# A Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds

Alessandro Carotenuto  Mathematical Institute of Charles University,
Sokolovská 83, Prague, Czech Republic<EMAIL_ADDRESS>,  Fredy Díaz
García  Mathematical Institute of Charles University, Sokolovská 83, Prague,
Czech Republic<EMAIL_ADDRESS>and  Réamonn Ó Buachalla
Mathematical Institute of Charles University, Sokolovská 83, Prague, Czech
Republic<EMAIL_ADDRESS>
###### Abstract.

We establish a noncommutative generalisation of the Borel–Weil theorem for the
Heckenberger–Kolb calculi of the irreducible quantum flag manifolds
$\mathcal{O}_{q}(G/L_{S})$, generalising previous work of a number of authors
(including the first and third authors of this paper) on the quantum
Grassmannians $\mathcal{O}_{q}(\mathrm{Gr}_{n,m})$. As a direct consequence we
get a novel noncommutative differential geometric presentation of the quantum
coordinate rings $S_{q}[G/L_{S}]$ of the irreducible quantum flag manifolds.
The proof is formulated in terms of quantum principal bundles, and the
recently introduced notion of a principal pair, and uses the Heckenberger and
Kolb first-order differential calculus for the quantum Possion homogeneous
spaces $\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$.

RÓB, AC, and FDG are supported by the Charles University PRIMUS grant
_Spectral Noncommutative Geometry of Quantum Flag Manifolds_
PRIMUS/21/SCI/026. AC was also supported by the GAČR project 20-17488Y and
RVO: 67985840. FDG was also supported by Conacyt (Consejo Nacional de Ciencia
y Tecnología, México) and PAEP (Programa de apoyos para estudios de posgrado,
UNAM, México). RÓB also acknowledges FNRS support through a postdoctoral
fellowship within the framework of the MIS Grant “Antipode” grant number
F.4502.18, and was also supported by GAČR through the framework of the grant
GA19-06357S.

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of the Paper
  2. 2 Preliminaries on Quantum Principal Bundles
    1. 2.1 First-Order Differential Calculi
    2. 2.2 Connections and Line Modules
    3. 2.3 Hopf–Galois Extensions
    4. 2.4 Quantum Principal Bundles
    5. 2.5 Principal Connections
  3. 3 Preliminaries on Drinfeld–Jimbo Quantum Groups
    1. 3.1 Quantised Enveloping Algebras
    2. 3.2 Type-1 Representations
    3. 3.3 Quantum Coordinate Algebras
    4. 3.4 Quantum Flag Manifolds
    5. 3.5 The Associated Quantum Poisson Homogeneous Spaces
  4. 4 Liouville’s Theorem for $\mathcal{O}_{q}(G/L_{S})$
    1. 4.1 Tangent spaces
    2. 4.2 Quantum Root Vectors
    3. 4.3 The Heckenberger–Kolb $(0,1)$-Calculus
    4. 4.4 Liouville’s Theorem
  5. 5 A Quantum Principal Bundle Presentation of $\Omega^{(0,1)}_{q}(G/L_{S})$
    1. 5.1 An Alternative Construction of $\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$
    2. 5.2 The Heckenberger–Kolb First-Order Differential Calculus on $\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$
    3. 5.3 Principal Pairs and Quantum Principal Bundles
    4. 5.4 A Quantum Principal Bundle
    5. 5.5 A Left $\mathcal{O}_{q}(G)$-Covariant Principal Connection
    6. 5.6 The Holomorphic Structure of the Relative Line Modules
  6. 6 A Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds
    1. 6.1 The Borel–Weil Theorem
    2. 6.2 A Holomorphic Description of the Quantum Homogeneous Coordinate Ring
    3. 6.3 The Opposite Borel–Weil Theorem
  7. A Relative Line Modules and Spherical Generators
    1. A.1 The Table of Spherical Weights
    2. A.2 Spherical Generators for Line bundles
  8. B Noncommutative Complex and Holomorphic Structures
    1. B.1 Differential Calculi
    2. B.2 Complex Structures
    3. B.3 Holomorphic Structures
    4. B.4 The Irreducible Quantum Flag Manifolds

##  1\. Introduction

The classical Borel–Weil theorem is a foundational result in geometric
representation theory which realises each irreducible representation of a
complex semisimple Lie algebra $\mathfrak{g}$ as the space of holomorphic
sections of a line bundle over a flag manifold. Extensions of this result to
the setting of quantum groups came soon after the discovery of quantum groups
themselves. These generalisations took a variety of different forms [1, 34,
19, 13, 29, 30], as discussed for example, in the introduction to [8].

In all of the above works no explicit noncommutative formulation of
holomorphicity was used. In recent years, however, a framework for describing
noncommutative holomorphic sections has emerged, expressed in terms of
differential graded algebras and generalising the Koszul–Malgrange
presentation of holomorphic vector bundles [24]. Indeed, this approach was in
part developed with a view to establishing a differential geometric
$q$-deformation of the Borel–Weil theorem. This direction of research was
initiated by Majid in his influential paper on the Podleś sphere [27], see
also [3, §7.4.1]. It was continued by Khalkhali, Landi, van Suijlekom, and
Moatadelro in [20, 21, 22] where the definitions of complex structure and
noncommutative holomorphic vector bundle were introduced and the family of
examples extended to include quantum projective space
$\mathcal{O}_{q}(\mathbb{CP}^{n})$. An equivalent definition of holomorphic
structure would later appear independently in the work of Beggs and Smith on
noncommutative coherent sheaves [4].

The differential graded algebras underlying all of these constructions are the
remarkable Heckengerber–Kolb differential calculi, arguably the most important
family of noncommutative differential structures in the theory of quantum
groups. These calculi are an essentially unique covariant $q$-deformation of
the de Rham complex of the irreducible quantum flag manifolds. This family
contains the quantum Grassmannians as the $A$-series special case. In
particular, it contains quantum projective space
$\mathcal{O}_{q}(\mathbb{CP}^{n})$, which reduces to the celebrated Podleś
sphere for the $n=1$ case. Thus it is very natural to try and extend the
Borel–Weil theorem from quantum projective space to the general irreducible
quantum flag manifolds. A significant step in this direction was the proof, by
the second and third authors and Krutov, Somberg, and Strung in [11], that
each relative line module over an irreducible quantum flag manifold admits a
unique covariant holomorphic structure. Building on this work, the Borel–Weil
theorem was generalised to the quantum Grassmannians by Morzinski and the
first and third authors in [8]. A major element of the proof was the
construction of a differential calculus, extending the Heckenberger–Kolb
calculus quantum Grassmannian calculus, for the Poisson quantum homogeneous
space $\mathcal{O}_{q}(S^{n,m})$, that is to say, the direct sum of the line
modules over $\mathcal{O}_{q}(\mathrm{Gr}_{n,m})$. This was achieved by taking
a suitable quotient of the bicovariant calculus of $\mathcal{O}_{q}(SU_{n})$
associated to its coquasitriangular structure. This extended the work of the
third author in [31] for the special case of quantum projective space
$\mathcal{O}_{q}(\mathbb{CP}^{n})$, and provided a novel coquasitriangular
presentation of the Heckenberger–Kolb calculus of
$\mathcal{O}_{q}(\mathrm{Gr}_{n,m})$.

While this approach is almost certainly extendable to the general irreducible
setting, it is at present not clear how to do so without a concerted technical
effort. Hence in this paper we adopt a different approach and take advantage
of the work of Heckenberger and Kolb who already constructed a left
$\mathcal{O}_{q}(G)$-covariant first-order differential calculus over
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$, the quantum Poisson homogeneous
space of $\mathcal{O}_{q}(G/L_{S})$. We show that these calculi give a quantum
principal bundles over the irreducible quantum flag manifolds using the formal
framework of principal pairs, as introduced by Mrozinski and the first and
third authors in [8]. Principal pairs, a special type of principal comodule
algebra constructed from a nested pair of quantum homogeneous spaces, and
provide a simple framework in which to verify the requirements of a quantum
principal bundle.

We then build upon this quantum principal presentation and construct principal
connections. The Heckenberger–Kolb calculus over
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ deforms a certain subspace of the
complexified one-forms of $G/L^{\,\mathrm{s}}_{S}$, namely the subspace whose
Lie algebra is spanned by positive root vectors. This means the bundle has no
horizontal forms, and that the zero map is a strong principal connection. This
results in a particularly simple realisation of the holomorphic structures
$\overline{\partial}_{{\mathcal{E}}_{k}}:{\mathcal{E}}_{k}\to\Omega^{(0,1)}\otimes_{\mathcal{O}_{q}(G/L_{S})}{\mathcal{E}}_{k}$.
Indeed, it enables us to simplify the approach of [8], and to finally prove
the Borel–Weil theorem for _all_ the irreducible quantum flag manifolds.

An important motivation of the paper is to further explore the connections
between the quantum coordinate algebras $\mathcal{O}_{q}(G/L_{S})$ and their
quantum homogeneous coordinate ring counterparts $S_{q}[G/L_{S}]$ studied in
noncommutative projective geometry, see for example [2, 35, 37, 39]. For the
special case of the quantum Grassmannians, these rings are $q$-deformations of
the Plücker embedding homogeneous coordinate ring and are important examples
in the theory of quantum cluster algebras [21, 22]. The quantum Borel–Weil
theorem gives us a direct $q$-deformation of the classical ample line bundle
presentation of $S_{q}[G/L_{S}]$. This directly generalises the work of [20,
21, 22] for quantum projective space, and gives us an important point of
contact between noncommutative differential geometry and noncommutative
projective geometry.

The Borel–Weil theorem has a number of important applications in associated
works. One example is to the study of the noncommutative Kähler geometry of
$\mathcal{O}_{q}(G/L_{S})$. The cohomological information given by the
Borel–Weil theorem allows us to identify which line modules over
$\mathcal{O}_{q}(G/L_{S})$ are positive and which are negative. Through an
application of the Kodaira vanishing theorem [33] we can then establish
vanishing of higher cohomologies for positive line modules. Moreover, the
Borel–Weil theorem allows us to conclude that twisting the Dolbeault–Dirac
operator of $\mathcal{O}_{q}(G/L_{S})$ by a negative line module produces a
Fredholm operator, which is to say, it allows us to conclude analytic
behaviour from purely geometric information [9].

###  1.1. Summary of the Paper

In §2, we recall the basic notions of Hopf–Galois extensions and quantum
homogeneous spaces. We recall how such structures interact with first-order
differential calculi, focusing on the theory of quantum principal bundles.

In §3 we treat the Drinfeld–Jimbo quantised enveloping algebras, quantum
coordinate algebras, and the quantum flag manifolds.

In §4 we recall the well-known tangent space formulation of covariant first-
order calculi over quantum homogeneous $\mathcal{O}_{q}(G)$-spaces. We then
use the tangent space formulation of the Heckenberger–Kolb calculi to
establish a direct $q$-deformation of Liouville’s theorem for the irreducible
quantum flags, which is to say, we show that the first anti-holomorphic
cohomology group of the calculus is of dimension $1$.

In §5 we give a quantum principal bundle description of the Heckenberger–Kolb
calculus, construct a principal connection, and use it to give an explicit
realisation of the covariant holomorphic structures of the relative line
modules ${\mathcal{E}}_{k}$ over $\mathcal{O}_{q}(G/L_{S})$.

In §6 we establish the main result of the paper, namely the Borel–Weil theorem
for the irreducible quantum flag manifolds.

######  Theorem 1.1.

For any irreducible quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$, it holds
that

  1. 1.

$H^{0}(\mathcal{E}_{k})$ is an irreducible $U_{q}(\mathfrak{g})$-module of
highest weight $-k\varpi_{x}$,

  2. 2.

$H^{0}(\mathcal{E}_{-k})=0$.

for all $k\in\mathbb{Z}_{>0}$, where $\Pi\backslash S$ consists of the simple
root $\alpha_{x}$, and $\varpi_{x}$ is the corresponding fundamental weight.

We then present an application of the Borel–Weil theorem and give a novel
noncommutative differential geometric presentation of the quantum homogeneous
coordinate rings $S_{q}(G/L_{S})$ of the irreducible quantum flag manifolds.
We finish by describing the situation for the opposite complex structure.

### Acknowledgments

We would like to thank to Elmar Wagner for suggesting the proof of Theorem
4.4.

##  2\. Preliminaries on Quantum Principal Bundles

We begin with a presentation of the necessary results on quantum group
noncommutative geometry, namely the theory of covariant differential calculi
and quantum principal bundles. All this material is by now quite well-know,
and a more detailed presentation can be found in the recent monograph [3].

###  2.1. First-Order Differential Calculi

A first-order differential calculus, or simply a _fodc_ , over an algebra $B$
is a pair $(\Omega^{1},\mathrm{d})$, where $\Omega^{1}$ is a $B$-bimodule and
$\mathrm{d}:B\to\Omega^{1}$ is a derivation such that $\Omega^{1}$ is
generated as a left $B$-module by those elements of the form $\mathrm{d}b$,
for $b\in B$. We call $\mathrm{d}$ the _exterior derivative_ of the fodc. The
universal fodc over $B$ is the pair $(\Omega^{1}_{u}(B),\mathrm{d}_{u})$,
where $\Omega^{1}_{u}(B)$ is the kernel of the multiplication map
$m_{B}:B\otimes B\to B$ endowed with the obvious bimodule structure, and
$\mathrm{d}_{u}$ is the map defined by

$\displaystyle\mathrm{d}_{u}:B\to\Omega^{1}_{u}(B),$ $\displaystyle b\mapsto
1\otimes b-b\otimes 1.$

Every fodc over $B$ is of the form
$\left(\Omega^{1}_{u}(B)/N,\,\mathrm{proj}\circ\mathrm{d}_{u}\right)$, where
$N$ is a $B$-sub-bimodule of $\Omega^{1}_{u}(B)$, and
$\mathrm{proj}:\Omega^{1}_{u}(B)\to\Omega^{1}_{u}(B)/N$ is the canonical
quotient map. For any subalgebra $B^{\prime}\subseteq B$, the _restriction_ of
a fodc over $B$ to $B^{\prime}$ is the fodc
$\Omega^{1}(B^{\prime})\subseteq\Omega^{1}(B)$ over $B^{\prime}$ generated by
the elements $\mathrm{d}b$, for $b\in B^{\prime}$.

For $H$ a Hopf algebra $H$, and $P$ a right $H$-comodule algebra, a fodc over
$P$ is said to be right $H$-covariant if the following (necessarily unique)
map is well defined

$\displaystyle\Delta_{R}:\Omega^{1}(P)\to\Omega^{1}(P)\otimes H,$
$\displaystyle p\mathrm{d}q\mapsto p_{(0)}\mathrm{d}q_{(0)}\otimes
p_{(1)}q_{(1)}.$

Covariance for a fodc over a left comodule algebra is defined similarly.

###  2.2. Connections and Line Modules

For $\Omega^{1}$ a fodc over an algebra $B$, and $\mathcal{F}$ a left
$B$-module, a _connection_ on ${\mathcal{F}}$ is a $\mathbb{C}$-linear map
$\nabla:\mathcal{F}\to\Omega^{1}\otimes_{B}{\mathcal{F}}$ satisfying

$\displaystyle\nabla(bf)=\mathrm{d}b\otimes f+b\nabla f,$
$\displaystyle\textrm{ for all }b\in B,f\in{\mathcal{F}}.$

In this paper, we are concerned not with connections for general $B$-modules,
but for modules which generalise the space of sections of a classical line
bundle: a _line module_ over $B$ is be an invertible $B$-bimodule
${\mathcal{E}}$, where _invertible_ means that there exists another
$B$-bimodule ${\mathcal{E}}^{\vee}$ such that

(1)
$\displaystyle{\mathcal{E}}\otimes_{B}{\mathcal{E}}^{\vee}\simeq{\mathcal{E}}^{\vee}\\!\otimes_{B}{\mathcal{E}}\simeq
B.$

Note that any such ${\mathcal{E}}$ is automatically projective as a left, and
right, $B$-module.

###  2.3. Hopf–Galois Extensions

A right $H$-comodule algebra $(P,\Delta_{R})$ is said to be an $H$-Hopf–Galois
extension of $B:=P^{\mathrm{co}(H)}$ if for $m_{P}:P\otimes_{B}P\to P$ the
multiplication of $P$, a bijection is given by

$\mathrm{can}:=(m_{P}\otimes\mathrm{id})\circ(\mathrm{id}\otimes\Delta_{R}):P\otimes_{B}P\to
P\otimes H.$

The Hopf–Galois condition is equivalent to exactness of the sequence

(2) $\displaystyle 0\longrightarrow
P\Omega^{1}_{u}(B)P{\buildrel\iota\over{\longrightarrow}}\Omega^{1}_{u}(P){\buildrel\mathrm{ver}\over{\longrightarrow}}P\otimes
H^{+}\longrightarrow 0,$

where $\iota$ is the inclusion map, and
$\mathrm{ver}:=\mathrm{can}\circ\mathrm{proj}_{B}$ with $\mathrm{proj}_{B}$
the restriction to $\Omega^{1}_{u}(P)$ of the canonical projection $P\otimes
P\to P\otimes_{B}P$.

We now consider a special type of Hopf–Galois extension which plays an
important role in this paper. Let $A$ be a Hopf algebra, then a _quantum
homogeneous space_ is a coideal subalgebra $B\subseteq A$ such that $A$ is
faithfully flat as a right $B$-module, and $AB^{+}=B^{+}A$, where we have
written $B^{+}:=\ker(\varepsilon|_{B}:B\to\mathbb{C})$. Denoting by
$\pi_{B}:A\to A/B^{+}A$ the canonical surjective Hopf algebra map, the
associated coaction

$\Delta_{R,\pi_{B}(A)}:=(\mathrm{id}\otimes\pi_{B})\circ\Delta:A\to
A\otimes\pi_{B}(A),$

gives $A$ the structure of a right $\pi_{B}(A)$-comodule algebra. Moreover, as
shown in [41], faithful flatness implies that $B$ is the space of coinvariants
of this coaction. As is well-known, $P$ is automatically a Hopf–Galois
extension of $B$, see for example [40, Lemma 3.9]. A _relative line module_
over a quantum homogeneous $A$-space $B$ is a $B$-sub-bimodule, left
$A$-comodule, ${\mathcal{E}}\subseteq A$ that is also a line module over $B$,
and for which the isomorphisms in (1) are left $A$-comodule maps.

###  2.4. Quantum Principal Bundles

The following definition, due to Brzeziński and Majid [3, 6], presents
sufficient criteria for the existence of a non-universal version of the
sequence (2).

######  Definition 2.1.

Let $H$ be a Hopf algebra. A quantum principal $H$-bundle is a pair
$(P,\Omega^{1}(P))$, consisting of a right $H$-comodule algebra
$(P,\Delta_{R})$ and a right-$H$-covariant calculus $\Omega^{1}(P)$, such
that:

  1. 1.

$P$ is a Hopf–Galois extension of $B=P^{\,\mathrm{co}(H)}.$

  2. 2.

If $N\subseteq\Omega^{1}_{u}(P)$ is the sub-bimodule of the universal calculus
corresponding to $\Omega^{1}(P)$, we have $\mathrm{ver}(N)=P\otimes I$, for
some $\mathrm{Ad}$-sub-comodule right ideal

$I\subseteq H^{+}:=\ker(\varepsilon:H\to\mathbb{C}),$

where $\mathrm{Ad}:H\to H\otimes H$ is defined by
$\mathrm{Ad}(h):=h_{(2)}\otimes S(h_{(1)})h_{(3)}.$

Denoting by $\Omega^{1}(B)$ the restriction of $\Omega^{1}(P)$ to $B$, and
$\Lambda^{1}(H):=H^{+}/I$, the quantum principal bundle definition implies
that an exact sequence is given by

(3) $\displaystyle 0\longrightarrow
P\Omega^{1}(B)P{\buildrel\iota\over{\longrightarrow}}\,\Omega^{1}(P){\buildrel\leavevmode\nobreak\
\mathrm{ver}\leavevmode\nobreak\ \leavevmode\nobreak\
\over{\longrightarrow}}P\otimes\Lambda^{1}(H)\longrightarrow 0,$

where by abuse of notation $\mathrm{ver}$ denotes the map induced on
$\Omega^{1}(P)$ by identifying $\Omega^{1}(P)$ as a quotient of
$\Omega_{u}^{1}(P)$. We denote $\Omega^{1}(P)_{\mathrm{hor}}:=P\Omega^{1}(B)P$
and call this subspace the _horizontal forms_ of the bundle.

###  2.5. Principal Connections

In this subsection, we briefly recall the theory of principal connections for
quantum principal bundle at a level of generality suitable to the paper. For
the full picture see [3, §5], or the accompanying paper [8, §2].

######  Definition 2.2.

A principal connection for a quantum principal $H$-bundle $(P,\Omega^{1}(P))$
is a left $P$-module, right $H$-comodule, projection
$\Pi:\Omega^{1}(P)\to\Omega^{1}(P)$ satisfying

$\ker(\Pi)=P\Omega^{1}(B)P.$

A principal connection $\Pi$ is called strong if
$(\mathrm{id}-\Pi)\big{(}\mathrm{d}P\big{)}\subseteq\Omega^{1}(B)P$. We denote
$\Omega^{1}(P)_{\mathrm{ver}}:=\mathrm{im}(\Pi)$ and call this subspace the
space of _vertical forms_ associated to $\Pi$.

For any left $P$-submodule right $H$-subcomodule ${\mathcal{E}}\subseteq P$,
we can use principal connections to define a connection
$\nabla:{\mathcal{E}}\to\Omega^{1}(B)\otimes_{B}{\mathcal{E}}$: Note first
that an isomorphism

$\displaystyle
j:\Omega^{1}(B)\otimes_{B}{\mathcal{E}}\hookrightarrow\Omega^{1}(B){\mathcal{E}},$

is given by the multiplication map. A strong principal connection $\Pi$
defines a connection $\nabla$ on ${\mathcal{E}}$ by

$\displaystyle\nabla:=j^{-1}\circ(\mathrm{id}-\Pi)\circ\mathrm{d}:{\mathcal{E}}\to\Omega^{1}(B)\otimes_{B}{\mathcal{E}}.$

Consider now the special case where $P$ is endowed with a left $A$-coaction
giving it the structure of an $(A,H)$-bicomodule. If the principal connection
$\Pi$ is a left $A$-comodule map, then we see that the connection $\nabla$
will also be a left $A$-comodule map. This will be the case for all the
principal connections considered in this paper.

##  3\. Preliminaries on Drinfeld–Jimbo Quantum Groups

In this section we treat the Drinfeld–Jimbo quantised enveloping algebras,
quantum coordinate algebras, quantum flag manifolds, and their associated
quantum Poisson homogeneous spaces.

###  3.1. Quantised Enveloping Algebras

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 $\Pi=\\{\alpha_{1},\dots,\alpha_{\ell}\\}$ 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
$\alpha_{i}^{\vee}:=2\alpha_{i}/(\alpha_{i},\alpha_{i})$.

Let $q\in{\mathbb{R}}$ such that $q\neq-1,0,1$, 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
$i=1,\ldots,l$, subject to 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\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $

and the quantum Serre relations which we omit (see [23, §6.1.2] for details).
The formulae

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

define a Hopf algebra structure on $U_{q}(\mathfrak{g})$, satisfying
$\varepsilon(E_{i})=\varepsilon(F_{i})=0$, and $\varepsilon(K_{i})=1$.

###  3.2. Type-1 Representations

The set of fundamental weights $\\{\varpi_{1},\dots,\varpi_{l}\\}$ of
$\mathfrak{g}$ is the dual basis of simple coroots
$\\{\alpha_{1}^{\vee},\dots,\alpha_{l}^{\vee}\\}$. We denote by
${\mathcal{P}}$ the ${\mathbb{Z}}$-span of the fundamental weights, and by
${\mathcal{P}}^{+}$ the $\mathbb{Z}_{\geq 0}$-span of the fundamental weights.
For any $U_{q}(\mathfrak{g})$-module $V$, a vector $v\in V$ is called a
_weight vector_ of _weight_ $\mathrm{wt}(v)\in\mathcal{P}$ if

(4) $\displaystyle K_{i}\triangleright v=q^{(\alpha_{i},\mathrm{wt}(v))}v,$
$\displaystyle\textrm{ for all }i=1,\ldots,r.$

For each $\lambda\in\mathcal{P}^{+}$ there exists an irreducible finite-
dimensional $U_{q}(\mathfrak{g})$-module $V_{\lambda}$, uniquely defined by
the existence of a weight vector $v_{\mathrm{hw}}\in V_{\lambda}$ of weight
$\lambda$, which we call a highest weight vector, satisfying
$E_{i}\triangleright v_{\mathrm{hw}}=0,$ for all $i=1,\dots,l$. We call any
finite direct sum of such $U_{q}(\mathfrak{g})$-representations a type-$1$
representation. Each type-$1$ module $V_{\lambda}$ decomposes into a direct
sum of _weight spaces_ , which is to say, subspaces of $V_{\lambda}$ spanned
by weight vectors of any given weight.

We denote by ${}_{U_{q}(\mathfrak{g})}\mathbf{type}_{1}$ the full subcategory
of ${U_{q}(\mathfrak{g})}$-modules whose objects are finite sums of type-$1$
modules $V_{\lambda}$, for $\lambda\in\mathcal{P}^{+}$. This category admits a
braided monoidal structure. Explicitly, for $V$ and $W$ two finite-dimensional
irreducible objects in ${}_{U_{q}(\mathfrak{g})}\mathbf{type}_{1}$, the
braiding is completely determined if one demands that $R_{V,W}$ is a
$U_{q}(\mathfrak{g})$-module homomorphism satisfying

$R_{V,W}(v\otimes w)=q^{(\mathrm{wt}(v),\mathrm{wt}(w))}w\otimes v+\sum_{i\in
I}w_{i}\otimes v_{i},$

where $I$ is some finite index set, and $w_{i}\in W$, $v_{i}\in V$ such that
$\mathrm{w}\succ\mathrm{wt}(w_{i})$ and $\mathrm{w}\succ\mathrm{wt}(w_{i})$,
with respect to the partial order $\succ$ on $\mathcal{P}$. Given a choice of
weight bases $\\{e_{i}\\}_{i=1}^{\mathrm{dim}(V)}$, and
$\\{f_{i}\\}_{i=1}^{\mathrm{dim}(W)}$, for two finite-dimensional
$U_{q}(\mathfrak{g})$-modules $V,W$,

$\displaystyle R_{V,W}(e_{i}\otimes
f_{j})=:\sum_{k=1}^{\dim(W)}\sum_{l=1}^{\mathrm{dim}(V)}(R_{V,W})^{kl}_{ij}f_{k}\otimes
e_{l},$

defines the associated $R$-matrix $(R_{V,W})^{kl}_{ij}$.

###  3.3. Quantum Coordinate Algebras

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 easily checked 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

(5)
$\displaystyle\mathcal{O}_{q}(G):=\bigoplus_{\lambda\in\mathcal{P}^{+}}C(V_{\lambda}).$

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 (5) the _Peter–Weyl decomposition_ of
$\mathcal{O}_{q}(G)$.

###  3.4. Quantum Flag Manifolds

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

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

Consider now the coideal subalgebra of $U_{q}(\mathfrak{l}_{S})$-invariants

$\displaystyle\mathcal{O}_{q}\big{(}G/L_{S}\big{)}:={}^{U_{q}(\mathfrak{l}_{S})}\mathcal{O}_{q}(G),$

with respect to the natural left $U_{q}(\mathfrak{g})$-module structure on
$\mathcal{O}_{q}(G)$. Cosemisimplicity of the Hopf dual of
$U_{q}(\mathfrak{l}_{S})$ implies that $\mathcal{O}_{q}(G/L_{S})$ is a quantum
homogeneous space. We call it the quantum flag manifold associated to $S$.

###  3.5. The Associated Quantum Poisson Homogeneous Spaces

Classically the algebra $\mathfrak{l}_{S}$ is reductive, and hence decomposes
into a direct sum $\mathfrak{l}_{S}^{\,\mathrm{s}}\oplus\mathfrak{u}_{1}$,
comprised of a semisimple part and a commutative part, respectively. In the
quantum setting, we are thus motivated to consider the Hopf subalgebra

$\displaystyle
U_{q}(\mathfrak{l}^{\,\mathrm{s}}_{S}):=\big{<}K_{i},E_{i},F_{i}\,|\,i\in
S\big{>}\subseteq U_{q}(\mathfrak{l}_{S}).$

Consider now the coideal subalgebra of
$U_{q}(\mathfrak{l}^{\,\mathrm{s}}_{S})$-invariants

$\displaystyle\mathcal{O}_{q}\big{(}G/L^{\,\mathrm{s}}_{S}\big{)}:={}^{U_{q}(\mathfrak{l}^{\,\mathrm{s}}_{S})}\mathcal{O}_{q}(G).$

Just as for the quantum flag manifolds, each
$\mathcal{O}_{q}\big{(}G/L^{\,\mathrm{s}}_{S}\big{)}$ is a quantum homogeneous
space. We call $\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$ the _quantum homogeneous
Poisson space associated_ to $S$.

Let $\\{v_{i}\\}_{i=1}^{N_{x}}$ be a weight basis of $V_{\varpi_{x}}$ such
that $v_{N_{x}}$ is a highest weight vector. We denote the dual basis of
$V_{-w_{0}(\varpi_{x})}$ by $\\{f_{i}\\}_{i=1}^{N_{x}}$. As shown in [38,
§Theorem 4.1], a set of generators for
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ is given by

$\displaystyle z_{i}:=c^{\varpi_{x}}_{f_{i},v_{N}},$
$\displaystyle\overline{z}_{j}:=c^{-w_{0}(\varpi_{x})}_{v_{j},f_{N}}$
$\displaystyle\text{ for
}i,j=1,\dots,N_{x}:=\dim(V_{\varpi_{x}}),\,x\in\Pi\backslash S,$

where to lighten notation we have respectively written $\varpi_{x}$, and
$-w_{0}(\varpi_{x})$, as superscripts instead of $V_{\varpi_{x}}$, and
$V_{-w_{0}(\varpi_{x})}$.

##  4\. Liouville’s Theorem for $\mathcal{O}_{q}(G/L_{S})$

In this section we prove a direct $q$-deformation of Liouville’s theorem for
the irreducible flag manifolds. More explicitly, we show that the kernel of
the anti-holomorphic differential $\overline{\partial}$ of the
$(0,1)$-Heckenberger–Kolb calculus contains only scalar multiples of the
identity. The proof uses the tangent space formulation of a fodc and the
quantum root vector presentation of the Heckenberger–Kolb calculi.

###  4.1. Tangent spaces

In this subsection we recall the tangent space formulation of covariant fodc,
over quantum homogeneous spaces, as formulated in [14]. For sake of
simplicity, we present the material for the special case of Drinfeld–Jimbo
quantum groups. In particular, we consider a quantum homogeneous
$\mathcal{O}_{q}(G)$-space of invariants $B:={}^{K}\mathcal{O}_{q}(G)$, for
some Hopf subalgebra $K\subseteq U_{q}(\mathfrak{g})$.

######  Definition 4.1.

A finite-dimensional subspace $T\subseteq B^{\circ}$ is called a _tangent
space_ for $B$ if

$\displaystyle 1.\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta(T)\subseteq
T_{\varepsilon}\otimes B^{\circ},$ $\displaystyle 2.\leavevmode\nobreak\
\leavevmode\nobreak\ KT\subseteq T,$

where we have denoted $T_{\varepsilon}:=T\oplus\mathbb{C}\varepsilon$.

For every tangent space $T$, the subspace

$\displaystyle I:=\left\\{b\in B^{+}\,|\,X(b)=0,\mbox{for all }X\in
T\right\\}$

is a right $B$-submodule, and a right $K$-submodule, of $B^{+}$. Thus the
quotient $V:=B^{+}/I$ has the structure of a right $B$-module, and a right
$K$-module.

We denote by $\Omega^{1}(B)$ the $K$-invariant subspace of
$\mathcal{O}_{q}(G)\otimes V$, that is to say,

$\left\\{\sum_{i}f_{i}\otimes v_{i}\in\mathcal{O}_{q}(G)\otimes
V\,|\,\sum_{i}(Y\triangleright f_{i})\otimes
v_{i}=\sum_{i}f_{i}\otimes(v_{i}\triangleleft Y),\textrm{ \, for all }Y\in
K\right\\}\\!.$

We endow $\mathcal{O}_{q}(G)\otimes V$ with a $B$-bimodule structure by taking
left multiplication on the first tensor factor, and by taking the diagonal
action on the right. With respect to this choice of bimodule structure,
$\Omega^{1}(B)$ is a $B$-sub-bimodule of $\mathcal{O}_{q}(G)\otimes V$.
Moreover, the obvious left $\mathcal{O}_{q}(G)$-comodule structure of
$\mathcal{O}_{q}(G)\otimes V$ restricts to a left
$\mathcal{O}_{q}(G)$-coaction on $\Omega^{1}(B)$.

Consider next the linear map

$\displaystyle\mathrm{d}:B\to\Omega^{1}(B),$ $\displaystyle b\mapsto
b_{(1)}\otimes[b_{(2)}^{+}],$

where $[-]$ denotes the coset in $V$ of an element in $B$. As is readily
checked, the pair $(\Omega^{1}(B),\mathrm{d})$ is a left
$\mathcal{O}_{q}(G)$-covariant fodc over $B$. For any choice of basis
$\\{X_{i}\\}_{i=1}^{n}$ of $T$, an explicit formula for $\mathrm{d}$ is given
by

(6) $\displaystyle\mathrm{d}b=\sum_{i=1}^{n}(X_{i}\triangleright b)\otimes
e_{i},$

where $\\{e_{i}\\}_{i=1}^{n}$ is the dual basis of $V$. Moreover, as shown by
Heckenberger and Kolb in [14], every left $\mathcal{O}_{q}(G)$-covariant fodc
$(\Omega^{1},B)$, for which $\Omega^{1}$ is finitely-generated as a left
$B$-module, arises in this way. This gives a bijective correspondence between
these two structures.

######  Remark 4.2.

Tangent spaces over quantum homogeneous spaces can be given a more formal
treatment in terms of Takeuchi’s categorical equivalence. See [14] for more
details.

###  4.2. Quantum Root Vectors

Let $w_{0}=w_{i_{1}}\cdots w_{i_{k}}$ be a reduced decomposition of the
longest element of the Weyl group of $\mathfrak{g}$. It can be shown that the
sequence

$\displaystyle\beta_{1}:=\alpha_{i_{1}},\leavevmode\nobreak\
\beta_{2}:=w_{i_{1}}(\alpha_{i_{2}}),\leavevmode\nobreak\
\dots,\leavevmode\nobreak\ \beta_{k}:=w_{i_{1}}\cdots
w_{i_{k}-1}(\alpha_{i_{k}})$

exhausts all the positive roots of $\mathfrak{g}$. This motivates the
following definition of root vector in the noncommutative setting, as
presented in detail in [26], or [23, §6.2].

For $q\in\mathbb{C}\backslash\\{0\\}$, the quantum integer $[m]_{q}$ is the
complex number

$[m]_{q}:=q^{-m+1}+q^{-m+3}+\cdots+q^{m-3}+q^{m-1}.$

Moreover, we have the _quantum factorials_

$\displaystyle[n]_{q}!=[n]_{q}[n-1]_{q}\cdots[2]_{q}[1]_{q},$
$\displaystyle[0]_{q}!=1.$

To every $i=1,\ldots,l$, there corresponds an algebra automorphism $T_{i}$ of
$U_{q}(\mathfrak{g})$ which acts on the generators as

$\displaystyle T_{i}(K_{j})=K_{j}K_{i}^{-a_{ij}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ T_{i}(E_{i})=-F_{i}K_{i},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ T_{i}(F_{i})=-K_{i}^{-1}E_{i},$ $\displaystyle
T_{i}(E_{j})=\sum_{t=0}^{-a_{ij}}(-1)^{t-a_{ij}}q_{i}^{-t}(E_{i})^{(-a_{ij}-t)}E_{j}(E_{i})^{(t)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{ for }i\neq
j,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \,$ $\displaystyle
T_{i}(F_{j})=\sum_{t=0}^{-a_{ij}}(-1)^{t-a_{ij}}q_{i}^{t}(F_{i})^{(t)}E_{j}(F_{i})^{(-a_{ij}-t)},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{ for
}i\neq j,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \,\,$

where $(E_{i})^{(n)}:=E_{i}^{n}/[n]_{q}!$, and
$(F_{i})^{(n)}:=F_{i}^{n}/[n]_{q}!$. The mapping $w_{i}\rightarrow T_{i}$
determines a homomorphism of the braid group $\mathfrak{B}_{\mathfrak{g}}$
into the group of algebra automorphisms of $U_{q}(\mathfrak{g})$. Associated
to a choice of reduced decomposition of $w_{0}=w_{i_{1}}\cdots w_{i_{k}}$ of
the longest element of the Weyl group, we have the elements

$\displaystyle E_{\beta_{r}}:=T_{i_{1}}T_{i_{2}}\cdots
T_{i_{r-1}}(E_{i_{r}}),$ $\displaystyle\textrm{ for }r=1,\dots,k.$

We call any such element a _root vector_ of $U_{q}(\mathfrak{g})$. Just as in
the classical case, the root vectors of $U_{q}(\mathfrak{g})$ have an
associated PBW basis [23, §6.2.3].

###  4.3. The Heckenberger–Kolb $(0,1)$-Calculus

We now present the Heckenberger-Kolb calculus, or more precisely the
$(0,1)$-summand of the Heckenberger–Kolb calculus (the $(1,0)$-summand will be
presented in §6.3).

We begin by introducing some notation: let $R\subseteq\mathfrak{h}^{*}$ denote
the root system associated to our choice of Cartan sub-algebra $\mathfrak{h}$
and let $R^{+}$, respectively $R^{-}$, be the set of positive, respectively
negative, roots. Denote

$\displaystyle R_{S}^{\pm}:=\mathbb{Z}S\cap R^{\pm},$
$\displaystyle\overline{R^{\pm}_{S}}:=R^{\pm}\setminus R_{S}^{\pm}.$

######  Definition 4.3.

The _Heckenberger–Kolb calculus_ is the left $\mathcal{O}_{q}(G)$-covariant
fodc calculus associated to the tangent space

$T^{(0,1)}:=\mathrm{span}_{\mathbb{C}}\\!\left\\{E_{\beta}\,|\,\beta\in\overline{R^{+}_{S}}\right\\}\subseteq\mathcal{O}_{q}(G/L_{S})^{\circ},$

where by abuse of notation, $E_{\beta}$ denotes the coset of the root vector
in $\mathcal{O}_{q}(G/L_{S})^{\circ}$.

Here we have chosen to describe the calculus explicitly in terms of its
corresponding tangent space. Alternatively the calculus can be described as
one of the two left $\mathcal{O}_{q}(G)$-covariant finite-dimensional
irreducible fodc over $\mathcal{O}_{q}(G/L_{S})$, see [15], or [12] for a
presentation in the notation of the present paper.

###  4.4. Liouville’s Theorem

Since the classical homogeneous space $G/L_{S}$ is a compact complex manifold,
it follows from Liouville’s theorem that its only globally holomorphic
functions are scalar multiples of the identity. In this subsection we show
that this result extends to the quantum setting using the tangent space
description of the $(0,1)$-Heckenberger–Kolb calculus. This result will be
used in §6 when we prove the Borel–Weil theorem for negative line bundles
${\mathcal{E}}_{-k}$.

######  Theorem 4.4.

For any irreducible quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$, with
associated $(0,1)$-Heckenberger-Kolb calculus $\Omega_{q}^{(0,1)}(G/L_{S})$,
it holds that

$\displaystyle
H^{0}_{q}(G/L_{S})=\mathrm{ker}\left(\overline{\partial}:\mathcal{O}_{q}(G/L_{S})\rightarrow\Omega_{q}^{(0,1)}(G/L_{S})\right)=\mathbb{C}1.$

###### Proof.

Since $\alpha_{x}$ cannot be contained in the $\mathbb{Z}$-span of $S$, we
must have that $\alpha_{s}\in\overline{R_{S}^{+}}$. Hence we must have that
$E_{x}\in T^{(0,1)}$. Next take an arbitrary $b\in\mathcal{O}_{q}(G/L_{S})$.
It follows from the definition of the exterior derivative given in (6), and
the fact that $E_{x}\in T^{(0,1)}$, that $\overline{\partial}(b)=0$ only if
$E_{x}\triangleright b=0.$ However, since $\mathcal{O}_{q}(G/L_{S})$ is by
definition the space of invariants of $U_{q}(\mathfrak{l}_{S})$, we know that
$E_{s}\triangleright b=0$, for all $s\neq x$, and moreover, that $b$ is a
weight vector of weight zero. Taken together, these two facts imply that $b$
is a highest weight vector of degree zero. However, it follows from the
Peter–Weyl decomposition of $\mathcal{O}_{q}(G)$ that the only such elements
are of the form $\lambda 1$, for some $\lambda\in\mathbb{C}$. ∎

######  Remark 4.5.

As established in [28], the Heckenberger–Kolb calculi all possess
noncommutative Kähler structures in the sense of [32, Definition 7.1]. Since
the Dolbeault cohomology of any differential calculus endowed with a Kähler
structure refines its de Rham cohomology [32, Corollary 7.7], we have that the
three cohomology groups $H^{0}_{\mathrm{d}}$, $H^{0}_{\partial}$, and
$H^{0}_{\overline{\partial}}$ all coincide. This motivates our decision to
denote the cohomology in Theorem 4.4 as $H^{0}_{q}(G/L_{S})$, without any
choice of reference to the exterior derivative involved.

##  5\. A Quantum Principal Bundle Presentation of
$\Omega^{(0,1)}_{q}(G/L_{S})$

In this subsection we recall Heckenberger and Kolb’s construction of a left
$\mathcal{O}_{q}(G)$-covariant fodc over the Poisson quantum homogeneous space
$\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$. We then show that this calculus is
right $\mathcal{O}(U_{1})$-covariant and hence gives a quantum principal
bundle presentation of the anti-holomorphic Heckenberger-Kolb calculi on the
irreducible quantum flag manifolds.

###  5.1. An Alternative Construction of $\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$

We now recall the alternative description of
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$ given by Heckenberger and Kolb in
[16, §3.1]. This alternative description is needed for the construction of
Heckenberger and Kolb’s fodc over $\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$ in the
following subsection. We begin by introducing the quantum homogeneous
coordinate rings.

######  Definition 5.1.

For any irreducible quantum flag manifold, the _homogeneous coordinate ring_
$S_{q}[G/L_{S}]$ is the subalgebra of $\mathcal{O}_{q}(G/L_{S})$ generated by
the elements $z_{i}$. Moreover, the _opposite homogeneous coordinate ring_
$S_{q}[G/L^{\mathrm{op}}_{S}]$ is the subalgebra of $\mathcal{O}_{q}(G)$
generated by the elements $\overline{z}_{i}$.

It is well known [5] that $S_{q}[G/L_{S}]$ is a quadratic algebra. Explicitly

$S_{q}[G/L_{S}]\simeq\mathbb{C}\langle z_{1},\dots
z_{N_{x}}\rangle/\langle\sum_{k,l=1}^{N_{x}}(R_{V_{\varpi_{x}},V_{\varpi_{x}}})^{ij}_{kl}z_{k}z_{l}-q^{(\lambda,\lambda)}z_{i}z_{j}\rangle.$

Similarly, the opposite homogeneous coordinate ring
$S_{q}[G/L^{\mathrm{op}}_{S}]$ is a quadratic algebra, admitting an analogous
$R$-matrix presentation. Consider the vector space

$S_{q}[G/L_{S}]_{\mathbb{C}}:=S_{q}[G/L_{S}]\otimes
S_{q}[G/L^{\mathrm{op}}_{S}].$

Endow $S_{q}[G/L_{S}]_{\mathbb{C}}$ with the multiplication

(7) $\displaystyle(1\otimes\overline{z}_{i})(z_{j}\otimes
1):=q^{(\lambda,\lambda)}\sum_{k,l=1}^{N_{x}}(R^{-1}_{V_{\varpi_{x}},V_{-w_{0}(\varpi_{x})}})^{ij}_{kl}\,z_{k}\otimes\overline{z}_{l}.$

It follows that

$c:=\sum_{i=1}^{N_{x}}(1\otimes\overline{z}_{i})(z_{i}\otimes 1)$

is a central right $U_{q}(\mathfrak{g})$-invariant element of
$S_{q}[G/L_{S}]_{\mathbb{C}}$. Consider the quotient algebra

$\displaystyle
S_{q}[G/L_{S}]_{\mathbb{C}}^{c=1}=S_{q}[G/L_{S}]_{\mathbb{C}}/\langle
c-1\rangle,$

where $\langle c-1\rangle$ denotes the two-sided ideal of
$S_{q}[G/L_{S}]_{\mathbb{C}}$ generated by the element $c-1$. It follows from
[16, Lemma 3.1] that the multiplication map

$\displaystyle
m:S_{q}[G/L_{S}]_{\mathbb{C}}^{c=1}\to\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S}),$

is an isomorphism, giving us the claimed alternative description of
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$. As an immediate consequence, we get
the identity

(8) $\displaystyle\sum_{i=1}^{N_{x}}\overline{z}_{i}z_{i}=1.$

As quadratic algebras both $S_{q}[G/L_{S}]$ and $S_{q}[G/L^{\mathrm{op}}_{S}]$
have a natural $\mathbb{Z}_{\geq 0}$-grading. This allows us to define a
$\mathbb{Z}$-grading on $S_{q}[G/L_{S}]_{\mathbb{C}}$ by setting

(9) $\displaystyle\mathrm{deg}(z_{i})=1,$
$\displaystyle\mathrm{deg}(\overline{z}_{i})=-1.$

Since $c$ is of degree zero, the grading descends to a $\mathbb{Z}$-grading on
$S_{q}[G/L_{S}]_{\mathbb{C}}^{c=1}$, giving us a $\mathbb{Z}$-grading on
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$.

###  5.2. The Heckenberger–Kolb First-Order Differential Calculus on
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$

Define $\Gamma$ to be the left $S_{q}[G/L_{S}]$-module generated by the
elements $\overline{\partial}z_{i}$, for $i=1,\dots,N_{x}$, subject to the
relations

$\sum_{k,l=1}^{N_{x}}\left[R^{2}_{V_{\varpi_{x}},V_{\varpi_{x}}}+q^{(\varpi_{x},\varpi_{x})}(q^{(\alpha_{x},\alpha_{x})}-1)R_{V_{\varpi_{x}},V_{\varpi_{x}}}+q^{2(\varpi_{x},\varpi_{x})-(\alpha_{x},\alpha_{x})}\mathrm{id}\right]^{kl}_{ij}z_{i}\overline{\partial}z_{j}=0,$

for all $i,j=1,\dots,N_{x}$. As shown in [16, §3.2.3], we can endow $\Gamma$
with a left $S_{q}[G/L_{S}]_{\mathbb{C}}$-module structure such that a
derivation $\overline{\partial}:S_{q}[G/L_{S}]_{\mathbb{C}}\to
S_{q}[G/L_{S}]\otimes\Gamma,$ is uniquely determined by

$\overline{\partial}:f\otimes\overline{z}_{i}\mapsto
f\otimes\overline{\partial}\overline{z}_{i}.$

This gives us a left $\mathcal{O}_{q}(G)$-covariant fodc over
$S_{q}[G/L_{S}]_{\mathbb{C}}$.

Consider the sub-bimodule of $S_{q}[G/L_{S}]\otimes\Gamma$ generated by the
form $\overline{\partial}c$ and the subsets

$\displaystyle(c-1)(S_{q}[G/L_{S}]\otimes\Gamma),$
$\displaystyle(S_{q}[G/L_{S}]\otimes\Gamma)(c-1).$

Quotienting $S_{q}[G/L_{S}]\otimes\Gamma$ by this sub-bimodule, we get a left
$\mathcal{O}_{q}(G)$-covariant fodc over
$\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})$, which we denote by

$\left(\,\Omega^{(0,1)}_{q}(G/L^{\,\mathrm{s}}_{S}),\,\overline{\partial}\,\right)\\!.$

We see that by construction $\overline{\partial}f=0$, for all $f\in
S_{q}[G/L_{S}]$.

###  5.3. Principal Pairs and Quantum Principal Bundles

We now recall the notion of a principal pair, introduced in [8] as a framework
in which to construct quantum principal bundles. We use the equivalent, yet
simpler, formulation introduced in [7, Definition 3.8].

######  Definition 5.2.

For a Hopf algebra $A$, a _principal $A$-pair_ is a pair of quantum
homogeneous $A$-spaces $B\subseteq P$, such that
$\pi_{B}(P)=\pi_{B}(A)^{\mathrm{co}(\pi_{P}(A))}$ is a Hopf subalgebra of
$\pi_{B}(A)$, with respect to the coaction

$\displaystyle\pi_{B}(P)\to\pi_{B}(P)\otimes\pi_{P}(A),$
$\displaystyle\pi_{B}(p)\mapsto\pi_{B}(p_{(1)})\otimes\pi_{P}(p_{(2)}).$

For principal pairs, the quantum principal bundle condition reduces to a much
simpler covariance requirement on the total space calculus, as we now recall.

######  Proposition 5.3.

For a principal pair $B\subseteq P$, let $\Omega^{1}(P)$ be a left
$A$-covariant, and right $\pi_{B}(A)$-covariant, fodc over $P$. It holds that
the pair $(P,\Omega^{1}(P))$ is a quantum principal bundle.

We will use this result in the following subsection to produce a quantum
principal bundle presentation of the Heckenberger–Kolb calculi.

###  5.4. A Quantum Principal Bundle

The motivating example of a principal pair was the Poisson circle bundle over
the quantum Grassmannians. In [7] this was extended to include all Poisson
torus bundles over quantum flag manifolds, as we now recall.

######  Theorem 5.4.

For any Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{g})$, and any subset
$S\subseteq\Pi$ of simple roots of $\mathfrak{g}$, the pair

$\left(\mathcal{O}_{q}(G/L_{S}),\,\mathcal{O}_{q}(G/L^{\,\mathrm{s}}_{S})\right)$

is a principal pair of quantum homogeneous $\mathcal{O}_{q}(G)$-spaces.

This means that we can now use Proposition 5.3 to produce a quantum principal
bundle from the Heckenberger–Kolb calculi. We first establish right
$\mathcal{O}(U_{1})$-covariance of the calculus over
$\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$, and then conclude the quantum principle
structure.

######  Lemma 5.5.

The fodc $\Omega^{(0,1)}_{q}(G/L^{\,\mathrm{s}}_{S})$ is right
$\mathcal{O}(U_{1})$-covariant.

###### Proof.

Consider the $\mathbb{Z}$-grading on $\Gamma$ defined by

$\displaystyle\deg(z_{i})=\deg(\overline{\partial}z_{i})=1.$

Since the defining relations of $\Gamma$ are of degree $2$ with respect to
this grading, it must descend to a well-defined $\mathbb{Z}$-grading on
$\Gamma$. Moreover, since $\overline{\partial}$ is clearly a degree zero map,
we have that the fodc is right $\mathcal{O}(U_{1})$-covariant.

With respect to the grading on $S_{q}[G/L^{\mathrm{op}}_{S}]$, inherited from
the grading of $\mathcal{O}_{q}(L^{\mathrm{s}}_{S})$, we take the tensor
product grading on

(10) $\displaystyle S[G/L^{\mathrm{op}}_{S}]\otimes\Gamma.$

With respect to this grading, the differential $\overline{\partial}$ is again
a degree zero map, meaning that we have a right $\mathcal{O}(U_{1})$-covariant
fodc.

Finally, we note that the elements $c-1$ and $\overline{\partial}c$ are
homogeneous and of degree zero. This means that the sub-bimodule they generate
is homogeneous, and so, the $\mathbb{Z}$-grading of (10) descends to a
$\mathbb{Z}$-grading on $\Omega^{(0,1)}_{q}(G/L^{\,\mathrm{s}}_{S})$, giving
us a right $\mathcal{O}(U_{1})$-covariant fodc. ∎

This result, taken together with Proposition 5.3 implies the following
proposition.

######  Proposition 5.6.

The pair

$\left(\Omega^{(0,1)}_{q}(G/L_{S}^{\,\mathrm{s}}),\Delta_{R,\,\mathcal{O}(U_{1})}\right)\\!$

is a quantum principal bundle.

###  5.5. A Left $\mathcal{O}_{q}(G)$-Covariant Principal Connection

In this subsection we construct a strong principal connection for our bundle.
We begin with some results about the horizontal forms of the bundle.

######  Lemma 5.7.

For the quantum principal bundle
$(\Omega^{(0,1)}_{q}(G/L_{S}^{\mathrm{s}}),\Delta_{R})$, it holds that

  1. 1.

$\Omega^{(0,1)}_{q}(G/L^{\mathrm{s}}_{S})_{\mathrm{hor}}=\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$,

  2. 2.

$\Omega^{(0,1)}_{q}(G/L^{\mathrm{s}}_{S})_{\mathrm{ver}}=0$.

###### Proof.

1\. An arbitrary form in $\Omega^{(0,1)}_{q}(G/L_{S})$ is a sum of elements of
the form $b\,\overline{\partial}b^{\prime}$, for
$b,b^{\prime}\in\mathcal{O}_{q}(G/L_{S})$. Consider now the form

$\displaystyle
z_{i}b\,\overline{\partial}b^{\prime}\in\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})\Omega^{(0,1)}_{q}(G/L_{S}),$
$\displaystyle\textrm{ for any }i=1,\dots,N_{x}.$

It follows from (7) and (8) that there exist elements $f_{j}\in
S_{q}[G/L_{S}]$ and $v_{j}\in S_{q}[G/L^{\mathrm{op}}_{S}]$ such that
$\sum_{j}f_{j}v_{j}=1.$ Then it follows from the Leibniz rule that

$\displaystyle
z_{i}b\overline{\partial}b^{\prime}=\overline{\partial}(z_{i}bb^{\prime})-\overline{\partial}(z_{i}b)b^{\prime}=$
$\displaystyle\,\sum_{j}\overline{\partial}(z_{i}bb^{\prime})f_{j}v_{j}-\sum_{j}\overline{\partial}(z_{i}b)f_{j}v_{j}b^{\prime}.$

Moreover, since $\overline{\partial}f_{j}=0$, we have that

$\displaystyle\sum_{j}\overline{\partial}(z_{i}bb^{\prime})f_{j}v_{j}-\sum_{j}\overline{\partial}(z_{i}b)f_{j}v_{j}b^{\prime}=\sum_{j}\overline{\partial}(z_{i}bb^{\prime}f_{j})v_{j}-\sum_{j}\overline{\partial}(z_{i}bf_{j})v_{j}b^{\prime}.$

Thus we see that $z_{i}b\overline{\partial}b^{\prime}$ is an element of
$\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$, for any
$i=1,\dots,N_{x}$. An analogous argument establishes that

$\overline{z}_{i}b\overline{\partial}b^{\prime}\in\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S}).$

Thus we can conclude that

$\displaystyle\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})\Omega^{(0,1)}_{q}(G/L_{S})\subseteq\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S}).$

2\. Consider now a general element
$\omega\in\Omega^{(0,1)}_{q}(G/L_{S}^{\,\mathrm{s}})$. It follows from the
Leibniz rule, and the fact that $\overline{\partial}z_{i}=0$, for all
$i=1,\dots,N_{x}$, that $\omega$ is a sum of the form

$\displaystyle\sum_{i}p_{i}(\overline{\partial}\overline{z}_{i})p^{\prime}_{i},$
$\displaystyle\textrm{ for
}p_{i},p^{\prime}_{i}\in\mathcal{O}_{q}(G/L_{S}^{\,\mathrm{s}}).$

Using the same argument as above, we see that

$\displaystyle p_{i}(\overline{\partial}\overline{z}_{i})p^{\prime}_{i}=$
$\displaystyle\,\sum_{j}p_{i}(\overline{\partial}\overline{z}_{i})f_{j}v_{j}p_{i}^{\prime}=\sum_{j}p_{i}\overline{\partial}(\overline{z}_{i}f_{j})v_{j}p_{i}^{\prime}.$

It now follows from the presentation of the horizontal forms given in 1. that

$\displaystyle\Omega^{(0,1)}_{q}(G/L_{S}^{\,\mathrm{s}})=\Omega^{(0,1)}_{q}(G/L_{S})\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S}).$

Thus we see that $\Omega^{(0,1)}_{q}(G/L^{\mathrm{s}}_{S})_{\mathrm{ver}}=0$.
∎

From this lemma, and the definition of a strong principal connection, we get
the following proposition.

######  Proposition 5.8.

The zero map on $\Omega^{(0,1)}_{q}(G/L^{\mathrm{s}}_{S})_{\mathrm{hor}}$ is a
left $\mathcal{O}_{q}(G)$-covariant strong principal connection.

######  Remark 5.9.

The zero projection can in fact be realised as the restriction of any
universal connection, just as for the quantum Grassmannians case [8]. For
example, an explicit construction of a universal connection is given in [7,
Proposition 5.9].

###  5.6. The Holomorphic Structure of the Relative Line Modules

We denote by

$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})\simeq\bigoplus_{k\in\mathbb{Z}}{\mathcal{E}}_{k}$

the decomposition of $\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$ into homogeneous
subspaces given by the $\mathbb{Z}$-grading defined in §5.1. It follows from
[7, Proposition 5.7] that this is a decomposition of
$\mathcal{O}_{q}(G/L^{\mathrm{s}}_{S})$ into covariant line modules, and
moreover, that every relative line module is of this form.

The principal connection identified in the subsection above associates to each
${\mathcal{E}}_{k}$ a left $\mathcal{O}_{q}(G)$-covariant connection, which by
uniqueness must coincide with the holomorphic structure
$\overline{\partial}_{{\mathcal{E}}_{k}}$. Thus we have produced a principal
connection presentation of the holomorphic structure of ${\mathcal{E}}_{k}$.
Explicitly, the holomorphic structure can be described as

(11)
$\displaystyle\overline{\partial}_{{\mathcal{E}}_{k}}:{\mathcal{E}}_{k}\to\Omega^{(0,1)}\otimes_{\mathcal{O}_{q}(G/L_{S})}{\mathcal{E}}_{k},$
$\displaystyle e\mapsto j^{-1}\circ\overline{\partial}(e),$

where $j$ is the map defined in §2.5.

##  6\. A Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds

In this section we prove the main result of the paper, namely the Borel-Weil
theorem for $(0,1)$-Heckenberger–Kolb calculi of the irreducible quantum flag
manifolds $\mathcal{O}_{q}(G/L_{S})$. We also discuss the situation for the
opposite complex structure of $\mathcal{O}_{q}(G/L_{S})$, and give a
noncommutative geometric presentation of the quantum homogeneous ring of
$S_{q}[G/L_{S}]$.

###  6.1. The Borel–Weil Theorem

In this subsection we prove a direct noncommutative generalisation of the
Borel-Weil theorem for the irreducible flag manifolds.

######  Theorem 6.1.

For any irreducible quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$, it holds
that

  1. 1.

$H^{0}(\mathcal{E}_{k})=z^{k}U_{q}(\mathfrak{g})$ is an irreducible
$U_{q}(\mathfrak{g})$-module of highest weight $k\varpi_{x}$,

  2. 2.

$H^{0}(\mathcal{E}_{-k})=0$.

for all $k\in\mathbb{Z}_{>0}$, where $\Pi\backslash S$ consists of the simple
root $\alpha_{x}$.

###### Proof.

1\. From the construction of the Heckenberger–Kolb calculus
$\Omega^{(0,1)}_{q}(G/L^{\,\mathrm{s}}_{S})$, we know that $z^{k}$ is
contained in the kernel of the differential $\overline{\partial}$, for all
$k\in\mathbb{Z}_{\geq 0}$. Thus it follows from (11) that $z^{k}$ is contained
in the kernel of the holomorphic structure map
$\overline{\partial}_{{\mathcal{E}}_{k}}$. Since
$\overline{\partial}_{{\mathcal{E}}_{k}}$ is a right
$U_{q}(\mathfrak{g})$-module map, the inclusion

$z^{k}U_{q}(\mathfrak{g})\subseteq H^{0}(\mathcal{E}_{k})$

now follows from Schur’s lemma.

To show the opposite inclusion, we need to consider the decomposition of
${\mathcal{E}}_{k}$ into irreducible right $U_{q}(\mathfrak{g})$-modules. As
established in Proposition A.1, each irreducible subcomodule contains an
element of the form $bz^{k}$, for some $b\in\mathcal{O}_{q}(G/L_{S})$. Let us
assume that one of these elements $bz^{k}$, for $b\notin\mathbb{C}1$, were
holomorphic. This would imply that

$\displaystyle
0=\overline{\partial}_{{\mathcal{E}}_{k}}\\!(bz^{k})=\overline{\partial}b\otimes
z^{k}+b\otimes\overline{\partial}_{{\mathcal{E}}_{k}}(z^{k})=\overline{\partial}b\otimes
z^{k}.$

Noting that $\Omega^{(0,1)}_{q}\\!\left(G/L^{\mathrm{\,s}}_{S}\right)$ is a
torsion-free right $\mathcal{O}_{q}(G/L_{S})$-module, that $\mathcal{E}_{k}$
is projective as left $\mathcal{O}_{q}(G/L_{S})$-module and that
$\mathcal{O}_{q}(G/L_{S})$ has no zero-divisors, we see that Liouville’s
theorem implies that

$\overline{\partial}b\otimes z^{k}\neq 0.$

Thus we see that no such holomorphic element exists, and so, the claimed
identity $z^{k}U_{q}(\mathfrak{g})\subseteq H^{0}(\mathcal{E}_{k})$ follows
from Schur’s lemma. Moreover, it is clear that $z^{k}U_{q}(\mathfrak{g})$ is
an irreducible $U_{q}(\mathfrak{g})$-module of highest weight $k\varpi_{s}$.

2\. We now come to the line bundles $\mathcal{E}_{-k}$, for
$k\in\mathbb{Z}_{>0}$. Assume there exists a non-zero holomorphic element
$e\in\mathcal{E}_{-k}$. This implies that, for any $i=1,\dots,N_{x}$, we have

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \overline{\partial}\big{(}ez^{\,k}_{i}\big{)}=$
$\displaystyle\,\overline{\partial}\big{(}ez^{\,k}_{i}\big{)}=\,\overline{\partial}(e)z^{\,k}_{i}+e\overline{\partial}(z^{\,k}_{i})=0,$

which is to say

$ez^{\,k}_{i}\in
H^{0}_{q}(G/L_{S}):=\ker(\overline{\partial}:\mathcal{O}_{q}(G/L_{S})\to\Omega^{(0,1)}_{q}(G/L_{S})).$

Liouville’s Theorem 4.4 implies that $ez_{i}^{k}$ must be a non-zero scalar
multiple of $1$, and so, we must have a distinct right inverse to $e$, for
each $i=1,\dots,N_{x}$. However, since $\mathcal{O}_{q}(G)$ has no zero
divisors, right inverses are unique. To avoid contradiction we are forced to
conclude that ${\mathcal{E}}_{-k}$ contains no holomorphic elements. ∎

###  6.2. A Holomorphic Description of the Quantum Homogeneous Coordinate Ring

Any positive line bundle ${\mathcal{E}}$ over a compact Kähler manifold $M$
gives an embedding of $M$ into complex projective space $\mathbb{CP}^{n}$.
Moreover, the associated homogeneous coordinate ring $S(M)$ is isomorphic to

(12) $\displaystyle\bigoplus_{k\in\mathbb{Z}_{\geq
0}}H^{0}({\mathcal{E}}^{\otimes k}),$

as a graded algebra. For the flag manifolds, the line bundle
${\mathcal{E}}_{1}$ is well-known to be positive, and hence we have an
associated projective embedding. For the special case of the Grassmannians,
this embedding reduces to the celebrated Plücker embedding.

The following proposition establishes a quantum generalisation of this result,
giving a noncommutative differential geometric realisation of the quantum
homogeneous coordinate ring $S_{q}[G/L_{S}]$ presented in the Definition 5.1.
This generalises earlier work in [20, 21, 22] for the case of quantum
projective space, and work by Mrozinski and the first and third authors in [8]
for the more general family of the quantum Grassmannians.

######  Proposition 6.2.

For any irreducible quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$, it holds
that

(13) $\displaystyle\bigoplus_{k\in\mathbb{Z}_{\geq
0}}H^{0}(\mathcal{E}_{k})=S_{q}[G/L_{S}],$

where ${\mathcal{E}}_{\pm k}$ is a line module over
$\mathcal{O}_{q}(G/L_{S})$, with $k\in\mathbb{Z}_{\geq 0}$, and both algebras
are considered as subalgebras of $\mathcal{O}_{q}(G)$.

###### Proof.

Recall from Theorem 6.1 that
$H^{0}({\mathcal{E}}_{k})=z^{k}U_{q}(\mathfrak{g})$. Thus since
$S_{q}[G/L_{S}]$ is a right $U_{q}(\mathfrak{g})$-submodule of
$\mathcal{O}_{q}(G)$, it follows that

$\displaystyle\bigoplus_{k\in\mathbb{Z}_{\geq
0}}H^{0}({\mathcal{E}}_{k})\subseteq S_{q}[G/L_{S}].$

We now establish the opposite inclusion. For $e\in
H^{0}({\mathcal{E}}_{k}),\,e^{\prime}\in H^{0}({\mathcal{E}}_{l})$, observe
that

$\displaystyle\overline{\partial}_{{\mathcal{E}}_{k+l}}(ee^{\prime})=$
$\displaystyle\,j^{-1}\\!\left(\overline{\partial}(ee^{\prime})\right)=\,j^{-1}\\!\left((\overline{\partial}e)e^{\prime}+e\overline{\partial}(e^{\prime})\right)=\,j^{-1}\left(\overline{\partial}e)e^{\prime}+e\overline{\partial}e^{\prime}\right)=\,0.$

Thus the right hand side of the equality (13) is a subalgebra of
$\mathcal{O}_{q}(G)$. Now $z_{i}\in zU_{q}(\mathfrak{g})$, for each
$i=1,\dots,N_{x}$, which is to say, the generators of $S_{q}[G/L_{S}]$ are
contained in $H^{0}(\mathcal{E}_{1})$. Thus we have the opposite inclusion

$S_{q}[G/L_{S}]\subseteq\bigoplus_{k\in\mathbb{Z}_{0}}H^{0}(\mathcal{E}_{k}),$

and hence equality of the two algebras. ∎

######  Remark 6.3.

This differential geometric description of the quantum homogeneous coordinate
ring of the irreducible quantum flag manifolds hints at the existence of a
noncommutative generalisation of the classical _géométrie algébrique et
géometrie analytique_ correspondence [36]. See [4, §7] for a detailed
discussion of how such a general picture might look.

###  6.3. The Opposite Borel–Weil Theorem

As discussed in §4.3, the Heckenberger–Kolb $(0,1)$-fodc is one of two calculi
identified in the classification of left $\mathcal{O}_{q}(G)$-covariant
finite-dimensional irreducible calculi over the irreducible quantum flag
manifolds. We call this other fodc the _Heckenberger–Kolb $(1,0)$-calculus_
and denote it by $\Omega^{(1,0)}_{q}(G/L_{S})$. Its tangent space $T^{(1,0)}$
is given by

$\displaystyle
T^{(1,0)}=\mathrm{span}\left\\{F_{\beta}\,|\,\beta\in\overline{R^{-}_{S}}\right\\}.$

Just as for the $(0,1)$-calculus, each line bundle ${\mathcal{E}}_{k}$ over
$\mathcal{O}_{q}(G/L_{S})$ admits a unique left $\mathcal{O}_{q}(G)$-covariant
connection

$\partial_{{\mathcal{E}}_{k}}:{\mathcal{E}}_{k}\to\Omega^{(0,1)}\otimes_{\mathcal{O}_{q}(G/L_{S})}{\mathcal{E}}_{k}.$

Using a construction dual to that of
$(\Omega^{(0,1)}_{q}(G/L^{\,\mathrm{s}}_{S}),\overline{\partial})$, a
covariant fodc $(\Omega^{(1,0)}_{q}(G/L^{\,\mathrm{s}}_{S}),\partial)$ was
introduced in [16, 3.2.3]. This fodc restricts to the Heckenberger–Kolb
$(1,0)$-calculus on $\mathcal{O}_{q}(G/L_{S})$, and satisfies $\partial v=0$,
for all $v\in S_{q}[G/L_{S}^{\mathrm{op}}]$. Just as for the $(0,1)$-case,
this fodc can be used to give quantum principal bundle presentation of the
Heckenberger–Kolb $(1,0)$-fodc, as well as a principal connection description
of the connections $\overline{\partial}_{{\mathcal{E}}_{k}}$, leading to the
following $(1,0)$-version of the Borel–Weil theorem.

######  Theorem 6.4.

For any irreducible quantum flag manifold $\mathcal{O}_{q}(G/L_{S})$, it holds
that

  1. 1.

$H^{0}(\mathcal{E}_{k})=0$,

  2. 2.

$H^{0}(\mathcal{E}_{-k})=\overline{z}^{k}U_{q}(\mathfrak{g})$ is an
irreducible $U_{q}(\mathfrak{g})$-module of highest weight
$-w_{0}(k\varpi_{x})$,

for all $k\in\mathbb{Z}_{>0}$, where $\Pi\backslash S$ consists of the simple
root $\alpha_{x}$, and $H^{0}:=\ker(\partial)$.

Just as for the $(0,1)$-case, we can use this result to produce a
noncommutative differential geometric description of the opposite quantum
homogeneous coordinate ring.

######  Corollary 6.5.

It holds that

(14) $\displaystyle\bigoplus_{k\in\mathbb{Z}_{\geq
0}}H^{0}(\mathcal{E}_{-k})=S_{q}[G/L^{\mathrm{op}}_{S}],$

where both algebras are considered as subalgebras of $\mathcal{O}_{q}(G)$.

##  Appendix A Relative Line Modules and Spherical Generators

To prove the Borel–Weil theorem we used the fact that every highest weight
vector in ${\mathcal{E}}_{k}$ is for the form $bz^{k}$, for some
$b\in\mathcal{O}_{q}(G/L_{S})$. The proof is essentially the same as in the
classical case, but for the reader’s convenience, we give an explicit proof.

###  A.1. The Table of Spherical Weights

In Table 1 below we recall the precise subsets $S$ of simple roots $\Pi$
defining the irreducible quantum flag manifolds. We represent
$S^{c}:=\Pi\backslash S$ graphically by coloured nodes in the Dynkin diagram
of $\mathfrak{g}$.

The set of highest weights of the $U_{q}(\mathfrak{g})$-algebra
$\mathcal{O}_{q}(G/L_{S})$ forms an additive submonoid
$Z_{S}\subseteq\mathfrak{h}^{*}$ under addition. A distinguished minimal set
of generators for $Z_{S}$, the spherical weights, was presented in the
classical case by Krämer in [25, Tabelle 1].

Table 1. Irreducible quantum flag manifolds with defining crossed node

$A_{n}$ |  | $\mathcal{O}_{q}(\mathrm{Gr}_{n,m})$
---|---|---
$B_{n}$ |  | $\mathcal{O}_{q}(\mathbf{Q}_{2n+1})$
$C_{n}$ |  | $\mathcal{O}_{q}(\mathbf{L}_{n})$
$D_{n}$ |  | $\mathcal{O}_{q}(\mathbf{Q}_{2n})$
$D_{n}$ |  | $\mathcal{O}_{q}(\textbf{S}_{n})$
$E_{6}$ |  | $\mathcal{O}_{q}(\mathbb{OP}^{2})$
$E_{7}$ |  | $\mathcal{O}_{q}(\textbf{F})$

Table 2.  We use Humphrey’s numbering of the Dynkin nodes [17, §11.4] of
$\mathfrak{g}$. For the spherical weights of
$\mathcal{O}_{q}(\mathbf{S}_{2m})$, the weight $2\varpi_{2m-1}$ or
$2\varpi_{2m}$ appears depending on the defining crossed node, as presented in
Table 1 above.

$\mathcal{O}_{q}(G/L_{S})$ | Spherical Weights
---|---
$\mathcal{O}_{q}(\text{Gr}_{n,m})$ | $\varpi_{1}+\varpi_{n-1},\,\varpi_{2}+\varpi_{n-2},\dots,\varpi_{m}+\varpi_{n-m}$
$\mathcal{O}_{q}(\mathbf{Q}_{2n+1})$ | $2\varpi_{1},\,\varpi_{2}$
$\mathcal{O}_{q}(\mathbf{L}_{n})$ | $2\varpi_{1},\,2\varpi_{2},\dots,2\varpi_{n}$
$\mathcal{O}_{q}(\mathbf{Q}_{2n})$ | $2\varpi_{1},\,\varpi_{2}$
$\mathcal{O}_{q}(\mathbf{S}_{2m})$ | $\varpi_{2},\,\varpi_{4},\dots,\varpi_{2m-2},\,2\varpi_{2m-1}\text{ or }2\varpi_{2m}$
$\mathcal{O}_{q}(\mathbf{S}_{2m+1})$ | $\varpi_{2},\,\varpi_{4},\dots,\varpi_{2m-2},\,\varpi_{2m}+\varpi_{2m+1}$
$\mathcal{O}_{q}(\mathbb{OP}^{2})$ |  $\varpi_{1}+\varpi_{6}$, $\varpi_{2}$
$\mathcal{O}_{q}(\mathbf{F})$ | $\varpi_{1},\,\varpi_{6},\,2\varpi_{7}$

Since the Weyl character formula is unchanged under $q$-deformation (see [23,
§7.1.4] for details) Krämer’s result carries over directly to the quantum
setting, and we present it as such in Table 2 above.

Let us now identify the properties of the spherical weights that are used
below. Firstly we highlight the fact that since $G/L_{S}$ is an Hermitian
symmetric space, $\mathcal{O}_{q}(G/L_{S})$ is multiplicity free as a
$U_{q}(\mathfrak{g})$-module. Secondly we note that the sum of the crossed
node $\varpi_{x}$ and its dual $-w_{0}(\varpi_{x})$ is a spherical generator
for each irreducible quantum flag, while neither the crossed node nor its dual
appear individually as generators. It is perhaps helpful to observe that in
the cases where the crossed node is self-dual (which is to say the spaces
$\mathcal{O}_{q}(\mathbf{Q}_{2n+1}),\mathcal{O}_{q}(\mathbf{L}_{n})$,
$\mathcal{O}_{q}(\mathbf{Q}_{2n})$, $\mathcal{O}_{q}(\mathbf{S}_{2m})$ and
$\mathcal{O}_{q}(\mathbf{F})$) we have that
$\varpi_{x}-w_{0}(\varpi_{x})=2\varpi_{x}$.

###  A.2. Spherical Generators for Line bundles

In this subsection, we produce the required presentation of the highest weight
elements of the line bundles ${\mathcal{E}}_{k}$, for $k\in\mathbb{Z}$. The
proof relies on the observation that since the product of two highest weight
elements of $\mathcal{O}_{q}(G)$ is again a highest weight element, the set of
highest weight elements of $\mathcal{O}_{q}(G)$ forms a submonoid of $A$. (For
a more detailed discussion of the properties of highest weight vectors in
$U_{q}(\mathfrak{g})$-algebras see [10, §4].)

######  Proposition A.1.

For every $k\in\mathbb{Z}_{\geq 0}$, it holds that

  1. 1.

the set of highest weights of ${\mathcal{E}}_{k}$ is given by

$\\{\lambda-kw_{0}(\varpi_{x})\,|\,\lambda\textrm{ a highest weight of
}{\mathcal{E}}_{0}\\},$

  2. 2.

every highest weight element of ${\mathcal{E}}_{k}$ is of the form $bz^{k}$,
for some $b\in\mathcal{O}_{q}(G/L_{S})$,

  3. 3.

the set of highest weights of ${\mathcal{E}}_{-k}$ is given by

$\\{\lambda+k\varpi_{x}\,|\,\textrm{ for }\lambda\textrm{ is a highest weight
of }{\mathcal{E}}_{0}\\},$

  4. 4.

every highest weight element of ${\mathcal{E}}_{-k}$ is of the form
$b\overline{z}^{k}$, for some $b\in\mathcal{O}_{q}(G/L_{S})$.

###### Proof.

1,2 Since the element $\overline{z}^{k}\in{\mathcal{E}}_{k}$ is a highest
weight vector, and the highest weight elements of $\mathcal{O}_{q}(G)$ are
closed under multiplication, we have a well-defined map

$\displaystyle
M_{\overline{z}^{k}}:({\mathcal{E}}_{k})_{\mathrm{hw}}\to({\mathcal{E}}_{0})_{\mathrm{hw}},$
$\displaystyle e\mapsto\overline{z}^{k}e.$

Moreover, since $\mathcal{O}_{q}(G)$ has no zero divisors,
$M_{\overline{z}^{k}}$ is injective. Injectivity implies that the dimensions
of the highest weight spaces are preserved. Thus since ${\mathcal{E}}_{0}$ is
multiplicity free, ${\mathcal{E}}_{k}$ must be multiplicity free.

Note next that since $\mathrm{wt}(\overline{z}^{k})=-kw_{0}(\varpi_{x})$, we
have, for any $e\in({\mathcal{E}}_{k})_{\mathrm{hw}}$, that

$\displaystyle\mathrm{wt}(M_{\overline{z}^{k}}(e))=\mathrm{wt}(\overline{z}^{k}e)=\mathrm{wt}(\overline{z}^{k})+\mathrm{wt}(e)=k\varpi_{x}+\mathrm{wt}(e).$

However, $M_{\overline{z}^{k}}(e)\in{\mathcal{E}}_{0}$, so there exists an
$l\in\mathbb{Z}_{\geq k}$ such that

$k\varpi_{x}+\mathrm{wt}(e)=l(\varpi_{x}-w_{0}(\varpi_{x}))+\beta,$

where $\beta\in\mathcal{P}^{+}$ is a weight contained in the $\mathbb{Z}$-span
of all the spherical weights apart from $\varpi_{x}-w_{0}(w_{x})$. This means
that

$\mathrm{wt}(e)=(l-k)\varpi_{x}-lw_{0}(\varpi_{x})+\beta,$

or equivalently, that every weight of ${\mathcal{E}}_{k}$ is of the form

(15) $\displaystyle\lambda-kw_{0}(\varpi_{x}),$ $\displaystyle\textrm{ for
$\lambda$ a highest weight of ${\mathcal{E}}_{0}$}.$

Let us now show that every such weight is realised. Let
$b\in{\mathcal{E}}_{0}$ be a highest weight vector of weight $\lambda$. We see
that $bz^{k}_{1}\in{\mathcal{E}}_{k}$, and that

$\mathrm{wt}(z^{k}b)=-kw_{0}(\varpi_{x})+\lambda.$

Thus we have found a weight vector for each weight of the form (15). Moreover,
we have shown that every highest weight vector of ${\mathcal{E}}_{k}$ is of
the form $bz^{k}$.

3,4. The proof for the case of the negative line bundles ${\mathcal{E}}_{-k}$
is analogous, and so, we omit it. ∎

##  Appendix B Noncommutative Complex and Holomorphic Structures

In order to keep the necessarily preliminaries to a minimum, this paper is
presented in terms of first-order differential calculi. However, these objects
find their fullest geometric presentation when considered as part of a
differential graded algebra. This is even more so when one is considered
noncommutative complex geometry. In this appendix, with a view to better
motivating the results of the paper, we recall the basic definitions and
results of the theory of differential calculi, complex structures, and
holomorphic structures, as well as a summary of the noncommutative complex
geometry of the Heckenberger–Kolb calculi.

###  B.1. Differential Calculi

A differential calculus
$\big{(}\Omega^{\bullet}\simeq\bigoplus_{k\in{\mathbb{Z}}_{\geq
0}}\Omega^{k},\mathrm{d}\big{)}$ is a differential graded algebra that is
generated as an algebra by the elements $a,\mathrm{d}b$, for
$a,b\in\Omega^{0}$. For a given algebra $B$, a differential calculus over $B$
is a differential calculus such that $B=\Omega^{0}$. 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
$\mathrm{d}(\omega^{*})=(\mathrm{d}\omega)^{*}$, and

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

Any fodc admits an extension to a differential calculus $\Omega^{\bullet}$
which is maximal in the sense that there exists a unique differential map from
$\Omega^{\bullet}$ onto any other extension of $\Omega^{1}$, see [3, §1.5] for
details. We call this extension the maximal prolongation of the first-order
calculus.

A differential calculus $\Omega^{\bullet}$ over a left $A$-comodule algebra
$P$ is said to be covariant if the coaction $\Delta_{L}:P\to A\otimes P$
extends to a (necessarily unique) $A$-comodule algebra structure
$\Delta_{L}:\Omega^{\bullet}\to A\otimes\Omega^{\bullet}$, with respect to
which the differential $\mathrm{d}$ is a left $A$-comodule map. Covariance for
a right $A$-comodule algebra is defined analogously. The maximal prolongation
of a covariant fodc is covariant. See [3, §1] for a more detailed discussion
of differential calculi.

###  B.2. Complex Structures

We now recall the definition of a complex structure as introduced in [4, 20].
This abstracts the properties of the de Rham complex of a classical complex
manifold [18].

######  Definition B.1.

A complex structure $\Omega^{(\bullet,\bullet)}$ for a differential
$*$-calculus $(\Omega^{\bullet},\mathrm{d})$ is a choice of $\mathbb{Z}_{\geq
0}^{2}$-algebra grading $\bigoplus_{(a,b)\in\mathbb{Z}_{\geq
0}^{2}}\Omega^{(a,b)}$ for $\Omega^{\bullet}$ such that

$\displaystyle 1.\,\Omega^{k}=\bigoplus_{a+b=k}\Omega^{(a,b)},$ $\displaystyle
2.\,\big{(}\Omega^{(a,b)}\big{)}^{*}=\Omega^{(b,a)},$ $\displaystyle
3.\,\mathrm{d}\Omega^{(a,b)}\subseteq\Omega^{(a+1,b)}\oplus\Omega^{(a,b+1)},$

for all $k\in\mathbb{Z}_{\geq 0}$, and $(a,b)\in\mathbb{Z}^{2}_{\geq 0}$.

We call an element of $\Omega^{(a,b)}$ an $(a,b)$-form, and denote

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

where $\mathrm{proj}_{\Omega^{(a+1,b)}}$, and
$\mathrm{proj}_{\Omega^{(a,b+1)}}$, are the projections from $\Omega^{a+b+1}$
to $\Omega^{(a+1,b)}$, and $\Omega^{(a,b+1)}$, respectively. It follows
directly from the definition of a complex structure that the triple
$(\Omega^{(\bullet,\bullet)},\partial,\overline{\partial})$ is a double
complex. Moreover, both $\partial$ and $\overline{\partial}$ are $*$-maps and
satisfy the graded Leibniz rule. The opposite complex structure of a complex
structure $\Omega^{(\bullet,\bullet)}$ is the $\mathbb{Z}^{2}_{\geq
0}$-algebra grading $\overline{\Omega}^{(\bullet,\bullet)}$, defined by
$\overline{\Omega}^{(a,b)}:=\Omega^{(b,a)}$, for $(a,b)\in\mathbb{Z}_{\geq
0}^{2}$.

For $\Omega^{\bullet}$ a covariant differential $*$-calculus
$\Omega^{\bullet}$ over a left $A$-comodule algebra $P$, we say that a complex
structure for $\Omega^{\bullet}$ is covariant if the $\mathbb{Z}_{\geq
0}^{2}$-decomposition is a decomposition in ${}^{A}\mathrm{Mod}$, the category
of left $A$-comodules.

###  B.3. Holomorphic Structures

Building on this idea we define noncommutative holomorphic vector bundles via
the classical Koszul–Malgrange characterisation of holomorphic bundles [24]

Any connection can be extended to a map
$\nabla:\Omega^{\bullet}\otimes_{B}\mathcal{F}\to\Omega^{\bullet}\otimes_{B}\mathcal{F}$
uniquely defined by

$\displaystyle\nabla(\omega\otimes f)=\mathrm{d}\omega\otimes
f+(-1)^{|\omega|}\,\omega\wedge\nabla f,$ $\displaystyle\textrm{for
}f\in{\mathcal{F}},\,\omega\in\Omega^{\bullet},$

for a homogeneous form $\omega$ with degree $|\omega|$. The _curvature_ of a
connection is the left $B$-module map
$\nabla^{2}:\mathcal{F}\to\Omega^{2}\otimes_{B}\mathcal{F}$. A connection is
said to be flat if $\nabla^{2}=0$. Since $\nabla^{2}(\omega\otimes
f)=\omega\wedge\nabla^{2}(f)$, a connection is flat if and only if the pair
$(\Omega^{\bullet}\otimes_{B}{\mathcal{F}},\nabla)$ is a complex.

######  Definition B.2.

For an algebra $B$, a _holomorphic vector bundle over $B$_ is a pair
$(\mathcal{F},\overline{\partial}_{\mathcal{F}})$, where $\mathcal{F}$ is a
finitely generated projective left $B$-module, and the
$\overline{\partial}_{\mathcal{F}}:\mathcal{F}\to\Omega^{(0,1)}\otimes_{B}\mathcal{F}$
is a flat $(0,1)$-connection, which we call the _holomorphic structure_ for
$({\mathcal{F}},\overline{\partial}_{{\mathcal{F}}})$.

The requirement of flatness means that any holomorphic vector bundle has an
associated noncommutative Dolbeault cohomology
$H^{\bullet}({\mathcal{F}})^{(\bullet,\bullet)}$. Moreover, as shown in [33],
there is an associated noncommutative generalisation of the Kodaira vanishing
theorem.

###  B.4. The Irreducible Quantum Flag Manifolds

A $*$-algebra structure on $U_{q}(\mathfrak{g})$, called the compact real
form, is defined by

$\displaystyle K^{*}_{i}:=K_{i},$ $\displaystyle E^{*}_{i}:=K_{i}F_{i},$
$\displaystyle F^{*}_{i}:=E_{i}K_{i}^{-1},$

and gives $U_{q}(\mathfrak{g})$ the structure of a Hopf $*$-algebra. This
$*$-structure dualises to a Hopf $*$-algebra structure on
$\mathcal{O}_{q}(G)$. The quantum flag manifolds $\mathcal{O}_{q}(G/L_{S})$
are $*$-subalgebras of $\mathcal{O}_{q}(G/L_{S})$ since
$U_{q}(\mathfrak{l}_{S})$ is a Hopf $*$-subalgebra of $U_{q}(\mathfrak{g})$.

The _Heckenberger–Kolb calculus_ of $\mathcal{O}_{q}(G/L_{S})$ is the direct
sum fodc

$\Omega^{1}_{q}(G/L_{S}):=\Omega^{(1,0)}_{q}(G/L_{S})\oplus\Omega^{(0,1)}_{q}(G/L_{S}).$

The maximal prolongation of $\Omega^{1}_{q}(G/L_{S})$ is a covariant
differential $*$-calculus of classical dimension. Moreover, it has a unique
pair of opposite left $\mathcal{O}_{q}(G)$-covariant complex structures
extending the decomposition of $\Omega^{1}_{q}(G/L_{S})$ into its $(1,0)$ and
$(0,1)$-summands. For the quantum flag manifolds $\mathcal{O}_{q}(G/L_{S})$,
and the associated line modules ${\mathcal{E}}_{k}$, for $k\in\mathbb{Z}$, the
left $\mathcal{O}_{q}(G)$-covariant $(0,1)$-connections presented in §5.6 are
in fact holomorphic structures [11, §4]. Thus they give a direct
$q$-deformation of the classical holomoprhic structures of the line bundles
over $G/L_{S}$. Moreover, the analogous results hold for the opposite complex
structure and the $(1,0)$-connections $\partial_{{\mathcal{E}}_{k}}$.

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