining the above
results we can conclude that the eigenvalues of $\Delta_{\overline{\partial}}$
tend to infinity, as $k,l\rightarrow\infty$. ∎

### Acknowledgments

FDG was supported by the Charles University PRIMUS grant _Spectral
Noncommutative Geometry of Quantum Flag Manifolds_ PRIMUS/21/SCI/026. The
author would like to thank Réamonn Ó Buachalla for his support and useful
discussions during the preparation of this paper.

## References

  * [1] A. Carotenuto, F. Díaz García, R. Ó Buachalla: _A Borel–Weil theorem for the irreducible quantum flag manifolds_. To appear in International Mathematics Research Notices.
  * [2] F. D’Andrea, L. Da̧browski: _Dirac operators on quantum projective spaces_. Commun. Math. Phys. 295, 731-790, 2010.
  * [3] F. D’Andrea, L. Da̧browski, G. Landi: _The noncommutative geometry of the quantum projective plane_. Rev. Math. Phys. 20, 979-1006, 2008.
  * [4] F. D’Andrea, L. Da̧browski, G. Landi, E. Wagner: _Dirac operators on all Podleś quantum spheres_. J. Noncommut. Geom. 1, 213-239, 2007.
  * [5] L. Da̧browski, G. Landi, A. Sitarz, W. van Suijlekom, J.C. Várilly: _The Dirac operator on $SU_{q}(2)$_. Commun. Math. Phys. 259, 729-759, 2005.
  * [6] L. Da̧browski, A. Sitarz: _Dirac operator on the standard Podleś quantum sphere_. Banach Cent. Publ. 61, 49-58, 2003.
  * [7] B. Das, R. Ó Buachalla, P. Somberg: _A Dolbeault–Dirac spectral triple for quantum proyective space_. Doc. Math. 25, 1079-1157, 2020.
  * [8] B. Das, R. Ó Buachalla, P. Somberg: _Compact quantum homogeneous Kähler spaces_. arXiv:1910.14007.
  * [9] B. Das, R. Ó Buachalla, P. Somberg: _Spectral gaps for twisted Dolbeault–Dirac operators over the irreducible quantum flag manifolds_. arXiv:2206.10719.
  * [10] F. Díaz García, R. Ó Buachalla, E. Wagner: _A Dolbeault–Dirac spectral triple for the $B_{2}$-irreducible quantum flag manifold_. arXiv:2109.09885.
  * [11] F. Díaz García, A. Krutov, R. Ó Buachalla, P. Somberg, K. R. Strung: _Positive line bundles over the irreducible quantum flag manifolds_. arXiv:1912.08802.
  * [12] I. Heckenberger, S. Kolb: _The locally finite part of the dual coalgebra of irreducible quantum flag manifolds_. Proc. London Math. Soc. 89, 457-484, 2004.
  * [13] I. Heckenberger, S. Kolb: _De Rham complex for the quantized irreducible flag manifolds_. J. Algebra, 305, 704-741, 2006.
  * [14] I. Heckenberger, S. Kolb: _Differential forms via the Bernstein–Gelfand–Gelfand resolution for quantized irreducible flag manifolds_. J. Geom. Phys. 57, 2316-2344, 2007.
  * [15] J. Humphreys: _An introduction to Lie algebras and representation theory_. Springer-Verlag, New York, 1972.
  * [16] D. Huybrechts: _Complex geometry: an introduction_. Universitext, Springer-Verlag, Heidelberg, 2005.
  * [17] A. Klimyk, K. Schmüdgen: _Quantum groups and their representations_. Texts and Monograph in Physics, Springer-Verlag, Berlin, 1998.
  * [18] U. Krähmer, M. Tucker-Simmons: _On the Dolbeault–Dirac operator on quantized symmetric spaces_. Trans. London Math. Soc. 2, 33-56, 2015.
  * [19] M. Krämer: _Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen_. Compisitio Math. 38, 129-153, 1979.
  * [20] S. Majid: _Noncommutative Rieamannian and spin geometry of the standard $q$-sphere_. Comm. Math. Phys. 256, 255-285, 2005.
  * [21] M. Matassa: _Kähler structures on quantum irreducible flag manifolds_. J. Geom. Phys. 145 (2019), 103477.
  * [22] M. Matassa: _The Parthasarathy formula and a spectral triple for the quantum Lagrangian Grassmanians of rank two_. Lett. Math. Phys. 109, 1703-1734, 2019.
  * [23] S. Neshveyev, L. Tuset: _The Dirac operator on compact quantum groups_. J. Reine. Angew. Math. 641, 1-20, 2010.
  * [24] R. Ó Buachalla: _Noncommutative Kähler structures of quantum homogeneous spaces_. Adv. Math. 322, 892-939, 2017.
  * [25] R. Ó Buachalla: _Noncommutative complex structures on quantum homogeneous spaces_. J. Geom. Phys. 99, 154-173, 2016.
  * [26] M. Takeuchi: _Relative Hopf modules - equivalence and freeness criteria_. J. Algebra, 60, 452-471, 1979.
  * [27] A.L. Onishchick, E. B. Vinberg: _Lie groups and algebraic groups_. Springer-Verlag, Heidelberg, 1990.