# The Capelli Eigenvalue Problem for Quantum Groups

Gail Letzter  Gail Letzter, Mathematics Research Group, National Security
Agency<EMAIL_ADDRESS>,  Siddhartha Sahi  Siddhartha Sahi, Department
of Mathematics, Rutgers University<EMAIL_ADDRESS>and  Hadi Salmasian
Hadi Salmasian, Department of Mathematics and Statistics, University of Ottawa
<EMAIL_ADDRESS>
###### Abstract.

We introduce and study quantum Capelli operators inside newly constructed
quantum Weyl algebras associated to three families of symmetric pairs ([19]).
Both the center of a particular quantized enveloping algebra and the Capelli
operators act semisimply on the polynomial part of these quantum Weyl
algebras. We show how to transfer well-known properties of the center arising
from the theory of quantum symmetric pairs to the Capelli operators. Using
this information, we provide a natural realization of Knop-Sahi interpolation
polynomials as functions that produce eigenvalues for quantum Capelli
operators.

##  1\. Introduction

In the mid 1980’s, Macdonald introduced a new family of parametrized
orthogonal polynomials, referred to as Macdonald polynomials. These
polynomials can be viewed as generalizations of the orthogonal polynomials
that appear as zonal spherical functions for real and $p$-adic symmetric
spaces ([20]). About a decade later, Knop ([9]) and Sahi ([26]) defined
parameterized versions of interpolation polynomials (in Type A) with close
connections to Macdonald polynomials. For certain parameters, the Knop-Sahi
interpolation polynomials can be viewed as $q$-analogs of polynomials that
produce eigenvalues for Capelli operators ([11],[12],[27]). Knop-Sahi
interpolation polynomials are referred to by a number of other names in the
literature including interpolation Macdonald polynomials ([25]), quantum
Capelli polynomials ([9]), and shifted Macdonald polynomials ([24]).

Drinfeld and Jimbo discovered quantized enveloping algebras, which are Hopf
algebra deformations of universal enveloping algebras of Lie algebras, around
the same time as Macdonald initiated the study of his new family of orthogonal
polynomials. Shortly afterwards, the quest began for realizing Macdonald
polynomials as zonal sperical functions on quantum symmetric spaces. This
realization was ultimately carried out in a series of papers ([4], [15],[16],
[21], [22], [23]). The theory relies on definitions of quantum symmetric pairs
using special coideal subalgebras constructed first for classical Lie algebras
using reflection equations ([20], [23]) and then in general via generators and
relations ([13], [14], [8]).

In this paper, we complete another part of the quantum picture, namely finding
a natural realization of the Knop-Sahi interpolation polynomials as functions
that produce eigenvalues for quantum Capelli operators. The crucial piece in
the story is identifying quantum Capelli operators, which are invariants with
respect to a particular quantized enveloping algebra, inside three families of
newly constructed quantum Weyl algebras $\mathscr{PD}_{\theta}$ ([19]). In
other words, we formulate and solve the Capelli eigenvalue problem in the
quantum case, thus providing quantum analogs of results in [27]. Similar
results for Capelli operators and their eigenvalues have been obtained in the
Lie superalgebra setting ([1], [28], [29], [30]). We also show that these two
quantum realizations, one for Macdonald polynomials and the other for Knop-
Sahi interpolation polynomials, are closely connected. In particular, we use
results from the Macdonald polyomial setting concerning central elements of
the quantized enveloping algebra in order to establish basic properties of
quantum Capelli operators.

The quantum Weyl algebra $\mathscr{PD}_{\theta}$ is associated to one of the
following three (infinitesimal) symmetric pairs $(\mathfrak{g},\mathfrak{k})$
where $\mathfrak{k}$ is the Lie subalgebra fixed by the involution $\theta$ of
$\mathfrak{g}$:

  * •

$(\mathfrak{g},\mathfrak{k})=(\mathfrak{gl}_{n},\mathfrak{so}_{n})$

  * •

$(\mathfrak{g},\mathfrak{k})=(\mathfrak{gl}_{2n},\mathfrak{sp}_{2n})$

  * •

$(\mathfrak{g},\mathfrak{k})=(\mathfrak{gl}_{n}\oplus\mathfrak{gl}_{n},\mathfrak{gl}_{n})$

We refer to the first pair as Type AI, the second as Type AII, and the third
as the Type A diagonal case, thus matching the language used in the
classification of symmetric pairs via Satake diagrams. These correspond to the
Jordan algebras of Hermitian matrices over $\mathbb{R}$, $\mathbb{H}$, and
$\mathbb{C}$ in Type AI, Type AII, and the Type A diagonal case respectively.
Indeed, in each case, the Riemannian symmetric space $G/K$ can be identified
naturally with the open subset of positive definite Hermitian matrices. We
focus on these three families precisely because the Jordan algebra setting is
central in Kostant and Sahi’s study of Capelli identities, Capelli operators,
and interpolation polynomials ([11], [12], [27]).

As explained in [19], the quantum Weyl algebra $\mathscr{PD}_{\theta}$ is a
deformation of the twisted tensor product of two algebras, the polynomial part
$\mathscr{P}_{\theta}$ and the constant coefficient differential part
$\mathscr{D}_{\theta}$. The polynomial part $\mathscr{P}_{\theta}$ is the
algebra of quantized functions on the space of $n\times n$ symmetric matrices
for Type AI, $n\times n$ skew symmetric matrices for Type AII, and all
$n\times n$ matrices in the Type A diagonal case. The algebra
$\mathscr{D}_{\theta}$ is isomorphic to the opposite algebra of
$\mathscr{P}_{\theta}$.

Let $\mathcal{B}_{\theta}$ denote the right coideal subalgebra of
$U_{q}(\mathfrak{g})$ which is a quantum analog of $U(\mathfrak{k})$ inside of
$U(\mathfrak{g})$ as defined in [14] (see also [8]). Recall that there is a
second root system, called the restricted root system, associated to a
symmetric pair $\mathfrak{g},\mathfrak{k}$. For each of the symmetric pairs
under consideration in this paper, the restricted root system $\Sigma$ is of
type $A_{n-1}$. Denote by $\Lambda^{+}_{\Sigma}$ the set of partitions of
length $n$ realized as weights associated to $\Sigma$. Note that these
partitions defined by $\Sigma$ form a subset of the partitions defined by the
root system of $\mathfrak{g}$. Thus, with respect to this inclusion, elements
of $\Lambda^{+}_{\Sigma}$ are also weights for $\mathfrak{g}$.

Let $M={\rm Mat}_{n}$ in type AI, $M={\rm Mat}_{2n}$ in type AII, and $M={\rm
Mat}_{n}\times{\rm Mat}_{n}$ in the diagonal case. The initial definition of
$\mathscr{P}_{\theta}$ (see [19]) is a subalgebra of $\mathcal{B}_{\theta}$
invariants inside the quantized function algebra $\mathcal{O}_{q}(M)$ on $M$.
Moreover, $\mathscr{P}_{\theta}$ inherits a left $U_{q}(\mathfrak{g})$-module
structure from $\mathcal{O}_{q}(M)$. As a left $U_{q}(\mathfrak{g})$-module,
$\mathscr{P}_{\theta}$ is multiplicity free and is isomorphic to a direct sum
of simple highest weight modules $L(2\lambda)$ where
$\lambda\in\Lambda^{+}_{\Sigma}$. The algebra $\mathscr{P}_{\theta}$ can be
viewed as a subalgebra of the quantum homogeneous space
$\mathcal{O}_{q}[G/K]$, which in turn is a subalgebra of $\mathcal{O}_{q}[G]$,
where $G,K$ is the symmetric pair of Lie groups associated to
$\mathfrak{g},\mathfrak{k}$. The algebra $\mathscr{D}_{\theta}$ also admits a
decomposition into left $U_{q}(\mathfrak{g})$-modules that is isomorphic to a
direct sum of the $L^{*}(2\lambda)$, for $\lambda\in\Lambda^{+}_{\Sigma}$
where $L^{*}(2\lambda)$ is a left $U_{q}(\mathfrak{g})$-module dual of
$L(2\lambda)$.

The quantum Weyl algebra $\mathscr{PD}_{\theta}$ inherits the structure of a
left $U_{q}(\mathfrak{g})$-module via its construction that is compatible with
the module actions on the subalgebras $\mathscr{P}_{\theta}$ and
$\mathscr{D}_{\theta}$. Thus, $\mathscr{PD}_{\theta}$ is isomorphic to the
direct sum of modules of the form $L(2\lambda)\otimes L^{*}(2\mu)$ where
$\lambda,\mu$ both run over partitions in $\Lambda^{+}_{\Sigma}$. Note that
$L(2\lambda)\otimes L^{*}(2\lambda)\cong{\rm End}\ L(2\lambda)$. Let
$C_{\lambda}$ be the vector corresponding to the identity in
$L(2\lambda)\otimes L^{*}(2\lambda)$ via this isomorphism. The space of left
$U_{q}(\mathfrak{g})$ invariants of $L(2\lambda)\otimes L^{*}(2\mu)$ is zero
if $\lambda\neq\mu$ and equal to the one-dimensioal space spanned by
$C_{\lambda}$ for $\lambda=\mu$. The quantum Capelli operators are the
elements $C_{\lambda},\lambda\in\Lambda^{+}_{\Sigma}.$

One can define an action of $\mathscr{PD}_{\theta}$ on $\mathscr{P}_{\theta}$
in a manner similar to the action of the classical Weyl algebra on its
polynomial subalgebra. Since $\mathscr{P}_{\theta}$ is a
$U_{q}(\mathfrak{g})$-module, elements of $U_{q}(\mathfrak{g})$ also act on
$\mathscr{P}_{\theta}$. These actions lead to $U_{q}(\mathfrak{g})$-module
maps from $\mathscr{PD}_{\theta}$ and $U_{q}(\mathfrak{g})$ into ${\rm End}\
\mathscr{P}_{\theta}$ where the module action for $\mathscr{P}_{\theta}$ is
the left action and the module action for $U_{q}(\mathfrak{g})$ is the (left)
adjoint action. Let $\mathcal{F}(U_{q}(\mathfrak{g}))$ denote the locally
finite subalgebra of $U_{q}(\mathfrak{g})$ with respect to the adjoint action.
We define a $U_{q}(\mathfrak{g})$-module map that takes certain elements of
$\mathcal{F}(U_{q}(\mathfrak{g}))$, including most of the center of
$U_{q}(\mathfrak{g})$, to the quantum Weyl algebra that is compatible with the
maps into ${\rm End}\ \mathscr{P}_{\theta}$. This enables us to establish the
following connection between the center and the Capelli operators.

Theorem A. There is an isomorphism between a polynomial subring $Z$ of the
center of $U_{q}(\mathfrak{g})$ and the algebra generated by the quantum
Capelli operators so that the action of the two agree on
$\mathscr{P}_{\theta}$.

In [18], quantum Weyl algebras over $m\times n$ matrices where $m\neq n$ are
studied. These quantum Weyl algebras come equipped with a left action of
$U_{q}(\mathfrak{gl}_{m})$ and a right action of $U_{q}(\mathfrak{gl}_{n})$. A
key tool in [18] is a mapping of most of the locally finite subalgebras of
$U_{q}(\mathfrak{gl}_{m})$ and of $U_{q}(\mathfrak{gl}_{n})$ to subalgebras of
the quantum Weyl algebra using very similar arguments to the ones found here.

By [17], there is a Harish-Chandra type map from the center to a subalgebra of
the Cartan subalgebra of $U_{q}(\mathfrak{g})$ so that the image of the center
$Z(U_{q}(\mathfrak{g}))$ consists of invariants with respect to a dotted
action of the Weyl group for the restricted root system. Note that elements of
$Z(U_{q}(\mathfrak{g}))$ act semisimply on $\mathscr{P}_{\theta}$ with
eigenspaces corresponding to simple modules $L(2\lambda)$,
$\lambda\in\Lambda^{+}_{\Sigma}$. The image of a central element inside of the
Cartan subalgebra can be viewed as an eigenvalue function: the eigenvalue for
a central element $z$ on $L(2\lambda)$ is obtained by evaluating the image of
$z$ under this Harish-Chandra map at $\lambda$, or more precisely, at
$q^{\lambda}$ (see Section 10.1 for a definition of this type of evaluation).

It follows from Theorem A that the eigenvalue functions for quantum Capelli
operators live inside the same subalgebra of the Cartan part and, moreover,
inherit the desired Weyl group invariance property from the center. Write
$\mathcal{E}_{\lambda}$ for the eigenvalue function associated to the Capelli
operator $C_{\lambda}$. We shall see that there is an interpretation for this
eigenvalue function as a polynomial in $n$ variables. The explicit map from
$Z$ to the quantum Capelli operators of Theorem A ensures that the degree of
$\mathcal{E}_{\lambda}$ is equal to $|\lambda|$.

Let $H_{2\lambda}$ denote the highest weight generating vector for the copy of
$L(2\lambda)$ inside of $\mathscr{P}_{\theta}$. The algebra
$\mathscr{P}_{\theta}$ has a natural degree function which turns it into a
graded algebra. Using a careful analysis involving the relations of
$\mathscr{PD}_{\theta}$, we show that: $C_{\lambda}\cdot H_{2\mu}=0$ for all
$H_{2\mu}$, $\mu\neq\lambda$, of degree less than or equal to that of
$C_{\lambda}$. Moreover, $C_{\lambda}\cdot H_{2\lambda}\neq 0$. These results
are used to show that the polynomials $\mathcal{E}_{\lambda}$ satisfy the
vanishing property, as well as a nondegeneracy condition, of the Knop-Sahi
interpolation polynomials. This leads to our main result where we refer to the
notation of [24] for the explicit definition of the polynomials
$P^{*}_{\lambda}({x;a,g})$.

Theorem B. For each $\lambda\in\Lambda^{+}_{\Sigma}$, $\mathcal{E}_{\lambda}$
is equal to the polynomial $P_{\lambda}^{*}(x;a,g)$ (up to a normalization
scalar) where

  * •

$(a,g)=(q^{4},q^{2})$ in Type AI

  * •

$(a,g)=(q^{2},q^{4})$ in Type AII,

  * •

$(a,g)=(q^{2},q^{2})$ in the Type A diagonal case.

It should be noted that the parameters obtained in Theorem B are precisely the
same parameters as those for the realization of Macdonald polynomials as zonal
spherical functions. This is due both to the connection between Macdonald
polynomials and Knop-Sahi interpolation polynomials as well as the
relationship between eigenspaces of $\mathscr{P}_{\theta}$ and quantum zonal
spherical functions. For more details on this connection, see Remark 10.5.

An original motivation for this paper was to understand and extend Bershtein’s
results on quantum Capelli operators in [3]. One can view the results in [3]
as a quantum analog of the classical setup relying on a Hermitian symmetric
pair of Type AIII. By [11], Section 1, the classical versions of Bershtein’s
approach and the Type A diagonal case of this paper produce the same Capelli
operator eigenvalues. This paper shows that the same happens in the quantum
setting. Indeed, the eigenvalues for the quantum Capelli operators in the Type
A diagonal case of Theorem B match those in [3]. Note that fixed Lie
subalgebra $\mathfrak{k}$ in [3] contains the entire Cartan subalgebra and so
the corresponding symmetric pair is maximally compact. An advantage of the
approach here is that the symmetric pairs are explicitly defined so that they
are in maximally split form. This allows us to use the theory of quantum
symmetric pairs and symmetric spaces as developed in [13],[14],[15], [16] and
[17].

The remainder of this paper is organized as follows. Section 2 sets notation
for root systems, quantized enveloping algebras, the three types of symmetric
pairs and their quantum versions. In Section 3, we give generators and
relations for the quantized function algebras $\mathcal{O}_{q}({\rm Mat}_{N})$
on $N\times N$ matrices and for the three quantum homogeneous spaces
$\mathscr{P}_{\theta}$. In Section 4, we obtain detailed descriptions of
highest weight vectors inside $\mathscr{P}_{\theta}$ with respect to the
action of $U_{q}(\mathfrak{g})$ and use them to write down explicit module
decompositions of the quantum homogeneous spaces $\mathscr{P}_{\theta}$.

In Section 5, we describe the quantum Weyl algebras $\mathscr{PD}_{\theta}$ of
[19] in terms of generators and relations. We then define a
$U_{q}(\mathfrak{g})$ equivariant action of $\mathscr{PD}_{\theta}$ on
$\mathscr{P}_{\theta}$ that resembles the action for the classical Weyl
algebra on the polynomial subalgebra. Using the relations for
$\mathscr{PD}_{\theta}$, we analyze the action of the quantum Weyl algebra on
the highest weight vectors $H_{2\lambda},\lambda\in\Lambda_{\Sigma}^{+},$
inside of $\mathscr{P}_{\theta}$. This information is used to determine the
action of the Capelli operators $C_{\lambda}$ on highest weight vectors
$H_{2\mu}$ needed for the proof of Theorem B.

Section 6 shows how to identify certain elements $u$ of the Cartan subalgebra
of $U_{q}(\mathfrak{g})$ with elements $a$ of $\mathscr{PD}_{\theta}$ so that
$u$ and $a$ agree with respect to their action on $\mathscr{P}_{\theta}$. The
elements of the Cartan subalgebra that have this property are part of the
locally finite subalgebra $\mathcal{F}(U_{q}(\mathfrak{g}))$ of
$U_{q}(\mathfrak{g})$. The locally finite subalgebra is a well-studied object
for quantized enveloping algebras of semisimple Lie algebras. In Section 7, we
give a complete description of $\mathcal{F}(U_{q}(\mathfrak{g}))$ by
translating the results for the locally finite part of
$U_{q}(\mathfrak{sl}_{N})$ to the $\mathfrak{gl}_{N}$ setting. This allows us
to define a mapping $\Upsilon$ on almost all of
$\mathcal{F}(U_{q}(\mathfrak{g}))$ into $\mathscr{PD}_{\theta}$ which is
compatible with the action on $\mathscr{P}_{\theta}$.

Section 8 is devoted to the center of $U_{q}(\mathfrak{g})$ and the image of a
special subalgebra $Z$ of the center (as in Theorem A) to
$\mathscr{PD}_{\theta}$ via $\Upsilon$. Generators for the image of $Z$ inside
${\rm End}\ \mathscr{P}_{\theta}$ are identified and bounds on the degree of
their image inside of $\mathscr{PD}_{\theta}$ are determined. Using the
standard Harish-Chandra map and a restricted version based on those in [17],
we explain the dotted restricted Weyl group invariance for central elements.
We further show that both $Z$ and its image under the restricted Harish-
Chandra map are isomorphic to a polynomial ring in $n$ generators.

Quantum Capelli operators, are defined in Section 9. Comparing the degree of
quantum Capelli operators with the degree of the image of generators of $Z$
under $\Upsilon$ yields Theorem A (Theorem 9.3). Eigenvalue functions
associated to quantum Capelli operators are introduced in Section 10.2 and
their relation to the image of central elements of $Z$ under the restricted
Harish-Chandra map is explained. We show that these eigenvalue functions
satisfy the defining properties of Knop-Sahi polynomials by combining facts
about the center of $U_{q}(\mathfrak{g})$ with structural properties of
$\mathscr{PD}_{\theta}$. This establishes Theorem B (Theorem 10.4).

Acknowledgements. A preliminary version of these results was presented by G.
Letzter at the Workshop on Symmetric Spaces, Their Generalizations, and
Special Functions, University of Ottawa, 2022, which was sponsored by grants
from the Fields Institute and the NSF. The research of S. Sahi was partially
supported by NSF grants DMS-1939600, DMS-2001537, and Simons foundation grant
509766. The research of H. Salmasian was partially supported by an NSERC
Discovery Grant (RGPIN-2018-04004).

##  2\. Quantized enveloping algebras

###  2.1. Roots and weights

Let $N$ be a positive integer and let $\epsilon_{1},\dots,\epsilon_{N}$ denote
a fixed orthonormal basis for $\mathbb{R}^{N}$ with respect to the standard
inner product $(\cdot,\cdot)$. Let $\Phi_{N}$ denote the root system of Type
$A_{N-1}$ with positive simple roots $\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}$
for $i=1,\dots,{N-1}$. Write $\omega_{1},\dots,\omega_{N-1}$ for the
fundamental weights associated to $\Phi_{N}$. Set
$P=\sum_{i=1}^{N-1}\mathbb{Z}\omega_{i}$ equal to the weight lattice and
$Q=\sum_{i=1}^{N-1}\mathbb{Z}\alpha_{i}$ equal to the root lattice. Write
$P^{+}=\sum_{i=1}^{N-1}\mathbb{N}\omega_{i}$ for the subset of dominant
integral weights. Let $Q^{+}$ denote the subset
$\sum_{i-1}^{N}\mathbb{N}\alpha_{i}$ of the root lattice.

Let $\Lambda_{N}$ denote the set of vectors
$\lambda=\lambda_{1}\epsilon_{1}+\cdots+\lambda_{N}\epsilon_{N}$ where each
$\lambda_{i}\in\mathbb{Z}$ and
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}$. Set $\Lambda_{N}^{+}$
equal to the subset of $\Lambda_{N}$ consisting of those weights
$\lambda\in\Lambda_{N}$ satisfying $\lambda_{1}\geq\cdots\geq\lambda_{N}\geq
0$. In other words, elements in $\Lambda^{+}_{N}$ correspond to the set of
partitions $(\lambda_{1},\dots,\lambda_{N})$. Recall that
$\epsilon_{1}+\cdots+\epsilon_{N}$ is orthogonal to each $\alpha_{i}$ with
respect to the given inner product. Moreover, it is straightforward to check
that the fundamental weight $\omega_{i}$ satisfies

(1)
$\displaystyle{\omega}_{i}=\epsilon_{1}+\cdots+\epsilon_{i}-{{i}\over{N}}(\epsilon_{1}+\cdots+\epsilon_{N})$

for each $i=1,\dots,N-1$. Set
$\hat{\omega}_{i}=\epsilon_{1}+\cdots+\epsilon_{i}$ for $i=1,\dots,N$ and so
the above equality becomes
$\omega_{i}=\hat{\omega}_{i}-{{i}\over{N}}\hat{\omega}_{N}$. We refer to the
weights $\hat{\omega}_{1},\dots,\hat{\omega}_{N}$ as the fundamental
partitions associated to the root system $\Phi_{N}$. Note that
$\Lambda_{N}^{+}$ is the $\mathbb{N}$ linear combination of the fundamental
partitions $\hat{\omega}_{1},\dots,\hat{\omega}_{N}$.

Let $w_{0}$ denote the longest element of the Weyl group in type $A_{N-1}$. We
have $w_{0}\alpha_{i}=-\alpha_{N-i}$ and $w_{0}\epsilon_{j}=\epsilon_{N+1-j}$
for $i=1,\dots,N-1$ and $j=1,\dots,N$. It follows that the set
$w_{0}\Lambda_{N}^{+}$ is the $\mathbb{N}$ linear combination of the elements
$w_{0}\hat{\omega}_{i}=\epsilon_{N-i+1}+\cdots+\epsilon_{N}$ for
$i=1,\dots,N$.

Let $\mathfrak{gl}_{N}$ denote the complex general linear Lie algebra
consisting of $N\times N$ matrices and let $\mathfrak{sl}_{N}$ be the Lie
subalgebra equal to the subspace of $N\times N$ matrices with trace $0$.
Recall that $\Phi_{N}$ is the root system for $\mathfrak{sl}_{N}$ and the
finite-dimensional simple $\mathfrak{sl}_{N}$-modules are parameterized by
their highest weights which are elements of $P^{+}$. For $\mathfrak{gl}_{N}$,
one uses $\Lambda_{N}$ to parameterize the finite-dimensional simple
$\mathfrak{gl}_{N}$-modules. More generally, the relationship between
$\omega_{i}$ and $\hat{\omega}_{i}$ in (1) will help us translate results from
the $\mathfrak{sl}_{N}$ setting to that of $\mathfrak{gl}_{N}$.

The root system for $\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$ is just the
disjoint union $\Phi^{(1)}_{N}\cup\Phi^{(2)}_{N}$ of two copies of the root
system for $\mathfrak{gl}_{N}$ and the corresponding Weyl group is just the
direct product of two copies of the Weyl group for $\Phi_{N}$ . Write
$\epsilon_{1},\dots,\epsilon_{N}$ for the orthonormal basis for the first copy
of $\mathfrak{gl}_{N}$ and $\epsilon_{N+1},\dots,\epsilon_{2N}$ for the second
copy. Set $\hat{\omega}_{i}=\epsilon_{1}+\cdots+\epsilon_{i}$ and
$\hat{\omega}_{i+N}=\epsilon_{1+N}+\cdots+\epsilon_{i+N}$ for $i=1,\dots,N$.
The longest Weyl group element is $w_{0}\times w_{0}$ which simply acts as
$w_{0}$ on each copy of $\Phi_{N}$. Other notions are extended from
$\mathfrak{gl}_{N}$ to $\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$ in a similar
fashion. Sometimes we denote weights for
$\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$ using a single symbol, say
$\lambda$. In other instances, we use the sum $\gamma\oplus\gamma^{\prime}$ to
represent the weight
$\gamma_{1}\epsilon_{1}+\cdots+\gamma_{N}\epsilon_{N}+\gamma^{\prime}_{1}\epsilon_{1+N}+\cdots+\gamma^{\prime}_{N}\epsilon_{2N}$.

###  2.2. The quantized enveloping algebra

Write $e_{1},\dots,e_{N-1},$ $f_{1},\dots,f_{N-1},$
$h_{\epsilon_{1}},\cdots,h_{\epsilon_{N}}$ for the standard Chevalley
generators for $\mathfrak{gl}_{N}$. For the direct sum
$\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$, we write
$e_{i},f_{i},h_{\epsilon_{j}}$ where
$i\in\\{1,\dots,N-1\\}\cup\\{N+1,\dots,2N-1\\}$ and $j\in\\{1,\dots,2N\\}$ for
the standard generators. Here, the $e_{i},f_{i},h_{\epsilon_{j}}$, $i\leq
N-1,j\leq N$ generate the first copy of $\mathfrak{gl}_{N}$ while
$e_{i},f_{i},h_{\epsilon_{j}}$, $i,j\geq N+1$ generate the second copy.

Let $q$ be an indeterminate. The quantized enveloping algebra
$U_{q}(\mathfrak{gl}_{N})$ is an algebra over $\mathbb{C}(q)$ generated by
$K_{\epsilon_{1}}^{\pm 1},\dots,K_{\epsilon_{N}}^{\pm 1}$,
$E_{1},\dots,E_{N-1},$ $F_{1},\dots,F_{N-1}$ subject to the algebra relations
as stated in [21] (see also [10], Section 10). Note that
$K_{\epsilon_{1}}\cdots K_{\epsilon_{N}}$ is a central element of
$U_{q}(\mathfrak{gl}_{N})$. Set
$K_{i}=K_{\epsilon_{i}}K_{\epsilon_{i+1}}^{-1}$ for $i=1,\dots,N-1$. The
subalgebra of $U_{q}(\mathfrak{gl}_{N})$ generated by

$K_{1}^{\pm 1},\dots,K_{N-1}^{\pm 1},E_{1},\dots,E_{N-1},F_{1},\dots,F_{N-1}$

is the quantized enveloping algebra $U_{q}(\mathfrak{sl}_{N}).$

Given an integer linear combination
$\beta=\sum_{i=1}^{N}\beta_{j}\epsilon_{j}$, write $K_{\beta}$ for the product
$K_{\epsilon_{1}}^{\beta_{1}}\cdots K_{\epsilon_{N}}^{\beta_{N}}$. The algebra
$U_{q}(\mathfrak{gl}_{N})$ is a Hopf algebra with coproduct $\Delta$, counit
$\epsilon$, and antipode $S$ defined on generators by

  * •

$\Delta(E_{i})=E_{i}\otimes 1+K_{i}\otimes E_{i}$, $\epsilon(E_{i})=0$ and
$S(E_{i})=-K_{i}^{-1}E_{i}$

  * •

$\Delta(F_{i})=F_{i}\otimes K_{i}^{-1}+1\otimes F_{i}$, $\epsilon(F_{i})=0$
and $S(F_{i})=-F_{i}K_{i}$

  * •

$\Delta(K)=K\otimes K$, $\epsilon(K)=1$ and $S(K)=K^{-1}$

for $i=1,\dots,N-1$ and for all
$K=K_{\beta},\beta\in\sum_{j}{\mathbb{Z}}\epsilon_{j}$. It follows from the
defining relations for $U_{q}(\mathfrak{gl}_{N})$ that

(2) $\displaystyle
K_{\beta}E_{i}K_{\beta}^{-1}=q^{(\beta,\alpha_{i})}E_{i}\quad{\rm and}\quad
K_{\beta}F_{i}K_{\beta}^{-1}=q^{-(\beta,\alpha_{i})}F_{i}$

for $i=1,\dots,N-1$ and all $\beta\in\sum_{j}{\mathbb{Z}}\epsilon_{j}$. The
subalgebra $U_{q}(\mathfrak{sl}_{N})$ is also a Hopf algebra using the same
coproduct, couinit, and antipode. Given an arbitrary element $u$ in one of
these Hopf algebras, we express the coproduct of $u$ by $\Delta(u)=\sum
u_{(1)}\otimes u_{(2)}$.

We also consider the quantized enveloping algebra of
$\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$. Note that

$U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})\cong
U_{q}(\mathfrak{gl}_{N})\otimes U_{q}(\mathfrak{gl}_{N})$

as Hopf algebras. Write $E_{i},F_{i},K_{\epsilon_{j}}^{\pm 1}$, $1\leq i\leq
N-1,1\leq j\leq N$ for the generators of the first copy of
$U_{q}(\mathfrak{gl}_{N})$ and $E_{i},F_{i},K_{\epsilon_{j}}^{\pm 1}$,
$N+1\leq i\leq 2N-1,N+1\leq j\leq 2N$ for the generators of the second copy.
Formally, we may identify $E_{i}$ with $E_{i}\otimes 1$ and $E_{i+N}$ with
$1\otimes E_{i}$ for $1\leq i\leq N-1$ with similar identifications for the
$F_{i}$ and $K_{\epsilon_{j}}^{\pm 1}$.

Let $U^{0}(\mathfrak{gl}_{N})$ denote the subalgebra of
$U_{q}(\mathfrak{gl}_{N})$ generated by $K_{\epsilon_{1}}^{\pm
1},\dots,K_{\epsilon_{N}}^{\pm 1}$. Similarly, we write
$U^{0}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$ for the subalgebra of
$U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$ generated by
$K_{\epsilon_{1}}^{\pm 1},\dots,K_{\epsilon_{2N}}^{\pm 1}$. We drop
$\mathfrak{gl}_{N}$ or $\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$ from the
notation and simply write $U^{0}$ when the associated Lie algebra can be
understood from context.

Let $U^{+}(\mathfrak{gl}_{N})$ denote the subalgebra of
$U_{q}(\mathfrak{gl}_{N})$ generated by $E_{1},\dots,E_{N-1}$. Similarly, let
$U^{+}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$ denote the subalgebra of
$U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$ generated by $E_{i},$ for
$i=1,\dots N-1$ and $i=N+1,\dots,2N-1$. Just as for $U^{0}$, we drop
$\mathfrak{gl}_{N}$ or $\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$ from the
notation and simply write $U^{+}$ when the associated Lie algebra can be
understood from context. Define the subalgebra $U^{-}$ in the same way with
each $E_{i}$ replaced by $F_{i}$.

In the study of $U_{q}(\mathfrak{sl}_{N})$, it is sometimes necessary to pass
to the simply connected quantized enveloping algebra
$\check{U}_{q}(\mathfrak{sl}_{N})$. This Hopf algebra is an extension of
$U_{q}(\mathfrak{sl}_{N})$ obtained by enlarging $U^{0}$ to $\check{U}^{0}$
where $\check{U}^{0}$ is the group algebra generated by the weight lattice $P$
(see for example [5], 3.2.10). In another words,
$\check{U}_{q}(\mathfrak{sl}_{N})$ is the Hopf algebra generated by
$U_{q}(\mathfrak{sl}_{N})$ and $\check{U}^{0}$ where the generators of
$U_{q}(\mathfrak{sl}_{N})$ and the elements $K_{\beta},\beta\in P$ satisfy
(2).

Recall that the augmentation ideal of a Hopf algebra $H$, denoted by $H_{+}$,
is the kernel of the counit $\epsilon$. Given a subset $M$ of $H$, we write
$M_{+}$ for the intersection of $M$ with $H_{+}$. For example, we write
$(U_{q}(\mathfrak{gl}_{N}))_{+}$ for the augmentation ideal of
$U_{q}(\mathfrak{gl}_{N})$. Similarly, we denote the intersection of $U^{+}$
(resp. $U^{-}$) with the augmentation ideal of $U_{q}(\mathfrak{gl}_{N})$ (or
$U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$ depending on context) by
$U^{+}_{+}$ (resp. $U^{-}_{+}$).

We write $L(\lambda)$ for the simple $U_{q}(\mathfrak{gl}_{N})$-module of
highest weight $\lambda$. In other words, $L(\lambda)$ is generated by a
highest weight vector $v_{\lambda}$ such that $E_{i}v_{\lambda}=0$ for all
$i=1,\dots,N-1$ and $K_{\beta}v_{\lambda}=q^{(\beta,\lambda})v_{\lambda}$ for
all weights $\beta$. Recall that $L(\lambda)$ is finite-dimensional viewed as
a $U_{q}(\mathfrak{sl}_{N})$-module if and only if $\lambda\in P^{+}$ and is
finite-dimensional viewed as a $U_{q}(\mathfrak{gl}_{N})$-module if and only
if $\lambda\in\Lambda_{N}$. These notions extend to the setting of
$U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$ in a straightforward manner
and we will similarly denote highest weight modules by $L(\lambda)$ where here
$\lambda$ is understood to be a weight for
$\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N}$.

###  2.3. The adjoint action

Given a Hopf algebra $H$ and a two-sided $H$-bimodule $M$, the bimodule $M$
admits an $H$-module structure via the adjoint action defined by

$\displaystyle({\rm ad}\ a)m=\sum a_{(1)}mS(a_{(2)})$

for all $a\in H$ and $m\in M$. The locally finite part $\mathcal{F}_{H}(M)$ of
$M$ is the submodule consisting of those elements that generate a finite
module with respect to the adjoint action of $H$. More precisely,

$\displaystyle\mathcal{F}_{H}(M)=\\{a\in M|\ \dim[({\rm ad}\ H)m]<\infty\\}.$

When $M$ is also a Hopf algebra, the locally finite part $\mathcal{F}_{H}(M)$
is a subalgebra of $M$ ([6]). We typically drop $H$ from the notation
$\mathcal{F}(M)$ when $H$ can be understood from context.

For $H=U_{q}(\mathfrak{gl}_{N})$, $U_{q}(\mathfrak{sl}_{N})$ or
$\check{U}_{q}(\mathfrak{sl}_{N})$, the adjoint action is determined by the
following formulas:

$\displaystyle({\rm ad}\ E_{i})\cdot m=E_{i}m-K_{i}mK_{i}^{-1}E_{i}$
$\displaystyle({\rm ad}\ F_{i})\cdot m=F_{i}mK_{i}-mF_{i}K_{i}$
$\displaystyle({\rm ad}\ K)\cdot m=KmK^{-1}$

for all $i=1,\dots,N-1$, $K=K_{\beta}$ for all weights $\beta$ with
$K_{\beta}$ in the specified quantized enveloping algebra, and $m\in M$. Note
that these formulas carry over easily to
$H=U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$.

###  2.4. Three symmetric pair families

We are interested in quantum homogeneous spaces associated to three families
of symmetric pairs $\mathfrak{g},\mathfrak{k}$ where $\mathfrak{k}$ is the Lie
subalgebra fixed by the involution $\theta$. Here,
$\mathfrak{g}=\mathfrak{gl}_{n}$ for the first family,
$\mathfrak{g}=\mathfrak{gl}_{2n}$ for the second family, and
$\mathfrak{g}=\mathfrak{gl}_{n}\oplus\mathfrak{gl}_{n}$ for the third family.
Below, we describe the involution for each family and then define the
generators for the associated quantum analog $\mathcal{B}_{\theta}$ of
$U(\mathfrak{g}^{\theta})$.

For each of the three families, we associate an $R$-matrix $R_{\mathfrak{g}}$
and a solution $J$ to the reflection equation
$R_{\mathfrak{g}}J_{1}R_{\mathfrak{g}}^{t_{1}}J_{2}=J_{2}R_{\mathfrak{g}}^{t_{1}}J_{1}R_{\mathfrak{g}}$
where $J$ is an $N\times N$ matrix, $J_{1}=J\otimes I$ and $J_{2}=I\otimes J$
where $I$ is the $N\times N$ identity matrix. These $R$-matrices are closely
related to the following matrix $R$ in ${\rm Mat}_{N}\times{\rm Mat}_{N}$
defined by

(3) $\displaystyle R$ $\displaystyle=\sum_{1\leq i\leq N}qe_{ii}\otimes
e_{ii}+\sum_{1\leq i<j\leq N}(e_{ii}\otimes e_{jj}+e_{jj}\otimes
e_{ii})+(q-q^{-1})\sum_{1\leq j<i\leq N}e_{ij}\otimes e_{ji}$

where ${\rm Mat}_{N}$ is the space of $N\times N$ matrices and the $e_{ij}$
are matrix units.

Type AI: $\mathfrak{g}=\mathfrak{gl}_{n}$ and $\theta$ is defined by
$\theta(e_{i})=-f_{i}$, $\theta(f_{i})=-e_{i}$ and
$\theta(h_{\epsilon_{j}})=-h_{\epsilon_{j}}$ each $i=1,\dots,n-1$ and
$j=1,\dots,n$. Hence $\mathfrak{k}$ is generated by $f_{i}-e_{i}$ for
$i=1,\dots,n-1$. Passing to the quantum case, $\mathcal{B}_{\theta}$ is
generated by $F_{i}-E_{i}K_{i}^{-1}$, for $i=1,\dots,n-1$. In this case,
$R_{\mathfrak{g}}=R$ as defined in (3) with $N=n$. The associated solution to
the reflection equation is $J=I_{n}$, the $n\times n$ identity matrix.

Type AII: $\mathfrak{g}=\mathfrak{gl}_{2n}$ and $\theta$ is defined by

  * •

$\theta(e_{i})=e_{i},\theta(f_{i})=f_{i},\theta(h_{i})=h_{i}$ for
$i=1,3,\dots,2n-1.$

  * •

$\theta(f_{i})=-[e_{i-1},[e_{i+1},e_{i}]]$ for $i=2,4,\dots,2n-2$.

  * •

$\theta(h_{\epsilon_{2i-1}})=-h_{\epsilon_{2i}}$ for $i=1,\dots,n$.

Hence $\mathfrak{k}$ is generated by $e_{i},f_{i},h_{i},$ for
$i=1,3,\dots,2n-1$ and $f_{i}-[e_{i-1},[e_{i+1},e_{i}]]$ for
$i=2,4,\dots,2n-2$. Passing to the quantum case, $\mathcal{B}_{\theta}$ is
generated by

  * •

$K_{i}^{\pm 1},E_{i},F_{i}$ for $i=1,3,\dots,2n-1.$

  * •

$B_{i}=F_{i}-q^{3}(({\rm ad}\ E_{i-1}E_{i+1})\cdot
E_{i})K_{i}^{-1}=F_{i}-q^{3}[E_{i-1},[E_{i+1},E_{i}]_{q}]_{q}K_{i}^{-1}$ for
$i=2,4,\dots,2n-2$ where $[a,b]_{q}$ denotes the $q$-commutator $ab-qba$ of
$a$ and $b$.

In this case, $R_{\mathfrak{g}}=R$ as defined in (3) with $N=2n$. The
associated solution to the reflection equation is
$J=\sum_{k=1}^{n}(e_{2k-1,2k}-qe_{2k,2k-1})$.

Type A diagonal case: $\mathfrak{g}=\mathfrak{gl}_{n}\oplus\mathfrak{gl}_{n}$
and $\theta$ is defined by $\theta(f_{i})=-e_{n+i},\ \theta(f_{n+i})=-e_{i},$
and $\theta(h_{\epsilon_{j}})=-h_{\epsilon_{n+j}}$ for $i=1,\dots,n-1$ and
$j=1,\dots,n$ and so $\mathfrak{k}$ is generated by $f_{i}-e_{n+i}$,
$f_{n+i}-e_{i}$, and $h_{\epsilon_{j}}-h_{\epsilon_{n+j}}$ for $i=1,\dots,n-1$
and $j=1,\dots,n$. Passing to the quantum case, the corresponding quantum
symmetric pair coideal subalgebra $\mathcal{B}_{\theta}$ is generated by

$\displaystyle B_{i}=F_{i}-qE_{n+i}K_{i}^{-1},\quad
B_{n+i}=F_{n+i}-qE_{i}K_{n+i}^{-1},{\rm\quad
and\quad}(K_{\epsilon_{j}}^{-1}K_{\epsilon_{n+j}})^{\pm 1}$

for $i=1,\dots,n-1$ and $j=1,\dots,n$. In this case, $R_{\mathfrak{g}}$ is the
$4n^{2}\times 4n^{2}$ block diagonal matrix with diagonal
$(R,I_{n^{2}},I_{n^{2}},R)$ were $R$ is the matrix defined by (3) with $N=n$.
The associated solution to the reflection equation is
$J=\sum_{k=1}^{n}(e_{k,n+k}+e_{n+k,k})$. We often drop the phrase Type A and
simply refer to this family as the diagonal case or diagonal type.

###  2.5. Restricted root systems

Write $\pi$ for the set of simple roots for $\mathfrak{g}$ and $\Phi$ for the
root system generated by $\pi$. By Section 2.1, for Types AI and AII, we have
$\pi=\\{\alpha_{1},\dots,\alpha_{N-1}\\}$ and the root system $\Phi=\Phi_{N}$
is of type $A_{N-1}$ where $N=n$ in Type AI and $N=2n$ in Type AII. For the
diagonal case,
$\pi=\\{\alpha_{1},\dots,\alpha_{n-1}\\}\cup\\{\alpha_{n+1},\dots,\alpha_{2n-1}\\}$
where $\\{\alpha_{1},\dots,\alpha_{n-1}\\}$ and
$\\{\alpha_{n+1},\dots,\alpha_{2n-1}\\}$ each separately generate a root
system of Type An and together generate a root system
$\Phi=\Phi^{(1)}_{n}\cup\Phi^{(2)}_{n}$.

Note that $\theta$ induces an involution, which we also call $\theta$, on the
root system $\Phi$. More generally, $\theta$ can be extended to an involution
on $\sum_{i=1}^{N}\mathbb{Z}\epsilon_{i}$ where $N=n$ in Type AI, and $N=2n$
in Type AII and the diagonal case. Set $\tilde{\beta}=(\beta-\theta(\beta))/2$
for each $\beta\in\sum_{i=1}^{N}\mathbb{Z}\epsilon_{i}$. The set of all
$\tilde{\beta}$ where $\beta$ runs over elements in $\Phi$, forms another root
system, called the restricted root system which we denote by $\Sigma$. For all
three families under consideration, the root system $\Sigma$ is of type An-1.
We denote the simple roots for $\Sigma$ by
$\alpha_{1}^{\Sigma},\dots,\alpha_{n-1}^{\Sigma}$ and explain below how these
are related to the simple roots for $\Phi$.

The restricted root system is contained in a vector space spanned by
orthonormal basis vectors that live inside the set $\\{\tilde{\beta}|\
\beta\in\sum_{i=1}^{n}\mathbb{Z}\epsilon_{i}\\}$. We denote this orthonormal
basis by $\epsilon_{1}^{\Sigma},\dots,\epsilon_{n}^{\Sigma}$. Write
$\eta_{1},\dots,\eta_{n-1}$ for the fundamental weights associated to $\Sigma$
and write $\hat{\eta}_{1},\dots,\hat{\eta}_{n}$ for the fundamental
partitions. Let $P_{\Sigma}$ denote the weight lattice and $P^{+}_{\Sigma}$
denote the dominant integral weights defined by the root system $\Sigma$. In
other words $P_{\Sigma}$ (resp. $P^{+}_{\Sigma}$) is just the set of
$\mathbb{Z}$ (resp. $\mathbb{N}$) linear combinations of the fundamental
weights $\eta_{1},\dots,\eta_{n-1}$. Let $\Lambda^{+}_{\Sigma}$ denote the
$\mathbb{N}$ span of the fundamental partitions $\hat{\eta}_{i},i=1,\dots,n$.
Let $W_{\Sigma}$ denote the Weyl group for the root system $\Sigma$. We
identify the $\epsilon^{\Sigma}_{i}$, $\eta_{i}$ and $\hat{\eta}_{i}$ with
elements of $\sum_{i=1}^{n}\mathbb{Z}\epsilon_{i}$ below for each of the three
families under consideration. With respect to this identification, it is
straightforward to check that the longest element of the Weyl group for $\Phi$
also acts as the longest element in the restricted Weyl group $W_{\Sigma}$.
Abusing notation slightly, we refer to $w_{0}$ as the longest element of
$W_{\Sigma}$ with the understanding that $w_{0}$ is the restriction of the
longest element of the Weyl group for $\Phi$ to restricted weights. In
particular,
$w_{0}\hat{\eta}_{i}=\epsilon^{\Sigma}_{n-i+1}+\cdots+\epsilon^{\Sigma}_{n}$
for each $i$ and for each of the three families.

Type AI: For Type AI,
$\tilde{\alpha}_{i}=(\alpha_{i}-\theta(\alpha_{i}))/2=\alpha_{i}$ for each
$i=1,\dots,n-1$. Hence $\alpha_{i}^{\Sigma}=\tilde{\alpha}_{i}=\alpha_{i}$ for
$i=1,\dots,n-1$ and the set of positive simple roots for the root system
$\Sigma$ is just the set $\\{\alpha_{1},\dots,\alpha_{n-1}\\}$. A similar
straightforward computation yields $\tilde{\epsilon}_{i}=\epsilon_{i}$ for
each $i$. In other words, the restricted root system $\Sigma$ in this case
equals $\Phi$. Thus, $\epsilon^{\Sigma}_{j}=\epsilon_{j}$,
$\hat{\eta}_{j}=\hat{\omega}_{j}$, ${\eta}_{k}={\omega}_{k}$, and
$\hat{\eta}_{k}=\hat{\omega}_{k}$ for $j=1,\dots,n$ and $k=1,\dots,n-1$. Note
further that $\tilde{\omega}_{k}=(\omega_{k}-\theta(\omega_{k}))/2=\omega_{k}$
and so $\eta_{k}$ equals $\tilde{\omega}_{k}$, the restricted weight
associated to $\omega_{k}$. Similarly,
$\hat{\eta}_{k}=(\hat{\omega}_{k}-\theta({\hat{\omega}}_{k}))/2$, the
restricted weight associated to $\hat{\omega}_{k}$, for each $k=1,\dots,n$.

Type AII: For Type AII, $\tilde{\alpha}_{2i-1}=0$ for $i=1,\dots,n$ and
$\tilde{\alpha}_{2j}=(\alpha_{2j-1}+2\alpha_{2j}+\alpha_{2j+1})/2$ for
$j=1,\dots,n-1$. The set of positive simple roots for $\Sigma$ is
$\\{\tilde{\alpha}_{2j}|\ j=1,\dots,n-1\\}$ and
$\alpha_{j}^{\Sigma}=\tilde{\alpha}_{2j}$ for $j=1,\dots,n-1$. Note that the
inner product on $\Phi$ needs to be scaled differently for the restricted root
system. This is because

$\displaystyle(\tilde{\alpha}_{2j},\tilde{\alpha}_{2j})=(\alpha_{2j-1}+2\alpha_{2j}+\alpha_{2j+1},\alpha_{2j-1}+2\alpha_{2j}+\alpha_{2j+1})/4=1.$

Hence the inner product for $\Sigma$ takes the form
$(\cdot,\cdot)_{\Sigma}=2(\cdot,\cdot)$ where $(\cdot,\cdot)$ is the usual
Cartan inner product for the root system of $\mathfrak{gl}_{2n}$. In other
words, $(\cdot,\cdot)_{\Sigma}$ is the normalization of $(\cdot,\cdot)$ chosen
so that $(\tilde{\alpha}_{2j},\tilde{\alpha}_{2j})=2$ for each $j$. Now
$\tilde{\epsilon}_{2j}=\tilde{\epsilon}_{2j-1}=(\epsilon_{2j-1}+\epsilon_{2j})/2$
and so $2(\tilde{\epsilon}_{2i},\tilde{\alpha}_{2j})=\delta_{ij}$. It follows
that the corresponding set of orthonormal vectors is
$\\{\tilde{\epsilon}_{2},\tilde{\epsilon}_{4},\dots,\tilde{\epsilon}_{2n}\\}$
and hence $\epsilon^{\Sigma}_{i}=\tilde{\epsilon}_{2i}$ for $i=1\dots,n$. The
fundamental partitions associated to this restricted root system take the form

$\displaystyle\hat{\eta}_{r}=\epsilon^{\Sigma}_{1}+\cdots\epsilon^{\Sigma}_{r}=\tilde{\epsilon}_{2}+\cdots+\tilde{\epsilon}_{2r}=(\epsilon_{1}+\epsilon_{2}+\cdots+\epsilon_{2r})/2=\hat{\omega}_{2r}/2$

for $r=1,\dots,n$. It follows that the fundamental weights are

$\displaystyle\eta_{r}=\hat{\eta}_{r}-({{r}/{N}})\hat{\eta}_{N}=(\epsilon_{1}+\epsilon_{2}+\cdots+\epsilon_{2r})/2-({{r}/{N}})(\epsilon_{1}+\epsilon_{2}+\cdots+\epsilon_{2n})/2={\omega}_{2r}/2$

for $r=1,\dots,n-1$. Note also that

$\displaystyle(\hat{\omega}_{2j}-\theta(\hat{\omega}_{2j}))/2=(\tilde{\epsilon}_{1}+\cdots+\tilde{\epsilon}_{2j})=2(\tilde{\epsilon}_{2}+\tilde{\epsilon}_{4}+\cdots+\tilde{\epsilon}_{2j})=2\hat{\eta}_{j}$

for each $j=1,\dots,n$. Similarly,

$\displaystyle(\hat{\omega}_{2j-1}-\theta(\hat{\omega}_{2j-1}))/2=(\tilde{\epsilon}_{1}+\cdots+\tilde{\epsilon}_{2j-1})=2(\tilde{\epsilon}_{2}+\tilde{\epsilon}_{4}+\cdots+\tilde{\epsilon}_{2j-2})+\tilde{\epsilon}_{2j}$

for $j=1,\dots,n$. Hence
$(\hat{\omega}_{1}-\theta(\hat{\omega}_{1}))/2=\hat{\eta}_{1}$ and

$\displaystyle(\hat{\omega}_{2j-1}-\theta(\hat{\omega}_{2j-1}))/2=\hat{\eta}_{j}+\hat{\eta}_{j-1}$

for each $j=2,\dots,n$. It follows that
$\omega_{1}-\theta(\omega_{1})=\eta_{1}$,
$\omega_{2j}-\theta(\omega_{2j})=2\eta_{j}$ and
$\omega_{2j-1}-\theta(\omega_{2j-1})=\eta_{j}+\eta_{j-1}$ for $j=2,\dots,n$.

Type A diagonal case: For the diagonal case, we have
$\tilde{\alpha}_{i}=(\alpha_{i}+\alpha_{n+i})/2=\tilde{\alpha}_{n+i}$. Thus
the set of positive simple roots for the root system $\Sigma$ is
$\\{\tilde{\alpha}_{1},\cdots,\tilde{\alpha}_{n-1}\\}$ and
$\alpha_{i}^{\Sigma}=\tilde{\alpha}_{i}$ for $i=1,\dots,n-1$. Note that
$(\tilde{\alpha}_{i},\tilde{\alpha}_{i})=((\alpha_{i}+\alpha_{n+i})/2,(\alpha_{i}+\alpha_{n+i})/2)=1$.
Hence, the inner product for the restricted root system in the diagonal case
is $(\cdot,\cdot)_{\Sigma}=2(\cdot,\cdot)$. Here, the corresponding
orthonormal basis is $\\{\tilde{\epsilon}_{1},\dots,\tilde{\epsilon}_{n}\\}$
where $\tilde{\epsilon}_{i}=(\epsilon_{i}+\epsilon_{n+i})/2$ for each $i$. In
particular, we have $\epsilon^{\Sigma}_{i}=\tilde{\epsilon}_{i}$ for
$i=1,\dots,n$. The fundamental partitions in this case are

$\displaystyle\hat{\eta}_{r}=\epsilon^{\Sigma}_{1}+\cdots\epsilon^{\Sigma}_{r}=\tilde{\epsilon}_{1}+\cdots+\tilde{\epsilon}_{r}=({\epsilon}_{1}+\epsilon_{1+n}+\cdots+{\epsilon}_{r}+\epsilon_{r+n})/2$

for $r=1,\dots,n$. It follows that the fundamental weights associated to
$\Sigma$ satisfy

$\displaystyle\eta_{r}=\hat{\eta}_{r}-({r}/{n})\hat{\eta}_{n}=(\hat{\omega}_{r}+\hat{\omega}_{r+n})/2-({r}/{n})(\hat{\omega}_{n}+\hat{\omega}_{2n})/2=\omega_{r}/2+\omega_{r+n}/2$

for $r=1,\dots,n-1$. We have

$\displaystyle(\hat{\omega}_{j}-\theta(\hat{\omega}_{j}))/2=\tilde{\epsilon}_{1}+\cdots+\tilde{\epsilon}_{j}=\hat{\eta}_{j}$

and, similarly,
$(\hat{\omega}_{n+j}-\theta(\hat{\omega}_{n+j}))/2=\hat{\eta}_{j}$. Thus
$\eta_{j}=\tilde{\omega}_{j}=\tilde{\omega}_{n+j}$ for $j=1,\dots,n$.

A finite-dimensional simple highest weight module is called spherical if it
contains a nonzero $\mathcal{B}_{\theta}$ invariant vector, i.e., a vector $v$
such that $x\cdot v=\epsilon(x)v$ for $x\in\mathcal{B}_{\theta}$. Note that
$2\Lambda^{+}_{\Sigma}=\\{2\lambda,\lambda\in\Lambda^{+}_{\Sigma}\\}$ is a
subset of $\Lambda_{N}^{+}$ where $N=n$ in Type AI and $N=2n$ in Type AII.
Moreover, given $\gamma\in\Lambda_{N}^{+}$, the module $L(\gamma)$ is
spherical if and only if $\gamma=2\lambda+s\hat{\eta}_{n}$ for some
$\lambda\in\Lambda^{+}_{\Sigma}$ and $s\in\mathbb{N}$ in Type AI and
$\gamma=2\lambda$ for $\lambda\in\Lambda^{+}_{\Sigma}$ in Type AII ([21],
(3.12)).

Now consider the diagonal case. For
$\gamma,\gamma^{\prime}\in\Lambda^{+}_{n}$, $L({\gamma\oplus\gamma^{\prime}})$
is spherical if and only if $\gamma=\gamma^{\prime}$. This follows from the
classification of spherical modules using $\mathfrak{sl}_{n}$ instead of
$\mathfrak{gl}_{n}$ (see [14], Section 7) along with the fact that weight
vectors admitting a trivial $K_{\epsilon_{i}}K_{\epsilon_{i+n}}^{-1}$ action
for each $i=1,\dots,n$ must have weights of the form $\gamma\oplus\gamma$.
Note that the set of $\gamma\oplus\gamma$ with $\gamma\in\Lambda_{n}^{+}$ is
precisely $2\Lambda^{+}_{\Sigma}$. Thus $L({\gamma\oplus\gamma^{\prime}})$ is
spherical if and only if $\gamma\oplus\gamma^{\prime}=2\lambda$ for some
$\lambda\in\Lambda^{+}_{\Sigma}$.

##  3\. Quantized function algebras

###  3.1. Quantized matrix functions

Let ${\rm Mat}_{N}$ denote the space of $N\times N$ matrices with basis
$e_{ij}$, $1\leq i,j\leq N$. The matrix $R$ defined by (3) can be written as
$R=\sum_{i,j,k,l}r^{ij}_{kl}e_{ik}\otimes e_{jl}$ where

  * •

$r^{ii}_{ii}=q,$ $r_{ij}^{ij}=1$ for all $i,j,$ with $i\neq j$.

  * •

$r_{ji}^{ij}=(q-q^{-1})$ for all $j<i$.

  * •

$r^{ij}_{kl}=0$ for all other choices of $i,j,k,l$.

We can view $R$ as the matrix in ${\rm Mat}_{N}\times{\rm Mat}_{N}$ with
$(i,k)\times(j,l)$ entry $r^{ij}_{kl}$. Given a matrix $A\in{\rm
Mat}_{N}\times{\rm Mat}_{N}$, write $A^{t_{s}}$ for the transpose in the first
term when $s=1$ and the second when $s=2$. For example,
$(R^{t_{1}})^{ij}_{kl}=r^{kj}_{il}.$

The quantized function algebra $\mathcal{O}_{q}({\rm Mat}_{N})$ is the
bialgebra over $\mathbb{C}(q)$ generated by $t_{ij},1\leq i,j\leq N$ where the
$t_{ij}$ satisfy the following relations

  * (i)

$t_{ki}t_{kj}=qt_{kj}t_{ki}$, $t_{ik}t_{jk}=qt_{jk}t_{ik}$ ($i<j$)

  * (ii)

$t_{il}t_{kj}=t_{kj}t_{il},t_{ij}t_{kl}-t_{kl}t_{ij}=(q-q^{-1})t_{kj}t_{il}$
($i<k;j<l$)

Set $T=(t_{ij})$, the $N\times N$ matrix with $ij$ entry equal to $t_{ij}$ and
set $T_{1}=T\otimes I$ and $T_{2}=I\otimes T$ where $I$ is the $N\times N$
identity matrix. These relations can be written in matrix form as
$RT_{1}T_{2}=T_{2}T_{1}R$, or equivalently, as the set of equations

$\displaystyle\sum_{j,k}r_{jk}^{ld}t_{ja}t_{kb}=\sum_{j,k}t_{dk}t_{lj}r^{jk}_{ab}.$

for all $a,b,d,l$. A straightforward computation shows that the map $\iota$
defined by

(4) $\displaystyle\iota(t_{ij})=t_{ji}$

for all $i,j=1,\dots,n$ defines an algebra automorphism of
$\mathcal{O}_{q}({\rm Mat}_{N})$.

The algebra $\mathcal{O}_{q}({\rm Mat}_{N})$ admits the structure of a
$U_{q}(\mathfrak{gl}_{N})$-bimodule algebra where the left action is
determined by

$\displaystyle E_{k}\cdot t_{ij}=\delta_{i-1,k}t_{i-1,j},\quad F_{k}\cdot
t_{ij}=\delta_{ik}t_{i+1,j},\quad K_{\epsilon_{r}}\cdot
t_{ij}=q^{\delta_{ir}}t_{ij}$

and the right action by

$\displaystyle t_{ij}\cdot E_{k}=\delta_{jk}t_{i,j+1},\quad t_{ij}\cdot
F_{k}=\delta_{j-1,k}t_{i,j-1},\quad t_{ij}\cdot
K_{\epsilon_{r}}=q^{\delta_{jr}}t_{ij}$

for $r=1,\dots,N$, $i,j=1,\dots,N$ and $k=1,\dots,N-1$. Here, we are using the
notation $t_{uv}=0$ for $u\in\\{0,N+1\\}$ or $v\in\\{0,N+1\\}$.

Let $\mathcal{O}_{q}({\rm Mat}_{N})^{op}$ denote the bialgebra with the same
coalgebra structure and opposite algebra structure of the bialgebra
$\mathcal{O}_{q}({\rm Mat}_{N})$. Write $\partial_{ij},1\leq i,j\leq N$ for
the generators of $\mathcal{O}_{q}({\rm Mat}_{N})^{op}$ so that the algebra
map defined by $t_{ij}\mapsto\partial_{ij}$ is an anti-automorphism. The
algebra $\mathcal{O}_{q}({\rm Mat}_{N})^{op}$ is also a
$U_{q}(\mathfrak{gl}_{N})$-bimodule algebra with action defined by

$\displaystyle
E_{k}\cdot\partial_{ij}=-\delta_{ik}q^{-1}\partial_{i+1,j},\quad
F_{k}\cdot\partial_{i,j}=-\delta_{i-1,k}q\partial_{i-1,j},\quad
K_{\epsilon_{r}}\cdot\partial_{ij}=q^{-\delta_{ir}}\partial_{ij}$

and right action defined by

$\displaystyle\partial_{ij}\cdot
E_{k}=-\delta_{j-1,k}q\partial_{i,j-1},\quad\partial_{ij}\cdot
F_{k}=-\delta_{jk}q^{-1}\partial_{i,j+1},\quad\partial_{ij}\cdot
K_{\epsilon_{r}}=q^{-\delta_{jr}}\partial_{ij}$

for $r=1,\dots,N$, $i,j=1,\dots,N$ and $k=1,\dots,N-1.$ Just as for the
$t_{uv}$, we are using the notation $\partial_{uv}=0$ for $u\in\\{0,N+1\\}$ or
$v\in\\{0,N+1\\}$.

Let $\mathcal{O}_{q}({\rm Mat}_{N})\otimes\mathcal{O}_{q}({\rm Mat}_{N})$
denote the algebra generated by two copies of $\mathcal{O}_{q}({\rm
Mat}_{N})$. The first is generated by $t_{ij},1\leq i,j\leq N$ and the second
is generated by $t_{i+N,j+N},1\leq i,j\leq N$ and $t_{ij}$ commutes with
$t_{k+N,l+N}$ for all $i,j,k,l\in\\{1,\dots,N\\}$. Formally, $t_{ij}$ can be
identified with $t_{ij}\otimes 1$ and $t_{i+N,j+N}$ with $1\otimes t_{ij}$.
The algebra $\mathcal{O}_{q}({\rm Mat}_{N})\otimes\mathcal{O}_{q}({\rm
Mat}_{N})$ is a $U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})\cong
U_{q}(\mathfrak{gl}_{N})\otimes U_{q}(\mathfrak{gl}_{N})$-bimodule algebra.
Here, the left action is given by $(g\otimes h)\cdot(a\otimes b)=(g\cdot
a)\otimes(h\cdot b)$ for all $g\otimes h\in U_{q}(\mathfrak{gl}_{N})\otimes
U_{q}(\mathfrak{gl}_{N})$ and $a\otimes b\in\mathcal{O}_{q}({\rm
Mat}_{N})\otimes\mathcal{O}_{q}({\rm Mat}_{N})$. The right action is defined
in the same way with the action on the right instead of the left. Similar
notions apply for the algebra $\mathcal{O}_{q}({\rm
Mat}_{N})^{op}\otimes\mathcal{O}_{q}({\rm Mat}_{N})^{op}$.

Consider the three families as described in Section 2.4. Set
$\mathscr{P}=\mathcal{O}_{q}({\rm Mat}_{N})$ in Type AI with $N=n$ and in Type
AII with $N=2n$. For the diagonal case, set $\mathscr{P}=\mathcal{O}_{q}({\rm
Mat}_{n})\otimes\mathcal{O}_{q}({\rm Mat}_{n})$. Similarly, set
$\mathscr{D}=\mathcal{O}_{q}({\rm Mat}_{N})^{op}$ in Type AI with $N=n$ and in
Type AII with $N=2n$. For the diagonal case, set
$\mathscr{D}=\mathcal{O}_{q}({\rm Mat}_{n})^{op}\otimes\mathcal{O}_{q}({\rm
Mat}_{n})^{op}$.

###  3.2. Functions on homogeneous spaces

Let $\mathfrak{g},\mathfrak{k}$ be a symmetric pair corresponding to one of
the three families described in Section 2.4. Recall that $J$ is the associated
solution to the reflection equation. Write $J_{r,s}$ for the coefficient of
$e_{rs}$ in $J$. Define elements $x_{ij}$ and $d_{ij}$ by

$\displaystyle x_{ij}=\sum_{r,s}t_{ir}J_{r,s}t_{js}{\rm\ and\
}d_{ij}=\sum_{r,s}q^{-2\hat{s}}\partial_{ir}J_{r,s}\partial_{js}$

for $1\leq i,j\leq N$ where $N=n$ in Type AI, $N=2n$ in Type AII and diagonal
type, $\hat{s}=s$ in Types AI and AII, and for the diagonal type $\hat{s}=s$
for $s\leq n$, and $\hat{s}=s-n$ when $s\geq n+1$.

Let $\mathscr{P}_{\theta}$ be the subalgebra of $\mathscr{P}$ generated by the
$x_{ij},1\leq i,j\leq N$, and let $\mathscr{D}_{\theta}$ be the subalgebra of
$\mathscr{D}$ generated by the $d_{ij},1\leq i,j\leq N$. As explained in [21]
and [19], the quantum homogeneous space $\mathcal{O}_{q}[G/K]$ associated to
$\mathfrak{g},\mathfrak{k}$ is generated by $\mathscr{P}_{\theta}$ and powers
of quantum determinants. Set $X=(x_{ij})_{1\leq i,j\leq N}$ and
$D=(d_{ij})_{1\leq i,j\leq N}$. Define $X_{1},X_{2},D_{1},$ and $D_{2}$ in the
same way as $T_{1},T_{2}$ (i.e. $X_{1}=X\otimes I$, etc.). The following
theorem summarizes properties of $\mathscr{P}_{\theta}$ from [19].

######  Theorem 3.1.

The relations satisfied by the generators $x_{ij}$ of $\mathscr{P}_{\theta}$
are determined by the following equations:

  * (i)

The matrix relation
$R_{\mathfrak{g}}X_{1}R_{\mathfrak{g}}^{t_{1}}X_{2}=X_{2}R_{\mathfrak{g}}^{t_{1}}X_{1}R_{\mathfrak{g}}$.

  * (ii)

The linear relations $x_{ij}=\gamma x_{ji}$ for $i<j$ and $x_{ab}=0$ for all
$(a,b)\in\mathcal{S}$ where

    * $\bullet$

$\gamma=q$ in Type AI, $\gamma=-q^{-1}$ in Type AII, $\gamma=1$ in the
diagonal type

    * $\bullet$

$\mathcal{S}$ is the empty set in Type AI,
$\mathcal{S}=\\{(i,i),i=1,\dots,2n\\}$ in Type AII, and
$\mathcal{S}=\\{(i,j),1\leq i,j\leq n\\}\cup\\{ij,n+1\leq i,j\leq 2n\\}$ in
diagonal type.

Moreover, $\mathscr{P}_{\theta}$ is a left $U_{q}(\mathfrak{g})$-module and
(trivial) right $\mathcal{B}_{\theta}$-module subalgebra of $\mathscr{P}$.

In the diagonal case, it turns out that the matrix relations for
$\mathscr{P}_{\theta}$ reduce to
$R\hat{X}_{1}\hat{X}_{2}=\hat{X}_{2}\hat{X}_{1}R$ where
$\hat{X}=(x_{i,j+n})_{1\leq i,j\leq n}$ and $R$ is the matrix defined by (3)
with $N=n$ ([19], Lemma 5.12). In particular,
$\mathscr{P}_{\theta}\cong\mathcal{O}_{q}({\rm Mat}_{n})$ as algebras via the
map that sends $x_{i,j+n}$ to $t_{ij}$ all $1\leq i,j\leq n$. For the other
two families, Types AI and AII, explicit relations are given in [19], Lemma
5.8. Here we provide some of these relations in special cases that will be
needed later in this paper. In particular, for Type AI we have

(5) $\displaystyle x_{en}x_{nn}=q^{2}x_{nn}x_{en}\quad{\rm
and\quad}x_{an}x_{en}=qx_{en}x_{an}$

for all $a<e<n$. Similarly, for Type AII, we have

(6) $\displaystyle x_{a,2n}x_{e,2n}=qx_{e,2n}x_{a,2n}$

for $a<e<2n$. More generally, it follows from [19], Lemma 5.8 that the
defining relations are $q$-analogs of commutativity relations and take the
form

(7) $\displaystyle
x_{ij}x_{kl}=q^{s}x_{kl}x_{ij}+\sum_{\\{a,b,c,d\\}=\\{i,j,k,l\\}}(q-q^{-1})wx_{ab}x_{cd}$

for some integer $s$ and some element $w\in\mathbb{C}[q,q^{-1}]$ where neither
$x_{ij}x_{kl}$ nor $x_{kl}x_{ij}$ appear in the final sum of the right hand
side. It follows from the formulas for the relations satisfied by the $t_{ij}$
and the fact that $\mathscr{P}_{\theta}\cong\mathcal{O}_{q}({\rm Mat}_{n})$ as
algebras in the diagonal case, that (7) holds for the diagonal family as well.

The following result, also from [19], holds for $\mathscr{D}_{\theta}$ and
shows that as an algebra, $\mathscr{D}_{\theta}$ is isomorphic to
$\mathscr{P}_{\theta}^{op}$. In analogy to the situation for
$\mathscr{P}_{\theta}$, the map sending $d_{i,j+n}$ to $\partial_{ij}$ for
$1\leq i,j\leq n$ defines an isomorphism of $\mathscr{D}_{\theta}$ onto
$\mathcal{O}_{q}({\rm Mat}_{n})^{op}$ for the diagonal family.

######  Theorem 3.2.

The relations satisfied by the generators $d_{ij}$ of $\mathscr{D}_{\theta}$
are determined by the following equations:

  * (i)

The matrix relation
$R_{\mathfrak{g}}D_{2}R_{\mathfrak{g}}^{t_{1}}D_{1}=D_{1}R_{\mathfrak{g}}^{t_{1}}D_{2}R_{\mathfrak{g}}$.

  * (ii)

The linear relations $d_{ij}=\gamma d_{ji}$ for $i<j$ and $d_{ab}=0$ for all
$(a,b)\in\mathcal{S}$ where

    * $\bullet$

$\gamma=q^{-1}$ in Type AI, $\gamma=-q$ in Type AII, $\gamma=1$ in the
diagonal type

    * $\bullet$

is the empty set in Type AI, $\mathcal{S}=\\{ii,i=1,\dots,2n\\}$ in Type AII,
and $\mathcal{S}=\\{ij,1\leq i,j\leq n\\}\cup\\{ij,n+1\leq i,j\leq 2n\\}$ in
diagonal type.

Moreover, $\mathscr{D}_{\theta}$ is a left $U_{q}(\mathfrak{g})$-module and
(trivial) right $\mathcal{B}_{\theta}$-module subalgebra of $\mathscr{D}$.

Note that the defining relations for $\mathscr{P}_{\theta}$ are homogeneous
and thus $\mathscr{P}_{\theta}$ has a natural degree function defined by
$\deg(x_{ij})=1$ for all $i,j$. Let $\mathcal{J}$ be the filtration on
$\mathscr{P}_{\theta}$ defined by $\deg$. In particular, for each $r$, we have

$\displaystyle\mathcal{J}_{r}(\mathscr{P}_{\theta})=\sum_{j\leq
r}\\{x\in\mathscr{P}_{\theta}|\ \deg(x)\leq r\\}.$

Since all the relations for $\mathscr{P}_{\theta}$ are homogeneous with
respect to degree, $\mathscr{P}_{\theta}$ is isomorphic to the graded algebra
defined by this filtration. Just as for $\mathscr{P}_{\theta}$, we can define
a degree function on $\mathscr{D}_{\theta}$ such that $\deg(d_{ij})=1$ for all
$i,j$. The resulting filtration yields a graded algebra isomorphic to
$\mathscr{D}_{\theta}$. For each $r$, set $\mathscr{P}^{r}_{\theta}$ equal to
the homogeneous subspace of $\mathscr{P}_{\theta}^{r}$ consisting of elements
of exactly degree $r$ and $\mathscr{D}_{\theta}^{r}$ equal to the homogeneous
subspace consisting of elements of exactly degree $r$.

The left $U_{q}(\mathfrak{g})$-module structures can be deduced directly from
the left actions of $U_{q}(\mathfrak{g})$ on $\mathscr{P}$ and $\mathscr{D}$.
The action of the generators of $U_{q}(\mathfrak{g})$ on the generators of
$\mathscr{P}_{\theta}$ and $\mathscr{D}_{\theta}$ is explicitly given in [19],
Lemma 5.4. As noted in [19], this action can be described as follows. The
vector space spanned by the generators $x_{ij}$ of $\mathscr{P}_{\theta}$
forms a simple left $U_{q}(\mathfrak{g})$-module generated by a highest weight
vector $x_{1r}$ of weight $L(\epsilon_{1}+\epsilon_{r})$ where $r=1$ in Type
AI, $r=2$ in Type AII, and $r=n+1$ in the diagonal type. Similarly,
$\sum_{i,j}\mathbb{C}(q)d_{ij}$ is a simple left $U_{q}(\mathfrak{g})$-module
generated by the lowest weight vector $d_{1r}$ of weight
$-\epsilon_{1}-\epsilon_{r}$ where $r=1$ in Type AI, $r=2$ in Type AII, and
$r=n+1$ in the diagonal type. Moreover, the action of the Cartan elements on
$\mathscr{P}_{\theta}$ and $\mathscr{D}_{\theta}$ is determined by

(8) $\displaystyle K_{\epsilon_{s}}\cdot
x_{ij}=q^{\delta_{is}+\delta_{js}}x_{ij}\quad\quad K_{\epsilon_{s}}\cdot
d_{ij}=q^{-\delta_{is}-\delta_{js}}d_{ij}$

for all $i,j,s$.

Note that the isomorphisms $\mathscr{P}_{\theta}\cong\mathcal{O}_{q}({\rm
Mat}_{n})$ and $\mathscr{D}_{\theta}\cong\mathcal{O}_{q}({\rm Mat}_{n})^{op}$
in the diagonal setting as described above are actually
$U_{q}(\mathfrak{gl}_{n})$-bimodule isomorphisms where the left action of
$U_{q}(\mathfrak{gl}_{n})$ on $\mathcal{O}_{q}({\rm Mat}_{n})$ (resp.
$\mathcal{O}_{q}({\rm Mat}_{n})^{op}$) is the same as the action of the first
copy of $U_{q}(\mathfrak{gl}_{n})$ inside
$U_{q}(\mathfrak{gl}_{n}\oplus\mathfrak{gl}_{n})$ on $\mathscr{P}_{\theta}$
(resp. $\mathscr{D}_{\theta}$). Similarly, the right action for
$\mathscr{P}_{\theta}$ (resp. $\mathscr{D}_{\theta})$ goes over to the action
of the second copy of $U_{q}(\mathfrak{gl}_{n})$ on $\mathcal{O}_{q}({\rm
Mat}_{n})$ (resp. $\mathcal{O}_{q}({\rm Mat}_{n})^{op}$). (See [19], Lemma
5.12 for details.)

It is well-known that the algebra $\mathcal{O}_{q}({\rm Mat}_{N})$ admits a
PBW basis using monomials in the $t_{ij}$. The next result from [19] shows
that the same is true for $\mathscr{P}_{\theta}$. Using the fact that
$x_{ij}\mapsto d_{ij}$ defines an antiautomorphism from $\mathscr{P}_{\theta}$
to $\mathscr{D}_{\theta}$, the next lemma also holds for
$\mathscr{D}_{\theta}$ with each $x_{ij}$ term replaced by $d_{ij}$.

######  Lemma 3.3.

The following monomials form a basis for $\mathscr{P}_{\theta}$:

  * (i)

Type AI:

$x_{11}^{m_{11}}x_{12}^{m_{12}}\cdots
x_{1n}^{m_{1n}}x_{22}^{m_{22}}x_{23}^{m_{23}}\cdots x_{2n}^{m_{2n}}\cdots
x_{n-1,n-1}^{m_{n-1,n-1}}x_{n-1,n}^{m_{n-1,n}}x_{nn}^{m_{nn}}$

  * (ii)

Type AII:

$x_{12}^{m_{12}}x_{13}^{m_{13}}\cdots
x_{1,2n}^{m_{1,2n}}x_{23}^{m_{23}}x_{24}^{m_{24}}\cdots
x_{2,2n}^{m_{2,2n}}\cdots x_{2n-1,2n}^{m_{2n-1,2n}}$

  * (iii)

Diagonal type:

$x_{1,n+1}^{m_{11}}x_{12}^{m_{12}}\cdots
x_{1,2n}^{m_{1n}}x_{2,n+1}^{m_{21}}x_{2,n+2}^{m_{22}}\cdots
x_{2,2n}^{m_{2n}}\cdots x_{n,n+1}^{m_{n1}}\cdots x_{n,2n}^{m_{nn}}$

as each $m_{ij}$ runs over nonnegative integers. Moreover, we also get a basis
if the order of the monomials in the terms above are reversed.

###  3.3. Module structure

It is well-known that $\mathcal{O}_{q}({\rm Mat}_{N})$ admits a decomposition
as $U_{q}(\mathfrak{gl}_{N})$-bimodules

(9) $\displaystyle\mathcal{O}_{q}({\rm
Mat}_{N})\cong\bigoplus_{\lambda\in\Lambda^{+}_{N}}L(\lambda)\otimes
L(\lambda)^{*}$

where $L(\lambda)$ is a left $U_{q}(\mathfrak{gl}_{N})$-module and
$L(\lambda)^{*}$ is a right $U_{q}(\mathfrak{gl}_{N})$-module. By restriction,
we can also view all these modules as $U_{q}(\mathfrak{sl}_{N})$-modules. Note
that $\mathcal{O}_{q}({\rm Mat}_{N})$ contains a nontrivial bi-invariant
submodule with respect to the action of $U_{q}(\mathfrak{sl}_{N})$. In
particular, the submodule of $\mathcal{O}_{q}({\rm Mat}_{N})$ consisting of
$U_{q}(\mathfrak{sl}_{N})$-bi-invariants corresponds to

$\bigoplus_{m\in\mathbb{N}}L(m\hat{\omega}_{N})\otimes
L(m\hat{\omega}_{N})^{*}.$

Decomposition (9) implies

(10) $\displaystyle\mathcal{O}_{q}({\rm Mat}_{N})\otimes\mathcal{O}_{q}({\rm
Mat}_{N})\cong\bigoplus_{(\lambda,\lambda^{\prime})\in\Lambda^{+}_{N}\times\Lambda^{+}_{N}}L(\lambda\oplus\lambda^{\prime})\otimes
L(\lambda\oplus\lambda^{\prime})^{*}$

as $U_{q}(\mathfrak{gl}_{N}\oplus\mathfrak{gl}_{N})$-bimodules. Moreover, the
submodule of $\mathcal{O}_{q}({\rm Mat}_{N})\otimes\mathcal{O}_{q}({\rm
Mat}_{N})$ consisting of $U_{q}(\mathfrak{sl}_{N}\oplus\mathfrak{sl}_{N})$-bi-
invariants is

$\bigoplus_{m,m^{\prime}\in\mathbb{N}}L(m\hat{\omega}_{N}\oplus
m^{\prime}\hat{\omega}_{N})\otimes L(m\hat{\omega}_{N}\oplus
m^{\prime}\hat{\omega}_{N})^{*}.$

Setting $N=n$ in Type AI and the diagonal case and $N=2n$ in Type AII, we can
read off of (9) and (10) the $U_{q}(\mathfrak{g})$-module decomposition of
$\mathscr{P}$. It follows from the description of spherical weights in Section
2.5 that the right $\mathcal{B}_{\theta}$ invariants
$\mathscr{P}^{\mathcal{B}_{\theta}}$ of $\mathscr{P}$ admits the following
decomposition as left $U_{q}(\mathfrak{g})$-modules

$\displaystyle\bigoplus_{\lambda\in\Lambda^{+}_{\Sigma},s\in\mathbb{N}}$
$\displaystyle L(2\lambda+s\hat{\omega}_{n}){\rm\ in\ Type\ AI,}$
$\displaystyle\bigoplus_{\lambda\in\Lambda^{+}_{\Sigma}}$ $\displaystyle
L(2\lambda){\rm\ in\ Type\ AII,\ and}$
$\displaystyle\bigoplus_{\lambda\in\Lambda^{+}_{\Sigma}}$ $\displaystyle
L(2\lambda)=\bigoplus_{\gamma\in\Lambda^{+}_{n}}L(\gamma\oplus\gamma){\rm\ in\
the\ diagonal\ type.\ }$

By [31] Lemma 5.3, the fact that $\mathscr{P}_{\theta}$ is generated by right
$\mathcal{B}_{\theta}$ invariant elements ensures that $\mathscr{P}_{\theta}$
is a subalgebra and submodule of $\mathscr{P}^{\mathcal{B}_{\theta}}$. We see
in Section 4 that $\mathscr{P}^{\mathcal{B}_{\theta}}$ agrees with
$\mathscr{P}_{\theta}$ in both Type AII and the diagonal case while it is
slightly larger in Type AI.

##  4\. Detailed module decompositions

###  4.1. Quantum determinants

There is a quantum analog of the determinant, denoted by $\det_{q}(T)$, which
is a central element in $\mathcal{O}_{q}({\rm Mat}_{N})$. The quantum
determinant can be expressed explicitly in terms of the $t_{ij}$ as

(11)
$\displaystyle{\det}_{q}(T)=\sum_{s\in\mathcal{S}_{n}}(-q)^{l(s)}t_{s(1),1}\cdots
t_{s(N),N}$

and satisfies $\iota(\det_{q}(T))=\det_{q}(T)$ where $\iota$ is the
antiautomorphism defined in Section 3.1 (see (1.26) of [21]). It is
straightforward to check that the quantum determinant $\det_{q}(T)$ satisfies
$\det_{q}(T)\cdot K_{i}=\det_{q}(T)$, $\det_{q}(T)\cdot F_{i}=0$ and
$\det_{q}(T)\cdot E_{i}=0$ for $i=1,\dots,N-1$. Hence $\det_{q}(T)$ is right
invariant with respect to the action of $U_{q}(\mathfrak{sl}_{N})$. The same
is true with respect to the left action of $U_{q}(\mathfrak{sl}_{N})$ and can
be easily verified with a similar computation using the fact that
$\iota(\det_{q}(T))=\det_{q}(T)$. Hence $\det_{q}(T)$ is both a right and left
invariant element with respect to the action of $U_{q}(\mathfrak{sl}_{N})$.
However, the same is not true upon passing to $U_{q}(\mathfrak{gl}_{N})$.
Indeed, we have

(12) $\displaystyle{\det}_{q}(T)\cdot
K_{\epsilon_{i}}=K_{\epsilon_{i}}\cdot{\det}_{q}(T)=q{\det}_{q}(T).$

Let $\det_{q}(T^{\prime})$ be the quantum determinant defined using the
elements $t_{i+N,j+N}$ instead of the $t_{ij}$ viewed as elements of
$\mathcal{O}_{q}({\rm Mat}_{n})\otimes\mathcal{O}_{q}({\rm Mat}_{n})$. The
properties for $\det_{q}(T)$ carry over to $\det_{q}(T^{\prime})$ where each
subscript $i$ is replaced by $i+N$.

Note that the weight of $\det_{q}(T)$ with respect to the action of
$U_{q}(\mathfrak{gl}_{N})$ is
$\hat{\omega}_{N}=\epsilon_{1}+\cdots+\epsilon_{N}$. Hence by (9), the
submodule of $U_{q}(\mathfrak{sl}_{N})$-bi-invariants inside
$\mathcal{O}_{q}({\rm Mat}_{N})$ is the polynomial ring
$\mathbb{C}(q)[\det_{q}(T)]$. Similarly, the submodule of
$\mathcal{O}_{q}({\rm Mat}_{N})\otimes\mathcal{O}_{q}({\rm Mat}_{N})$
consisting of $U_{q}(\mathfrak{sl}_{N}\oplus\mathfrak{sl}_{N})$-bi-invariants
is the polynomial ring $\mathbb{C}(q)[\det_{q}(T),\det_{q}(T^{\prime})]$.

######  Lemma 4.1.

The intersection of $\mathbb{C}(q)[\det_{q}(T)]$ and $\mathscr{P}_{\theta}$ is
$\mathbb{C}(q)[(\det_{q}(T))^{2}]$ in Type AI and equals
$\mathbb{C}(q)[\det_{q}(T)]$ in type AII. The intersection of
$\mathbb{C}(q)[\det_{q}(T),\det_{q}(T^{\prime})]$ and $\mathscr{P}_{\theta}$
is $\mathbb{C}(q)[\det_{q}(T)\det_{q}(T^{\prime})]$ in the diagonal case.

###### Proof.

Consider Type AI. Note that any element in $\mathscr{P}_{\theta}$ can be
written as a linear combination of monomials, say $t_{i_{1},j_{1}}\cdots
t_{i_{m},j_{m}}$. Moreover, by the form of the elements $x_{ab}$, each right
index $j_{k}$ must appear an even number of times in a particular monomial.
Thus, examining $\det_{q}(T)$, we see that
$\det_{q}(T)\notin\mathscr{P}_{\theta}$ since each right index $j_{k}$ shows
up exactly once in $t_{s(1),1}\cdots t_{s(n),n}$. The same holds for
$(\det_{q}(T))^{m}$ for $m$ odd. On other hand, by Remark 4.12 of [21],
$(\det_{q}(T))^{2}\in\mathscr{P}_{\theta}$. Hence the first assertion for Type
AI holds. For Type AII, note that Remark 4.12 of [21] also shows that
$\det_{q}(T)\in\mathscr{P}_{\theta}$ in this case. This completes the proof of
the first assertion of the lemma.

For the diagonal case, note that $\det_{q}(T)\det_{q}(T^{\prime})$ is right
invariant with respect to action of
$U_{q}(\mathfrak{sl}_{n}\oplus\mathfrak{sl}_{n})$ as well as with respect to
the action of $K_{\epsilon_{i}}K_{\epsilon_{n+i}}^{-1}$ for $i=1,\dots,n$.
Since $\mathcal{B}_{\theta}$ is a subalgebra of the algebra generated by both
$U_{q}(\mathfrak{sl}_{n}\oplus\mathfrak{sl}_{n})$ and the
$K_{\epsilon_{i}}K_{\epsilon_{n+i}}^{-1}$ for $i=1,\dots,n$, it follows that
$\det_{q}(T)\det_{q}(T^{\prime})$ is an element of
$\mathscr{P}^{\mathcal{B}_{\theta}}$. Now $\det_{q}(T)\det_{q}(T^{\prime})$
has weight $\hat{\omega}_{n}\oplus\hat{\omega}_{n}$ with respect to the left
action of $U_{q}(\mathfrak{gl}_{n}\oplus\mathfrak{gl}_{n})$ and, moreover,
generates a trivial left
$U_{q}(\mathfrak{sl}_{n}\oplus\mathfrak{sl}_{n})$-module. Hence from the
description of $\mathscr{P}^{\mathcal{B}_{\theta}}$ in the diagonal case given
in Section 3.3, we see that $(\det_{q}(T)\det_{q}(T^{\prime}))^{m}$ is a basis
vector for the one dimensional left module $L(m\hat{\omega}_{n}\oplus
m\hat{\omega}_{n})$. Thus
$\mathscr{P}^{\mathcal{B}_{\theta}}\cap\mathbb{C}(q)[\det_{q}(T),\det_{q}(T^{\prime})]=\mathbb{C}(q)[\det_{q}(T)\det_{q}(T^{\prime})]$.
Since $\mathscr{P}_{\theta}\subseteq\mathscr{P}^{\mathcal{B}_{\theta}}$, we
also have that the intersection
$\mathscr{P}_{\theta}\cap\mathbb{C}(q)[\det_{q}(T),\det_{q}(T^{\prime})]$ is
subset of $\mathbb{C}(q)[\det_{q}(T)\det_{q}(T^{\prime})]$.

Now consider the element $\det_{q}(X)$ defined by replacing each $t_{ij}$ in
the definition of $\det_{q}(T)$ by $x_{ij}$, again in the diagonal case.
Recall that $\mathscr{P}_{\theta}$ is isomorphic as an algebra and
$U_{q}(\mathfrak{gl}_{n})$-bimodule to $\mathcal{O}_{q}({\rm Mat}_{N})$ (see
the discussions following Theorems 3.1 and 3.2). It follows that $\det_{q}(X)$
is an element of $\mathscr{P}^{\mathcal{B}_{\theta}}$ invariant with respect
to the left action of $U_{q}(\mathfrak{sl}_{n}\oplus\mathfrak{sl}_{n})$.
Moreover, it is straightforward to see that the weight of $\det_{q}(X)$ is
$\hat{\omega}_{n}\oplus\hat{\omega}_{n}$. By the previous paragraph,
$\det_{q}(X)$ must be a nonzero scalar multiple of
$\det_{q}(T)\det_{q}(T^{\prime})$. This guarantees the inclusion
$\mathbb{C}(q)[\det_{q}(T)\det_{q}(T^{\prime})]\subseteq\mathscr{P}_{\theta}$
which combined with the previous paragraph yields the desired equality. ∎

Set $H_{n}=\det_{q}(T)\det_{q}(T^{\prime})$ in the diagonal case,
$H_{n}=\det_{q}(T)^{2}$ in Type AI, and $H_{n}=\det_{q}(T)$ in Type AII. By
Lemma 4.1, $H_{n}\in\mathscr{P}_{\theta}$. Moreover, since $\det_{q}(T)$ is a
central element in $\mathcal{O}_{q}({\rm Mat}_{N})$, we must have $H_{n}$ is
central in $\mathscr{P}_{\theta}$.

###  4.2. Chains of algebras

Consider the chain of quantized enveloping algebras

$\displaystyle U_{q}(\mathfrak{g}_{1})\subset
U_{q}(\mathfrak{g}_{2})\subset\cdots\subset
U_{q}(\mathfrak{g}_{n})=U_{q}(\mathfrak{g})$

where $\mathfrak{g}_{r}=\mathfrak{gl}_{r}$ in Type AI,
$\mathfrak{g}_{r}=\mathfrak{gl}_{2r}$ in Type AII,
$\mathfrak{g}_{r}=\mathfrak{gl}_{r}\oplus\mathfrak{gl}_{r}$ in the diagonal
case and $U_{q}(\mathfrak{g}_{r})$ is identified with the subalgebra of
$U_{q}(\mathfrak{g})$ generated by $E_{i},F_{i},K_{\epsilon_{j}}$ where

  * •

$i\in\\{1,\dots,r-1\\}$ and $j\in\\{1,\dots,r\\}$ in Type AI

  * •

$i\in\\{1,\dots,2r-1\\}$ and $j\in\\{1,\dots,2r\\}$ in Type AII

  * •

$i\in\\{1,\dots,r-1\\}\cup\\{n+1,\dots,n+r-1\\}$,
$j\in\\{1,\dots,r\\}\cup\\{n+1,\dots,n+r\\}$ in the diagonal case.

Note that $U_{q}(\mathfrak{g}_{1})$ is just a commutative Laurent polynomial
ring over $\mathbb{C}(q)$ in Types AI and the diagonal case. In Types AI, this
Laurent polynomial ring is generated by $K_{\epsilon_{1}}$and in the diagonal
case, it is generated by $K_{\epsilon_{1}}$ and $K_{\epsilon_{n+1}}$. In Type
AII, $U_{q}(\mathfrak{g}_{1})$ is the copy of the quantized enveloping algebra
of $\mathfrak{gl}_{2}$ generated by $E_{1},F_{1},$ and $K_{\epsilon_{1}}^{\pm
1},K_{\epsilon_{2}}^{\pm 1}$.

Similarly, we have a chain of quantized function algebras

$\displaystyle\mathscr{P}(\mathfrak{g}_{1})\subset\mathscr{P}(\mathfrak{g}_{2})\subset\cdots\subset\mathscr{P}(\mathfrak{g}_{n})=\mathscr{P}$

where $\mathscr{P}(\mathfrak{g}_{r})\cong\mathcal{O}_{q}({\rm Mat}_{r})$ in
Type AI, $\mathscr{P}(\mathfrak{g}_{r})\cong\mathcal{O}_{q}({\rm Mat}_{2r})$
in Type AII, and $\mathscr{P}(\mathfrak{g}_{r})\cong\mathcal{O}_{q}({\rm
Mat}_{r})\otimes\mathcal{O}_{q}({\rm Mat}_{r})$ in the diagonal case.
Moreover, this isomorphism is an equality for $r=n$ and so
$\mathscr{P}(\mathfrak{g}_{n})=\mathscr{P}$. For $r<n$,
$\mathscr{P}(\mathfrak{g}_{r})$ is identified with the subalgebra of
$\mathscr{P}$ generated by

  * •

$t_{ij}$ for $1\leq i,j\leq r$ in Type AI

  * •

$t_{ij}$ for $1\leq i,j\leq 2r$ in Type AII

  * •

$t_{ij}$ for $1\leq i,j\leq r$ and $n+1\leq i,j\leq n+r$ in the diagonal case.

Note that $\mathscr{P}(\mathfrak{g}_{r})$ is a
$U_{q}(\mathfrak{g}_{r})$-bimodule and, moreover, this bimodule structure is
compatible with the $U_{q}(\mathfrak{g})$-bimodule structure on $\mathscr{P}$.

Set $\mathcal{B}_{\theta}^{r}=\mathcal{B}_{\theta}\cap
U_{q}(\mathfrak{g}_{r})$ for $r=1,\dots,n$. Note that this gives us a chain of
subalgebras

$\displaystyle\mathcal{B}_{\theta}^{1}\subset\mathcal{B}_{\theta}^{2}\subset\cdots\subset\mathcal{B}_{\theta}^{n}=\mathcal{B}_{\theta}.$

For each $r$, it is straightforward to see that $\theta$ restricts to an
involution on $\mathfrak{g}_{r}$ with fixed Lie subalgebra
$\mathfrak{k}_{r}=\mathfrak{g}_{r}^{\theta}$. We write
$\mathcal{B}^{r}_{\theta}$ for the right coideal subalgebra of
$U_{q}(\mathfrak{g}_{r})$ that is a quantum analog of $U(\mathfrak{k}_{r})$.
Note that $\mathcal{B}_{\theta}^{r}$ belongs to the same family as
$\mathcal{B}_{\theta}$ with the only difference being the rank of the
underlying Lie algebra $\mathfrak{g}_{r}$.

Using $\mathcal{B}_{\theta}^{r},\mathscr{P}(\mathfrak{g}_{r})$, and
$U_{q}(\mathfrak{g}_{r})$, one can define the quantized function algebra
$\mathscr{P}_{\theta}(\mathfrak{g}_{r})$ generated by elements $x(r)_{ij}$
where $1\leq i,j\leq r$ in Type AI, $1\leq i,j\leq 2r$ in Type AII, and $1\leq
i,j-n\leq r$ or $1\leq i-n,j\leq r$ in the diagonal case. Here, we use the
notation $x_{ij}$ for the generators when $r=n$ (i.e. $x(n)_{ij}=x_{ij}$).
Note that the difference between $x(r)_{ij}$ and $x_{ij}$ has to do with which
$t_{kl}$ are involved in the expression of these elements in terms of elements
of $\mathscr{P}$. For example, in Type AI,

$\displaystyle x(r)_{ij}=\sum_{k=1}^{r}t_{ik}t_{jk}$

while

$\displaystyle x_{ij}=\sum_{k=1}^{n}t_{ik}t_{jk}.$

Nevertheless, we see from the next lemma that this distinction is not
important.

######  Lemma 4.2.

For each $r,s$ with $1\leq r<s\leq n$, the map
$\psi_{r,s}:\mathscr{P}_{\theta}(\mathfrak{g}_{r})\rightarrow\mathscr{P}_{\theta}(\mathfrak{g}_{s})$
defined by $\psi_{r,s}(x(r)_{ij})=x(s)_{ij}$ induces an algebra embedding
which preserves the left $U_{q}(\mathfrak{g}_{r})$-module and (trivial) right
$\mathcal{B}_{\theta}^{r}$-module structures.

###### Proof.

Note that the relations for both algebras are given by Theorem 3.1. Moreover,
there are two types of relations: linear and quadratic. The linear relations
take the form $x_{ab}=0$ for various conditions on $a,b$ and $x_{ij}=\gamma
x_{ji}$ for an appropriate scalar $\gamma$ and all $i,j$. Clearly these agree
for the two algebras and so $x(r)_{ab}=0$ if and only if $x(s)_{ab}=0$ and
$x_{ij}(r)=\gamma x_{ji}(r)$ if and only if $x_{ij}(s)=\gamma x_{ji}(s).$

By (7), the quadratic relations correspond to $q$-analogs of commutativity
relations between two generators, say $x_{ij}$ and $x_{ld}$. Moreover, the
only terms showing up in these relations are of the form $x_{ab}x_{cd}$ where
$\\{a,b,c,d\\}=\\{i,j,l,d\\}$. Now if $i,j$ and $l,d$ satisfy the necessary
conditions for $x(r)_{ij}$ and $x(r)_{ld}$ to be generators of
$\mathscr{P}_{\theta}(\mathfrak{g}_{r})$, then $x(r)_{ab}$ is also a valid
generator whenever $\\{a,b\\}\in\\{i,j,l,d\\}$. In other words, when $i,j,l,d$
are chosen so that $x(r)_{ij},x(r)_{ld}$ are among the generators for
$\mathscr{P}_{\theta}(\mathfrak{g}_{r})$, then the quadratic relations
involving $x(r)_{ij}$ and $x(r)_{ld}$ inside
$\mathscr{P}_{\theta}(\mathfrak{g}_{r})$ take exactly the same form as the
relations satisfied by $x_{ij}$ and $x_{ld}$ inside of $\mathscr{P}_{\theta}$.
The same holds with $r$ replaced by $s$. Thus, the quadratic relations
satisfied by the $x(r)_{ij}$ of $\mathscr{P}_{\theta}(\mathfrak{g}_{r})$ agree
with the relations coming from the larger algebra
$\mathscr{P}_{\theta}(\mathfrak{g}_{s})$ for the corresponding elements
$x(s)_{ij}$.

The module structures for both algebras can be deduced directly from the
actions of $U_{q}(\mathfrak{g})$ and $\mathcal{B}_{\theta}$ on $\mathscr{P}$
and these actions do not depend on $r$ or $s$, Thus, these module actions
agree which yields the desired module isomorphisms. ∎

An immediate consequence of Lemma 4.2 is that
$\mathscr{P}_{\theta}(\mathfrak{g}_{r})$ is isomorphic to a subalgebra and
left $U_{q}(\mathfrak{g}_{r})$-submodule of $\mathscr{P}_{\theta}$ where each
generator $x(r)_{ij}$ is mapped to $x_{ij}$. Moreover, it is straightforward
to see that the embeddings of this lemma are all compatible with each other
and so $\psi_{r,s}\circ\psi_{s,m}=\psi_{r,m}$ for all $1\leq r<s<m\leq n$.

###  4.3. Highest weight generators

Let $T_{(r)}$ be the submatrix of $T$ with entries $t_{ij}$ where $1\leq
i,j\leq r$ in Type AI and Type AII. Similarly, let $T^{\prime}_{(r)}$ be the
submatrix of $T^{\prime}$ with entries $t_{n+i,n+j}$ where again $1\leq
i,j\leq r$. Set

  * •

$\hat{H}_{r}=(\det_{q}T_{(r)})^{2}$ for Type AI

  * •

$\hat{H}_{r}=\det_{q}T_{({2r})}$ in Type AII and

  * •

$\hat{H}_{r}=(\det_{q}T_{(r)})(\det_{q}T^{\prime}_{(r)})$ in the diagonal
case.

Note that $\hat{H}_{r}\in\mathscr{P}(\mathfrak{g}_{r})$. Moreover, by Lemma
4.1, we have $\hat{H}_{r}\in\mathscr{P}_{\theta}(\mathfrak{g}_{r})$. Set
$H_{r}=\psi_{r,n}(\hat{H}_{r})$ for $r=1,\dots,n$.

Note that for each $r$, $\mathscr{P}_{\theta}(\mathfrak{g}_{r})$ has a natural
degree function compatible with the degree function on $\mathscr{P}_{\theta}$.
In particular, we have $\deg x(r)_{ij}=1$ all $r,i,j$.

######  Proposition 4.3.

The elements $H_{1},\dots,H_{n}$ generate a commutative subring of
$\mathscr{P}_{\theta}$ that is isomorphic to a polynomial ring in these $n$
generators. Moreover, $H_{r}$ is a highest weight vector of weight
$2\hat{\eta}_{r}$ with respect to the left action of $U_{q}(\mathfrak{g})$ and
$H_{r}\in\mathcal{J}_{r}(\mathscr{P}_{\theta})$.

###### Proof.

Since $\det_{q}(T_{(r)})$ is a central element of $\mathcal{O}_{q}({\rm
Mat}_{r})$, it follows that $\hat{H}_{r}$ is in the center of
$\mathscr{P}(\mathfrak{g}_{r})$. Hence $\hat{H}_{r}$ is in the center of
$\mathscr{P}_{\theta}(\mathfrak{g}_{r})$. It follows that $\hat{H}_{r}$
commutes with each $\psi_{m,r}(\hat{H}_{m})$ for $m\leq r$. By induction, we
get $\psi_{1,r}(\hat{H}_{r}),\dots,\psi_{r-1,r}(\hat{H}_{r}),\hat{H}_{r}$
generates a commutative subring of $\mathscr{P}_{\theta}(\mathfrak{g}_{r})$.
When $r=n$, this sequence is simply $H_{1},\dots,H_{n}$ and so these elements
generate a commutative subring of $\mathscr{P}_{\theta}$.

Note that for each of the three families, $H_{r}$ has degree $2r$ viewed as a
(noncommutative) polynomial in the $t_{ij}$. Since the $x_{ij}(r)$ are
quadratic in the $t_{ij}$ terms, it follows that $\hat{H}_{r}$ has degree $r$
viewed as a (noncommutative) polynomial in the $x_{ij}(r)$. Hence $H_{r}$ has
degree $r$ as a polynomial in the $x_{ij}$ and so
$H_{r}\in\mathcal{J}_{r}(\mathscr{P}_{\theta})$.

The fact that $H_{r}$ is left $U_{q}(\mathfrak{g}_{r})$ invariant follows from
the definition of $\hat{H}_{r}$ and properties of quantum determinants (see
Section 4.1). Since $\hat{H}_{r}$ only contains terms $t_{kl}$ with $1\leq
k,l\leq r$, it follows that $H_{r}$ only contains terms $x_{ij}$ for $1\leq
i,j\leq r$. The formulas for the left action of $U_{q}(\mathfrak{gl}_{N})$ on
$\mathcal{O}_{q}({\rm Mat}_{N})$ (in Section 3.1) ensure that $E_{i}\cdot
t_{jk}=0$ for all $i$ with $i>j$. Hence, $E_{s}\cdot a=0$ for all
$a\in\mathscr{P}(\mathfrak{g}_{r})$ where $s>r$ in Type AI, $s>2r$, and
$r<s\leq n,n+r<s\leq 2n$ in the diagonal case. This guarantees that
$E_{s}\cdot H_{r}=0$ for all $E_{s}\notin U_{q}(\mathfrak{g}_{r})$. But we
also know that $E_{s}\cdot\hat{H}_{r}=0$ for all $E_{s}\in
U_{q}(\mathfrak{g}_{r})$ because of the left $U_{q}(\mathfrak{g}_{r})$
invariant property of $H_{r}$. This proves that $E_{s}\cdot H_{r}=0$ for all
$E_{s}\in U_{q}(\mathfrak{g})$.

Note that $\det_{q}(T_{(r)})$ has weight $\epsilon_{1}+\cdots+\epsilon_{r}$ in
terms of the left action of $U_{q}(\mathfrak{gl}_{r})$. Thus, it is
straightforward to see from the definitions of $\hat{H}_{r}$ and of the
restricted weight $\hat{\eta}_{r}$ (see Section 2.5) that $\hat{H}_{r}$ has
weight $2\hat{\eta}_{r}$ for each $r$. The weight of $H_{r}$ is the same as
that of $\hat{H}_{r}$, hence by the previous paragraph, $H_{r}$ is a highest
weight vector of weight $2\hat{\eta}_{r}$ with respect to the left action of
$U_{q}(\mathfrak{g})$.

We finish the proof by arguing that the $H_{1},\dots,H_{n}$ are algebraically
independent and hence the ring they generate is a polynomial ring in these $n$
generators. Suppose

$\sum_{m}a_{m}H_{1}^{m_{1}}\cdots H_{n}^{m_{n}}=0$

where $m=(m_{1},\dots,m_{n})$. Since the monomials $H_{1}^{m_{1}}\cdots
H_{n}^{m_{n}}$ have distinct weights $\sum_{i}2m_{i}\hat{\eta}_{i}$, we can
separate the monomials using the action of $U_{q}(\mathfrak{g})$. Thus the
above equality implies

$a_{m}H_{1}^{m_{1}}\cdots H_{n}^{m_{n}}=0$

and hence $a_{m}=0$ each $m$. ∎

We frequently write $H_{2\mu}$ for the element $H_{1}^{m_{1}}\cdots
H_{n}^{m_{n}}$ for each $\mu=\sum_{i}m_{i}\hat{\eta}_{i}$, thus labeling this
element by its weight. In particular, by the above proposition, $H_{2\mu}$ is
a highest weight vector of weight $2\mu$ with respect to the action of
$U_{q}(\mathfrak{g})$ on $\mathscr{P}_{\theta}$.

###  4.4. Explicit module descriptions

The decompositions of Section 3.3 combined with Lemma 4.1 ensure that

(13)
$\displaystyle\mathscr{P}_{\theta}\cong\bigoplus_{\lambda\in\Lambda^{+}_{\Sigma}}L(2\lambda)$

as left $U_{q}(\mathfrak{g})$-modules and trivial right
$\mathcal{B}_{\theta}$-modules. (Note that this only differs in Type AI from
the decompositions in Section 3.3.)

######  Theorem 4.4.

We have

$\mathscr{P}_{\theta}=\bigoplus_{\lambda\in\Lambda^{+}_{\Sigma}}(U_{q}(\mathfrak{g}))\cdot
H_{2\lambda}$

where $(U_{q}(\mathfrak{g}))\cdot H_{2\lambda}$ is isomorphic to the simple
(left) $U_{q}(\mathfrak{g})$-module generated by the highest weight vector
$H_{2\lambda}$ with weight $2\lambda$ and is a trivial right
$\mathcal{B}_{\theta}$-module.

###### Proof.

We see from (13) that $\mathscr{P}_{\theta}$ is the direct sum of finite-
dimensional simple modules with respect to the left action of
$U_{q}(\mathfrak{g})$. Moreover, the decomposition is multiplicity free with
exactly one finite-dimensional simple module of the form $L(2\lambda)$ for
each $\lambda\in\Lambda^{+}_{\Sigma}$. By Proposition 4.3, $H_{2\lambda}$
generates a finite-dimensional simple (left) $U_{q}(\mathfrak{g})$-module with
highest weight $2\lambda$ for each $\lambda\in\Lambda^{+}_{\Sigma}$. Thus the
module $L(2\lambda)$ in the decomposition (13) must correspond to
$(U_{q}(\mathfrak{g}))\cdot H_{2\lambda}$. ∎

The entire construction of this section can be transferred to the differential
parts $\mathscr{D}_{\theta}$ by using the antiautomorphisms sending
$\mathscr{P}$ to $\mathscr{D}=\mathscr{P}^{op}$. Let $H^{*}_{2\mu}$ be the
image of $H_{2\mu}$ via this antiautomorphism. A comparison of the action of
$U_{q}(\mathfrak{g})$ on $\mathscr{P}$ and on $\mathscr{D}$ yields that
$H^{*}_{2\lambda}$ is a lowest weight vector of weight $-2\lambda$. Hence, we
have a similar decomposition as above for $\mathscr{D}_{\theta}$, again as
left $U_{q}(\mathfrak{g})$-modules and trivial right
$\mathcal{B}_{\theta}$-modules

$\displaystyle\mathscr{D}_{\theta}\cong\oplus_{\lambda\in\Lambda^{+}_{\Sigma}}(U_{q}(\mathfrak{g}))\cdot
H^{*}_{2\lambda}.$

Note that $(U_{q}(\mathfrak{g}))\cdot H^{*}_{2\lambda}$ can be viewed as the
left dual of $(U_{q}(\mathfrak{g}))\cdot H_{2\lambda}$.

######  Remark 4.5.

The generators of each module $L(2\lambda)$ is expressed using formulas in
terms of the $t_{ij}$ for Types AI and AII in [21] (see Lemma 4.10.A), Our
approach yields another concrete identification of these generators that also
applies to the diagonal case. Our methods, which rely directly on quantum
determinants also lead to formulas in the $t_{ij}$. However, because these
highest weight terms are elements of $\mathscr{P}_{\theta}(\mathfrak{g}_{r})$
for various choices of $r$, it is easier to read off the possible $x_{ij}$
that may appear. This will be helpful in describing and analyzing the quantum
Capelli operators in Section 9.1 of this paper.

##  5\. Quantum Weyl Algebras

###  5.1. Generators and relations

We associate a quantum Weyl algebra $\mathscr{PD}_{q}({\rm Mat}_{N})$ with
polynomials corresponding to $\mathcal{O}_{q}({\rm Mat}_{N})$ and constant
term differentials corresponding to $\mathcal{O}_{q}({\rm Mat}_{N})^{op}$ as
defined and studied in [31], [3], [2], and [19]. This Weyl algebra
$\mathscr{PD}_{q}({\rm Mat}_{N})$ is generated by $t_{ij}$ and $\partial_{ij}$
for $1\leq i,j\leq N$. The algebra $\mathcal{O}_{q}({\rm Mat}_{N})$ (resp.
$\mathcal{O}_{q}({\rm Mat}_{N})^{op}$) embeds inside $\mathscr{PD}_{q}({\rm
Mat}_{N})$ and can be identified with the subalgebra generated by the $t_{ij}$
(resp. $\partial_{ij}$). Moreover, the $t_{ij}$ and $\partial_{ij}$ satisfy
the following relation

$\displaystyle\partial_{ab}t_{ef}=\sum_{r,l,j,k}(R^{t_{2}})^{rl}_{ea}(R^{t_{2}})^{jk}_{fb}t_{rj}\partial_{lk}+\delta_{ae}\delta_{bf}$

for all $a,b,e,f$ in $\\{1,\dots,N\\}$. The quantum Weyl algebra
$\mathscr{PD}_{q}({\rm Mat}_{N})$ inherits the structure of a
$U_{q}(\mathfrak{gl}_{N})$-bimodule from the bimodules $\mathcal{O}_{q}({\rm
Mat}_{N})$ and $\mathcal{O}_{q}({\rm Mat}_{N})^{op}.$

We may view $\mathscr{P}_{\theta}$ and $\mathscr{D}_{\theta}$ as right
$\mathscr{B}_{\theta}$ invariant subalgebras of $\mathscr{PD}_{q}({\rm
Mat}_{N})$. However, together, they generate an algebra inside of
$\mathscr{PD}_{q}({\rm Mat}_{N})$ that is too large to be taken for a quantum
analog of the Weyl algebra with polynomial part equal to
$\mathscr{P}_{\theta}$. In particular, the subset
$\mathscr{P}_{\theta}\mathscr{D}_{\theta}$ consisting of sums of terms of the
form $pd$ with $p\in\mathscr{P}_{\theta}$ and $d\in\mathscr{D}_{\theta}$ is
strictly smaller than the subalgebra generated by $\mathscr{P}_{\theta}$ and
$\mathscr{D}_{\theta}$ (see [19] for more details). Instead, we use the
construction of [19] which starts with a twisted tensor product of
$\mathscr{P}_{\theta}$ and $\mathscr{D}_{\theta}$ and deforms it so as to add
constant terms to some of the relations. The result is the quantum Weyl
algebra $\mathscr{PD}_{\theta}$ associated to $\theta$ for each of the three
settings of this paper. As an algebra, $\mathscr{PD}_{\theta}$ is generated by
$x_{ij}$, $d_{ij}$, $1\leq i,j\leq N$ where $N=n$ in Type AI and $N=2n$ in
Type AII and the diagonal case. The algebra $\mathscr{P}_{\theta}$ (resp.
$\mathscr{D}_{\theta}$) embeds inside $\mathscr{PD}_{\theta}$ and can be
identified with the subalgebra generated by the $x_{ij}$ (resp. $d_{ij}$).
Moreover, the $x_{ij}$ and $d_{ij}$ satisfy the following relation

(14) $\displaystyle
d_{ab}x_{ef}=\sum_{w,r,x,q,p,m,y,l}(R_{\mathfrak{g}}^{t_{2}})^{wr}_{xq}(R_{\mathfrak{g}}^{t_{2}})^{pq}_{ma}(R_{\mathfrak{g}}^{t_{2}})^{xy}_{fl}(R_{\mathfrak{g}}^{t_{2}})^{ml}_{eb}x_{pw}d_{ry}+q^{-\delta_{ef}}\delta_{ae}\delta_{bf}$

for all $a,b,e,f\in{1,\dots,N}$. The map
$\mathscr{P}_{\theta}\otimes\mathscr{D}_{\theta}$ to $\mathscr{PD}_{\theta}$
defined by multiplication is an isomorphism of left
$U_{q}(\mathfrak{g})$-module algebra and (trivial) right
$\mathcal{B}_{\theta}$-module algebras. Relations (14) in the diagonal case
are equivalent to the simpler relations

(15) $\displaystyle
d_{a,b+n}x_{e,f+n}=\sum_{r,l,j,k}(R^{t_{2}})^{rl}_{ea}(R^{t_{2}})^{jk}_{fb}x_{r,j+n}d_{l,k+n}+\delta_{ae}\delta_{bf}$

for all $1\leq a,b,e,f\leq n$ where here we are taking into account linear
relations satisfied by the $d_{ij}$ and by the $x_{ij}$. In particular, this
relation combined with results from Section 3.2 ensure that in the diagonal
case $\mathscr{PD}_{\theta}$ and $\mathscr{PD}_{q}({\rm Mat}_{n})$ are
isomorphic as algebras via the map sending $x_{i,j+n}$ to $t_{ij}$ and
$d_{i,j+n}$ to $\partial_{ij}$ for each $1\leq i,j\leq n$. (For additional
details, see [19].)

The following result from [19] gives insight into the overall form of the
relations coming from the twisting map.

######  Theorem 5.1.

([19], Corollary 8.11) For each of the three families, the following
inclusions hold for the quantum Weyl algebra $\mathscr{PD}_{\theta}$

$\displaystyle
d_{ab}x_{ef}-q^{\delta_{af}+\delta_{ae}+\delta_{bf}+\delta_{be}}x_{ef}d_{ab}-q^{-\delta_{ef}}\delta_{ae}\delta_{bf}\in\sum_{(e^{\prime},f^{\prime},a^{\prime},b^{\prime})>(e,f,a,b)}\mathbb{C}(q)x_{e^{\prime}f^{\prime}}d_{a^{\prime}b^{\prime}}$

for all $a,b,e,f\in\\{1,\dots,{\rm rank}(\mathfrak{g})\\}$ where

  * •

$a\leq b$ and $e\leq f$ in Type AI

  * •

$a<b$ and $e<f$ in Type AII

  * •

$a\leq n<b$ and $e\leq n<f$ in diagonal type

and $(e^{\prime},f^{\prime},a^{\prime},b^{\prime})>(e,f,a,b)$ if and only if
$e^{\prime}\geq e,f^{\prime}\geq f,a^{\prime}\geq a,b^{\prime}\geq b$ and at
least one of these inequalities is strict.

It is also helpful for arguments later in the paper to express these relations
in special cases. We do this in the next lemma.

######  Lemma 5.2.

In Type AI and for $a<n$, we have

  * (i)

$d_{an}x_{en}=q^{1+\delta_{ae}}x_{en}d_{an}-\delta_{ae}\sum_{a^{\prime}>a}q^{2+\delta_{a^{\prime}n}}(q^{-2}-1)x_{a^{\prime}n}d_{a^{\prime}n}+\delta_{ae}$
where $e<n$,.

  * (ii)

$d_{nn}x_{ef}=q^{\delta_{nf}+\delta_{ne}}x_{ef}d_{nn}+q^{-\delta_{ef}}\delta_{ne}\delta_{nf}$
where $e\leq f$.

In Type AII and for $a<2n,e<2n$, we have

  * (iii)

$d_{a,2n}x_{e,2n}=q^{1+\delta_{ae}}x_{e,2n}d_{a,2n}-\delta_{ae}\sum_{a^{\prime}>a}q^{2+\delta_{a^{\prime},2n}}(q^{-2}-1)x_{a^{\prime},2n}d_{a^{\prime},2n}+\delta_{ae}$.

###### Proof.

Consider Type AI. Using the explicit formulas for the entries of $R$ (see
Section 3.1) $(R^{t_{2}})^{ul}_{vn}=r^{un}_{vl}\neq 0$ implies that $n=l$ and
$u=v$. Similarly, $(R^{t_{2}})^{ul}_{nv}\neq 0$ implies that $n=v=u=l$.
Moreover, if $v<n$, then $(R^{t_{2}})^{vn}_{vn}\neq 1$ while if $v=n$, then
$(R^{t_{2}})^{nn}_{nn}=q$.

Suppose that $a$ and $e$ are both strictly less than $n$. By the above
information about the entries for $R$, we get

(16) $\displaystyle d_{an}x_{en}$
$\displaystyle=(R^{t_{2}})^{na}_{na}(R^{t_{2}})^{na}_{na}(R^{t_{2}})^{ea}_{ea}(R^{t_{2}})^{nn}_{nn}(R^{t_{2}})^{en}_{en}x_{en}d_{an}$
(17)
$\displaystyle+\delta_{ae}\sum_{a^{\prime}>a}(R^{t_{2}})^{na^{\prime}}_{na^{\prime}}(R^{t_{2}})^{a^{\prime}a^{\prime}}_{aa}(R^{t_{2}})^{nn}_{nn}(R^{t_{2}})^{en}_{en}x_{a^{\prime}n}d_{a^{\prime}n}+\delta_{ae}$
(18)
$\displaystyle=q^{1+\delta_{ae}}x_{en}d_{an}-\delta_{ae}\sum_{a^{\prime}>a}q^{2+\delta_{a^{\prime}n}}(q^{-2}-1)x_{a^{\prime}n}d_{a^{\prime}n}+\delta_{ae}.$

This proves (i).

Using (14) and the above information about the entries for $R$, we see that

(19) $\displaystyle d_{nn}x_{ef}$
$\displaystyle=(R^{t_{2}})^{fn}_{fn}(R^{t_{2}})^{en}_{en}(R^{t_{2}})^{fn}_{fn}(R^{t_{2}})^{en}_{en}x_{ef}d_{nn}+q^{-\delta_{ef}}\delta_{ne}\delta_{nf}$
(20)
$\displaystyle=q^{2\delta_{nf}+2\delta_{ne}}x_{ef}d_{nn}+q^{-\delta_{ef}}\delta_{ne}\delta_{nf}$

for $e\leq f$. This proves (ii).

The argument for (iii) is the same as for (i) with $n$ replaced by $2n$
everywhere. ∎

We can define a degree function and related filtration on
$\mathscr{PD}_{\theta}$ that is compatible with the degree functions on
$\mathscr{P}_{\theta}$ and $\mathscr{D}_{\theta}$ (see Section 3.2). In
particular, the degree function for the quantum Weyl algebra
$\mathscr{PD}_{\theta}$ is $\deg d_{ij}=\deg x_{ij}=1$ for all $i,j$. We use
the same notation, namely $\mathcal{J}$, to denote the filtration on
$\mathscr{PD}_{\theta}$ defined by this degree function.

Recall that multiplication induces an isomorphism from
$\mathscr{P}_{\theta}\otimes\mathscr{D}_{\theta}$ to $\mathscr{PD}_{\theta}$.
Inductively define subspaces of $\mathscr{PD}_{\theta}$ by
$\tilde{\mathcal{J}}_{0}=\mathbb{C}(q)$ and

$\tilde{\mathcal{J}}_{r}=\sum_{u+v=r}\mathcal{J}_{u}(\mathscr{P}_{\theta})\mathcal{J}_{v}(\mathscr{D}_{\theta}).$

Since the map defined by multiplication from $\mathscr{P}\otimes\mathscr{D}$
to $\mathscr{PD}_{\theta}$ is a vector space isomorphism, we see that
$\tilde{\mathcal{J}}_{r-1}$ is a proper subspace of $\tilde{\mathcal{J}}_{r}$
for all $r\geq 0$. We get a filtration on $\mathscr{PD}_{\theta}$ by setting

$\mathcal{J}_{r}(\mathscr{PD}_{\theta})=\sum_{u\leq
r}\tilde{\mathcal{J}}_{r}(\mathscr{PD}_{\theta}).$

Note that the filtration $\mathcal{J}$ is preserved by the action of
$U_{q}(\mathfrak{g})$. Moreover, it agrees with the filtrations on
$\mathscr{P}_{\theta}$ and $\mathscr{D}_{\theta}$ defined via degree as
presented in Section 3.2. Since $\mathscr{PD}_{\theta}$ is a PBW deformation
of a twisted tensor product of $\mathscr{P}_{\theta}$ and
$\mathscr{D}_{\theta}$, the associated graded algebra defined by $\mathcal{J}$
is simply this twisted tensor product.

###  5.2. Action on polynomials

Let $\mathcal{L}$ be the left ideal of $\mathscr{PD}_{\theta}$ generated by
the elements $d_{ij}$ for $1\leq i\leq j\leq N$ where $N=n$ in Type AI and
$N=2n$ in the other two cases. Note that $\mathscr{PD}_{\theta}$ admits a
direct sum decomposition
$\mathscr{PD}_{\theta}=\mathscr{P}_{\theta}\oplus\mathcal{L}.$ Define an
action of $\mathscr{PD}_{\theta}$ on $\mathscr{P}_{\theta}$ via this
projection. In other words $a\cdot b=c$ where $c$ is the unique element of
$\mathscr{P}_{\theta}$ such that $ab-c\in\mathcal{L}$.

Recall (Section 3.2) that $\mathscr{P}^{r}_{\theta}$ equal to the homogeneous
space of degree $r$ with respect to the degree filtration $\mathcal{J}$. Note
that $\mathscr{P}^{0}_{\theta}=\mathbb{C}(q)$ and so
$\mathscr{P}_{\theta}=\mathbb{C}(q)\oplus\sum_{r>0}\mathscr{P}_{\theta}^{r}$.
This decomposition can be used to define a projection of
$\mathscr{P}_{\theta}$ onto the scalars. Define a bilinear form
$\langle\cdot,\cdot\rangle$ from
$\mathscr{D}_{\theta}\times\mathscr{P}_{\theta}$ to $\mathbb{C}(q)$ by

(21) $\displaystyle\langle d,p\rangle=(d\cdot p)_{0}$

where $(a)_{0}$ denotes the image of $a$ in $\mathbb{C}(q)$ via this
projection.

######  Lemma 5.3.

The bilinear form $\langle\cdot,\cdot\rangle$ on
$\mathscr{D}_{\theta}\times\mathscr{P}_{\theta}$ satisfies

$\displaystyle\sum\langle u_{(1)}\cdot d,u_{(2)}\cdot
p\rangle=\epsilon(u)\langle d,p\rangle$

for all $u\in U_{q}(\mathfrak{g})$, $d\in\mathscr{D}_{\theta}$ and
$p\in\mathscr{P}_{\theta}$ and hence is (left) $U_{q}(\mathfrak{g})$
invariant.

###### Proof.

Write $dp=\langle d,p\rangle+a$ where
$a\in\sum_{r>0}\mathscr{P}_{\theta}^{r}$. Since $\mathscr{PD}_{\theta}$ is a
left $U_{q}(\mathfrak{g})$-module, we have

$\displaystyle u\cdot(dp)=\sum(u_{(1)}\cdot d)(u_{(2)}\cdot p)=u\cdot(dp).$

On the other hand

$\displaystyle u\cdot(dp)=u\cdot(\langle d,p\rangle+a)=u\cdot(\langle
d,p\rangle)+u\cdot a=\epsilon(u)\langle d,p\rangle+u\cdot a$

since $\langle d,p\rangle$ is a scalar. Since the action of
$U_{q}(\mathfrak{g})$ on $\mathscr{P}_{\theta}$ preserves degree,
$\sum_{r>0}\mathscr{P}^{r}_{\theta}$ is a $U_{q}(\mathfrak{g})$-module. Thus,
$u\cdot a\in\sum_{r>0}\mathscr{P}^{r}_{\theta}$. Hence putting together the
two formulas for $u\cdot(dp)$ yields

$\displaystyle\sum(u_{(1)}\cdot d)\cdot(u_{(2)}\cdot p)\in\epsilon(u)\langle
d,p\rangle+\sum_{r>0}\mathscr{P}^{r}_{\theta}$

which implies the desired property. Thus $\langle\cdot,\cdot\rangle$ is (left)
$U_{q}(\mathfrak{g})$ invariant. ∎

The action of $\mathscr{PD}_{\theta}$ on $\mathscr{P}_{\theta}$ can be viewed
as a map of algebras, say $\phi$, from $\mathscr{PD}_{\theta}$ into ${\rm
End}\ \mathscr{P}_{\theta}$. (Here we write ${\rm End}\ \mathscr{P}_{\theta}$
for ${\rm End}_{\mathbb{C}(q)}\mathscr{P}_{\theta}$ which are endomorphisms
over the scalars. The field $\mathbb{C}(q)$ is dropped since it can be
understood from context.) Given $a\in\mathscr{PD}_{\theta}$ we frequently
write $\phi_{a}$ for $\phi(a)$ in order to make the exposition below clearer.
Since $\mathscr{P}_{\theta}$ is a left $U_{q}(\mathfrak{g})$-module, ${\rm
End}\ \mathscr{P}_{\theta}$ inherits the structure of a
$U_{q}(\mathfrak{g})$-bimodule in the standard way. Thus ${\rm End}\
\mathscr{P}_{\theta}$ is an $({\rm ad\ }U_{q}(\mathfrak{g}))$-module via

$\displaystyle(({\rm ad}\ a)\cdot b)(p)=\left(\sum
a_{(1)}bS(a_{(2)})\right)(p)=\sum a_{(1)}b\left(S(a_{(2)})(p)\right)$

for all $a\in U_{q}(\mathfrak{g})$, $b\in{\rm End}\ \mathscr{P}_{\theta}$ and
$p\in\mathscr{P}_{\theta}$.

######  Proposition 5.4.

The map $\phi$ is a $U_{q}(\mathfrak{g})$-module map with respect to the left
action on $\mathscr{PD}_{\theta}$ and the left adjoint action on ${\rm End}\
\mathscr{P}_{\theta}$ and so

(22) $\displaystyle\phi_{u\cdot a}=({\rm ad}\ u)\cdot\phi_{a}$

for all $a\in\mathscr{PD}_{\theta}$ and $u\in U_{q}(\mathfrak{g})$. Thus the
image of $\mathscr{PD}_{\theta}$ under $\phi$ is an $({\rm ad}\
U_{q}(\mathfrak{g}))$-submodule of ${\rm End}\ \mathscr{P}_{\theta}$.

###### Proof.

Consider $a\in\mathscr{PD}_{\theta}$, $b\in\mathscr{P}_{\theta}$, and
$c\in\mathscr{P}_{\theta}$ such that $ab-c\in\mathcal{L}$ and so
$\phi_{a}(b)=a\cdot b=c$. We prove the proposition by showing that (22) holds
for all generators of $U_{q}(\mathfrak{g})$ playing the role of $u$. Given $K$
in the Cartan part of $U_{q}(\mathfrak{g})$, we have

$\displaystyle(({\rm ad}\ K)\cdot\phi_{a})(K\cdot b)=(K\cdot\phi_{a}\cdot
K^{-1})(K\cdot b)=(K\cdot\phi_{a})(b)=K\cdot\phi_{a}(b).$

Since $\phi_{a}(b)=c$, it follows that
$K\cdot(\phi_{a}(b))-K\cdot(ab)\in\mathcal{L}$. On the other hand,
$K\cdot\mathcal{L}=\mathcal{L}$ and $K\cdot(ab)=(K\cdot a)(K\cdot b)$. Hence

$\displaystyle(({\rm ad}\ K)\cdot\phi_{a})(K\cdot b)-(K\cdot a)(K\cdot
b)\in\mathcal{L}$

for all $b\in U_{q}(\mathfrak{g})$. Since $(K\cdot a)(K\cdot b)-\phi_{K\cdot
a}(K\cdot b)\in\mathcal{L}$, it follows that

$\displaystyle(({\rm ad}\ K)\cdot\phi_{a})(K\cdot b)-\phi_{K\cdot a}(K\cdot
b)\in\mathcal{L}$

for all $b\in U_{q}(\mathfrak{g})$. Replacing $b$ with $K^{-1}\cdot b$ in the
above expression shows that $\phi_{K\cdot a}$ and $({\rm ad}\ K)\cdot\phi_{a}$
agree on all $b\in\mathscr{P}_{\theta}$. Thus $\phi_{K\cdot a}=({\rm ad}\
K)\cdot\phi_{a}$ for all $a\in\mathscr{PD}_{\theta}$.

Now fix $i$ and consider the action of $E_{i}$ on $a$ and $\phi_{a}$. Recall
the description of $({\rm ad}\ E_{i})$ from Section 2.3. Since
$\mathscr{PD}_{\theta}$ is a left $U_{q}(\mathfrak{g})$\- module algebra,
using the form of $\Delta(E_{i})$ from Section 2.2, we have $E_{i}\cdot
ab=(E_{i}\cdot a)b+(K_{i}\cdot a)(E_{i}\cdot b)$ for all
$a,b\in\mathscr{PD}_{\theta}$. Hence, for all $b\in\mathscr{P}_{\theta}$ we
have

$\displaystyle(({\rm ad}\ E_{i})\cdot\phi_{a})(b)$
$\displaystyle=(E_{i}\cdot\phi_{a})(b)-(K_{i}\cdot\phi_{a}\cdot
K_{i}^{-1}E_{i})(b)=E_{i}\cdot\phi_{a}(b)-K_{i}\cdot(\phi_{a}(K_{i}^{-1}E_{i}\cdot
b))$ $\displaystyle\in E_{i}\cdot ab-(K_{i}\cdot\phi_{a}\cdot
K_{i}^{-1})(E_{i}\cdot b)+\mathcal{L}$ $\displaystyle=E_{i}\cdot
ab-(K_{i}\cdot a)(E_{i}\cdot b)+\mathcal{L}=(E_{i}\cdot a)(b)+\mathcal{L}.$

Hence $({\rm ad}\ E_{i})\cdot\phi_{a}=\phi_{(E_{i}\cdot a)}$. A similar
arguent holds for $E_{i}$ replaced by $F_{i}$. ∎

###  5.3. Orthogonality conditions

As explained in Section 5.1, as an algebra, $\mathscr{PD}_{\theta}$ is
isomorphic to $\mathscr{PD}_{q}({\rm Mat}_{n})$ in the diagonal case. Thus,
the second assertion of the next result is a generalization of [3],
Proposition 1 to include the other two families.

######  Proposition 5.5.

For each $r$ and $s$ with $r\neq s$, $\mathscr{D}^{r}_{\theta}$ is equal to
the vector space dual of $\mathscr{P}^{r}_{\theta}$ and
$\mathscr{D}^{r}_{\theta}$ is orthogonal to $\mathscr{P}_{\theta}^{s}$ with
respect to the bilinear form defined by (21). Moreover, $\mathscr{P}_{\theta}$
is a faithful $\mathscr{PD}_{\theta}$-module with respect to the action
defined above.

###### Proof.

Note that the relation between the bilinear form and the action ensures that
the moreover part of the proposition is an immediate consequence of the main
assertion. Hence, we focus on proving the duality result.

By Theorem 5.1

$\displaystyle\mathscr{D}^{r+s}_{\theta}\mathscr{P}^{s}_{\theta}\subseteq\sum_{s\geq
k\geq 0}\mathscr{P}^{k}_{\theta}\mathscr{D}^{r+k}_{\theta}{\rm\ and\
}\mathscr{D}^{r}_{\theta}\mathscr{P}^{r+s}_{\theta}\subseteq\sum_{s\geq k\geq
0}\mathscr{P}^{r+k}_{\theta}\mathscr{D}^{k}_{\theta}$

for all $r\geq 0$ and $s\geq 0$. It follows that $(d\cdot p)_{0}=0$ for
$d\in\mathscr{D}^{r}_{\theta},p\in\mathscr{P}^{s}_{\theta}$ with $r\neq s$.

By Lemma 3.3, the set of monomials of the form

$\displaystyle x_{e_{1},f_{1}}^{m_{1}}x_{e_{2},f_{2}}^{m_{2}}\cdots
x_{e_{r},f_{r}}^{m_{r}}$

form a PBW basis for $\mathscr{P}_{\theta}^{m}$ where $m=m_{1}+\cdots+m_{r}$
and $(e_{1},f_{1})>(e_{2},f_{2})>\dots>(e_{r},f_{r})$. As explained preceding
Lemma 3.3, the analogous result holds for $\mathscr{D}_{\theta}$ with each
$x_{ij}$ replaced by $d_{ij}$. Moreover, by Lemma 3.3, we can switch the
ordering of the subscripts and still get a PBW basis. We do this below for
$\mathscr{D}_{\theta}$. Consider a sequence of ordered pairs
$(e_{1},f_{1}),\dots,(e_{r},f_{r})$ satisfying

$\displaystyle(a,b)\geq(e_{1},f_{1})>(e_{2},f_{2})>\dots>(e_{r},f_{r})$

where here $``>"$ is the standard lexicographic ordering (from left to right).
It follows from Theorem 5.1 that

$\displaystyle d_{ab}x_{e_{1},f_{1}}^{m_{1}}x_{e_{2},f_{2}}^{m_{2}}\cdots
x_{e_{r},f_{r}}^{m_{r}}\in c\
\delta_{a,e_{1}}\delta_{b,f_{1}}x_{e_{1}.f_{1}}^{m_{1}-1}x_{e_{2},f_{2}}^{m_{2}}\cdots
x_{e_{r},f_{r}}^{m_{r}}+\mathscr{P}_{\theta}^{m-1}+\sum_{(a^{\prime},b^{\prime})\geq(a,b)}\mathscr{P}_{\theta}^{m}d_{a^{\prime}b^{\prime}}$

where $m=m_{1}+\cdots+m_{r}$ and $c$ is a nonzero scalar. Hence given
$h\in\mathscr{D}_{\theta}^{m-1}$, we have

$\displaystyle(hd_{ab}\cdot
x_{e_{1},f_{1}}^{m_{1}}x_{e_{2},f_{2}}^{m_{2}}\cdots
x_{e_{r},f_{r}}^{m_{r}})_{0}\neq 0$

implies that $(a,b)=(e_{1},f_{1})$. Using induction, we obtain

$\displaystyle(d_{a_{k},b_{k}}^{s_{k}}d_{a_{k-1},b_{k-1}}^{s_{k-1}}\cdots
d_{a_{1},b_{1}}^{s_{1}}\cdot
x_{e_{1},f_{1}}^{m_{1}}x_{e_{2},f_{2}}^{m_{2}}\cdots
x_{e_{r},f_{r}}^{m_{r}})_{0}\neq 0$

with

$\displaystyle(a_{k},b_{k})<(a_{k-1},b_{k-1})<\cdots<(a_{1},b_{1})$

if and only if $r=k,s_{i}=m_{i},e_{i}=a_{i},$ and $f_{i}=b_{i}$ for
$i=1,\dots,r$. This proves the desired duality result. ∎

###  5.4. Action on highest weight terms

By Proposition 4.3, the highest weight vector
$H_{2\mu},\mu=\sum_{i}m_{i}\hat{\eta}_{i}$ is homogeneous of degree
$\sum_{r}m_{r}r$. Note that this degree is just the size of $\mu$ viewed as a
partition. In other words, $\sum_{r}m_{r}r=\sum_{r}\mu_{r}=|\mu|$ where $\mu$
is expressed as $\sum_{i}\mu_{i}\epsilon^{\Sigma}_{i}$, a linear combination
in terms of the orthonormal basis in the restricted root setting. Moreover,
the action of $U_{q}(\mathfrak{g})$ preserves the degree. Thus, for each
$\mu\in\Lambda_{\Sigma}^{+}$, the module $({\rm ad}\ U_{q}(\mathfrak{g}))\cdot
H_{2\mu}$ sits inside the homogeneous component $\mathscr{P}_{\theta}^{m}$ of
degree $m=|\mu|$. Recall the definition of the augmentation ideals $U^{+}_{+}$
and $U^{-}_{+}$ given in Section 2.2.

######  Proposition 5.6.

For all $\mu$ and $\gamma$ in $\Lambda_{\Sigma}^{+}$ with $\mu\neq\gamma$, the
space $U_{q}(\mathfrak{g})\cdot H_{2\mu}$ is equal to the
$U_{q}(\mathfrak{g})$-module dual of $U_{q}(\mathfrak{g})\cdot H^{*}_{2\mu}$
and orthogonal to $U_{q}(\mathfrak{g})\cdot H^{*}_{2\gamma}$ with respect to
the bilinear form defined by (21). Moreover,

$\langle H_{2\mu}^{*},H_{2\mu}\rangle\neq 0$

whereas

$\langle E\cdot H_{2\mu}^{*},H_{2\mu}\rangle=0=\langle H_{2\mu}^{*},F\cdot
H_{2\mu}\rangle$

for all $E\in U^{+}_{+}$ and $F\in U^{-}_{+}$.

###### Proof.

Let $\mu\in\Lambda_{\Sigma}^{+}$ and set $m=|\mu|$. Since the left action of
$U_{q}(\mathfrak{g})$ preserves degree, both $\mathscr{P}_{\theta}^{m}$ and
$\mathscr{D}_{\theta}^{m}$ are left $U_{q}(\mathfrak{g})$\- modules. By Lemma
5.3, the dualities of vector spaces in Proposition 5.5 are actually dualities
of left $U_{q}(\mathfrak{g})$-modules.

Note that there is only one way to express a weight
$\mu\in\Lambda^{+}_{\Sigma}$ as a linear combination of the $\hat{\eta}_{r}$.
This means that $U_{q}(\mathfrak{g})\cdot H_{2\mu}$ is the unique simple
module with highest weight $2\mu$ inside the decomposition of
$\mathscr{P}_{\theta}$, and thus inside of $\mathscr{P}^{m}_{\theta}$.
Similarly, $U_{q}(\mathfrak{g})\cdot H^{*}_{2\mu}$ is the unique simple module
with lowest weight $-2\mu$ inside the decomposition of $\mathscr{D}_{\theta}$,
and thus inside of $\mathscr{D}^{m}_{\theta}$. Hence, by the previous
paragraph, $U_{q}(\mathfrak{g})\cdot H_{2\mu}$ is equal to the
$U_{q}(\mathfrak{g})$-module dual of $U_{q}(\mathfrak{g})\cdot H_{2\mu}^{*}$
with respect to bilinear form defined by (21).

Since $H_{2\mu}^{*}$ is a lowest weight generating vector for
$U_{q}(\mathfrak{g})\cdot H_{2\mu}^{*}$, it follows that

$\displaystyle U_{q}(\mathfrak{g})\cdot H_{2\mu}^{*}=H_{2\mu}^{*}\oplus
U^{+}_{+}\cdot H_{2\mu}^{*}.$

The fact that $H_{2\mu}$ is a highest weight vector combined with the
$U_{q}(\mathfrak{g})$ invariance of the bilinear form
$\langle\cdot,\cdot\rangle$ (Lemma 5.3) ensures that $H_{2\mu}$ is
perpendicular to $U^{+}_{+}\cdot\mathscr{D}_{\theta}$. The argument showing
$H_{2\mu}^{*}$ is perpendicular to $U_{+}^{-}\cdot\mathscr{P}_{\theta}$
follows in a similar fashion. This completes the proof of the proposition. ∎

Note that by the above proposition, the action of $H_{2\mu}^{*}$ on $H_{2\mu}$
is nonzero. In the next result, we obtain more information as to when such an
action (and related ones) are possibly nonzero.

######  Lemma 5.7.

Given $\mu$ and $\gamma$ in $\Lambda_{\Sigma}^{+}$ such that
$|\gamma|\geq|\mu|$ and $\mu\neq\gamma$, we have $(u\cdot
H_{2\gamma}^{*})H_{2\mu}\in\mathcal{L}$ and so $(u\cdot H_{2\gamma}^{*})\cdot
H_{2\mu}=0$ for any $u\in U^{+}_{+}.$

###### Proof.

Consider $u\in U^{+}$ and note that $u-\epsilon(u)$ is in $U^{+}_{+}$.
Moreover, it follows from the form the coproduct takes on the generators
$E_{i}$ of $U^{+}$ that

$\displaystyle\Delta(u-\epsilon(u))\in u\otimes 1+U_{q}(\mathfrak{g})\otimes
U^{+}_{+}.$

Since $H_{2\mu}$ is a highest weight vector, it is annihilated by elements of
$U^{+}_{+}$. Hence

(23) $\displaystyle u\cdot\left(H_{2\gamma}^{*}H_{2\mu}\right)=(u\cdot
H_{2\gamma}^{*})H_{2\mu}+\epsilon(u)H_{2\gamma}^{*}H_{2\mu}.$

Assume that

(24) $\displaystyle H_{2\gamma}^{*}H_{2\mu}\in\mathcal{L}.$

Since $\mathcal{L}$ is a left $U_{q}(\mathfrak{g})$-module, It follows that
$u\cdot(H_{2\gamma}^{*}H_{2\mu})\in\mathcal{L}$ for any $u\in U^{+}$. Hence by
(23), the same holds true for $(u\cdot H_{2\gamma}^{*})H_{2\mu}$. Thus it is
sufficient to prove (24) given the assumptions on $\gamma$ and $\mu$.

Recall that $H_{2\gamma}^{*}$ has degree $|\gamma|$ and $H_{2\mu}$ has degree
$|\mu|$. It follows from the defining relations of $\mathscr{PD}_{\theta}$
that when we move the $d_{ij}$ terms to the right pass the $x_{ij}$ terms in
the expression of the left hand side of (24) we end up an expression of the
form $\sum_{j}u_{j}v_{j}$ where each $v_{j}$ is in
$\mathscr{D}^{r_{j}}_{\theta}$ and each $u_{j}$ is in
$\mathscr{P}^{w_{j}}_{\theta}$ and $r_{j}-w_{j}=|\gamma|-|\mu|$. In other
words, the relations cancel out the same number of $x_{ij}$ and $d_{ij}$
terms.

If $|\gamma|$ is strictly greater than $|\mu|$, then each $v_{j}$ has degree
at least $1$ and so we end up with an element in
$\sum_{i,j}\mathscr{PD}_{\theta}d_{ij}=\mathcal{L}.$ Now assume that
$|\gamma|=|\mu|$. In this case, we end up with $\deg v_{j}=\deg u_{j}$ each
$j$ and so

$\displaystyle H_{2\gamma}^{*}H_{2\mu}\in\mathbb{C}(q)+\mathcal{L}.$

Note that this scalar term must be $\langle H_{2\gamma}^{*},H_{2\mu}\rangle$.
By Proposition 5.6, this term is zero for $\gamma\neq\mu$. ∎

##  6\. Action of Cartan elements

###  6.1. Special Cartan elements

We turn our attention to understanding the action of the various elements of
the Cartan subalgebra on the generators of $\mathscr{P}_{\theta}$ and then
identifying them with elements of the appropriate quantum Weyl algebra.

######  Lemma 6.1.

Let $N=n$ in Type AI, $N=2n$ in Type AII and $N\in\\{n,2n\\}$ in the diagonal
type. The element $K_{2\epsilon_{N}}-1$ acts the same on
$\mathscr{P}_{\theta}$ as the element $X$ in $\mathscr{PD}_{\theta}$ where

$\displaystyle
X=\left\\{\begin{matrix}&(q^{2}-1)((q^{3}+q)x_{nn}d_{nn}+\sum_{a=1}^{n-1}x_{an}d_{an}){\rm\
in\ Type\ AI}\cr\cr&(q^{2}-1)(\sum_{a=1}^{2n-1}x_{a,2n}d_{a,2n}){\rm\ in\
Type\ AII}\cr\cr&(q^{2}-1)\sum_{a=1}^{n}x_{a,2n}d_{a,2n}{\rm\ in\ diagonal\
type\ }(N=n)\cr\cr&(q^{2}-1)\sum_{a=1}^{n}x_{n,a+n}d_{n,a+n}{\rm\ in\
diagonal\ type\ }(N=2n)\end{matrix}\right.$

###### Proof.

Consider first Type AI. It follows from Lemma 5.2 and the relations (5) and
(6) satisfied by the $x_{ij}$ that for any scalars $c_{1},\dots,c_{n}$,

$\displaystyle\sum_{a=1}^{e}c_{a}$ $\displaystyle
x_{an}d_{an}x_{en}=c_{e}x_{en}d_{en}x_{en}+\sum_{a<e}c_{a}x_{an}d_{an}x_{en}$
$\displaystyle=q^{-\delta_{en}}c_{e}x_{en}+q^{2+2\delta_{en}}c_{e}x_{en}x_{en}d_{en}+\sum_{a>e}\mathscr{P}_{\theta}d_{an}+q^{2+2\delta_{en}}c_{a}x_{en}\sum_{a<e}x_{an}d_{an}$

and so

$\displaystyle\sum_{a=1}^{e}c_{a}x_{an}d_{an}x_{en}^{2}$
$\displaystyle=q^{-\delta_{en}}c_{e}x_{en}^{2}+q^{2+2\delta_{en}}c_{e}x_{en}x_{en}d_{en}x_{en}$
$\displaystyle+\sum_{a>e}\mathscr{P}_{\theta}d_{an}+q^{2+2\delta_{en}}x_{en}^{2}\sum_{a<e}c_{a}x_{an}d_{an}$
$\displaystyle=q^{-\delta_{en}}(q^{2+2\delta_{en}}+1)c_{e}x_{en}^{2}+q^{4+4\delta_{en}}c_{e}x_{en}^{3}d_{en}$
$\displaystyle+\sum_{a>e}\mathscr{P}_{\theta}d_{an}+q^{4+4\delta_{en}}x_{en}^{2}\sum_{a<e}c_{a}x_{an}d_{an}.$

By induction, we have

$\displaystyle\sum_{a=1}^{e}c_{a}x_{an}d_{an}x_{en}^{s}=q^{-\delta_{en}}{{(q^{(2+2\delta_{en})s}-1)}\over{(q^{2+2\delta_{en}}-1)}}c_{e}x_{en}^{s}+\sum_{a\geq
e}\mathscr{P}_{\theta}d_{an}+q^{(2+2\delta_{en})s}x_{en}^{s}\sum_{a<e}c_{a}x_{an}d_{an}.$

Now consider $c_{n}=q^{3}+q$ and $c_{a}=1$ for $a<n$. We have

$\displaystyle\sum_{a=1}^{n}c_{a}$ $\displaystyle
x_{an}d_{an}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}={{(q^{4s_{n}}-1)}\over{(q^{2}-1)}}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}$
$\displaystyle+q^{4s_{n}}x_{nn}^{s_{n}}\left(\sum_{a=1}^{n-1}c_{a}x_{an}d_{an}\right)x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}+\mathscr{P}d_{nn}x_{n-1,n}^{s_{n-1}}\cdots x_{1n}^{s_{1}}.$

For the final term, we can move $d_{nn}$ to the right, past all the $x_{ij}$,
yielding an element of $\mathscr{P}_{\theta}d_{nn}.$ The penultimate term
equals

$\displaystyle q^{4s_{n}}{{(q^{2s_{n-1}}-1)}\over{(q^{2}-1)}}$ $\displaystyle
x_{nn}^{s_{n}}\cdots
x_{1n}^{s_{1}}+q^{4s_{n}+2s_{n-1}}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\left(\sum_{a=1}^{n-2}c_{a}x_{an}d_{an}\right)x_{n-2,n}^{s_{n-2}}\cdots
x_{1n}^{s_{1}}$
$\displaystyle+\left(\mathscr{P}_{\theta}d_{n-1,n}+\mathscr{P}_{\theta}d_{nn}\right)x_{n-2,n}^{s_{n}-2}\cdots
x_{1n}^{s_{1}}.$

Noting that this final term is in
$\mathscr{P}_{\theta}d_{n-1,n}+\mathscr{P}_{\theta}d_{nn}$ (here, we can once
again move the $d$ terms all the way to the right), and summing up the above
two expressions yields

$\displaystyle\sum_{a=1}^{n}c_{a}$ $\displaystyle
x_{an}d_{an}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}={{(q^{4s_{n}+2s_{n-1}}-1)}\over{(q^{2}-1)}}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}$
$\displaystyle+q^{4s_{n}+2s_{n-1}}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\left(\sum_{a=1}^{n-2}c_{a}x_{an}d_{an}\right)x_{n-2,n}^{s_{n-2}}\cdots
x_{1n}^{s_{1}}+\mathscr{P}_{\theta}d_{n-1,n}+\mathscr{P}_{\theta}d_{nn}.$

Using induction, we get

$\displaystyle\sum_{a=1}^{n}c_{a}$ $\displaystyle
x_{an}d_{an}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}={{(q^{4s_{n}+2s_{n-1}+\cdots
2s_{1}}-1)}\over{(q^{2}-1)}}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}+\sum_{a}\mathscr{P}_{\theta}d_{an}.$

By Theorem 5.1,
$\sum_{a}\mathscr{P}_{\theta}d_{an}x_{ef}\in\sum_{a^{\prime}}\mathscr{P}_{\theta}d_{a^{\prime}n}$
for all $e,f$ strictly less than $n$. Hence

$\displaystyle\sum_{a=1}^{n}c_{a}$ $\displaystyle
x_{an}d_{an}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}x_{e_{1},f_{1}}\cdots x_{e_{r},f_{r}}$
$\displaystyle={{(q^{4s_{n}+2s_{n-1}+\cdots
2s_{1}}-1)}\over{(q^{2}-1)}}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}x_{e_{1},f_{1}}\cdots
x_{e_{r},f_{r}}+\sum_{a}\mathscr{P}_{\theta}d_{an}.$

for all $(e_{1},f_{1})\geq\cdots\geq(e_{r},f_{r})$ with $e_{j}\leq f_{j}<n$
each $j$. It follows from the definition of $\mathcal{L}$ that
$\mathscr{P}_{\theta}d_{an}\subset\mathcal{L}$. Thus, we get

$\displaystyle\left(\sum_{a=1}^{n}c_{a}x_{an}d_{an}\right)$
$\displaystyle\cdot x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}x_{e_{1},f_{1}}\cdots x_{e_{r},f_{r}}$
$\displaystyle={{(q^{4s_{n}+2s_{n-1}+\cdots
2s_{1}}-1)}\over{(q^{2}-1)}}x_{nn}^{s_{n}}x_{n-1,n}^{s_{n-1}}\cdots
x_{1n}^{s_{1}}x_{e_{1},f_{1}}\cdots x_{e_{r},f_{r}}$

for all $(e_{i},f_{i}),i=1,\dots,r$ as described above. In other words,
$\sum_{a=1}^{n}c_{a}x_{an}d_{an}$ acts the same as
$(K_{2\epsilon_{n}}-1)/(q^{2}-1)$ on a basis for $\mathscr{P}_{\theta}$, and
thus on $\mathscr{P}_{\theta}$, as desired. The argument in Type AII is
exactly the same where we omit any terms involving $d_{nn}$ and $x_{nn}$ and
replace $n$ with $2n$ everywhere.

Now consider the diagonal case. Using the relations for
$\mathscr{PD}_{\theta}$, which in this case is the same as that for
$\mathscr{PD}_{q}[{\rm Mat}_{n}]$ with $t_{ij}$ replaced by $x_{i,j+n}$ and
$\partial_{uv}$ replaced by $d_{u,v+n}$ (see (15) and [19], Sections 6.3 and
8.3), gives us

$\displaystyle x_{a,2n}d_{a,2n}x_{e,f+n}\in\sum_{a^{\prime}\geq
a}\mathscr{P}d_{a^{\prime},2n}$

whenever $f<n$ or $e<a$. Hence, by induction,

$\displaystyle x_{a,2n}d_{a,2n}x_{e_{1},f_{1}+n}\cdots
x_{e_{r},f_{r}+n}\in\sum_{a^{\prime}>a}\mathscr{P}_{\theta}d_{a^{\prime},2n}$

for any choice of $(e_{1},f_{1}),\cdots,(e_{r},f_{r})$ with
$(e_{i},f_{i})<(a,n)$ each $i$. On the other hand,

$\displaystyle\sum_{a\leq
e}x_{a,2n}d_{a,2n}x_{e,2n}=q^{2}x_{e,2n}^{2}d_{e,2n}+x_{e,2n}+q^{2}x_{e,2n}\sum_{a<e}x_{a,2n}d_{a,2n}+\sum_{a^{\prime}\geq
e}\mathscr{P}_{\theta}d_{a^{\prime},2n}.$

By induction, we have

$\displaystyle
x_{a,2n}d_{a,2n}x_{a,2n}^{s}=q^{2s}x_{a,2n}^{2}d_{a,2n}+{{(q^{2s}-1)}\over{(q^{2}-1)}}x_{a,2n}+q^{2s}x_{e,2n}^{s}\sum_{a<e}x_{a,2n}d_{a,2n}+\sum_{a^{\prime}>a}\mathscr{P}_{\theta}d_{a^{\prime}n}.$

Hence

$\displaystyle\sum_{a=1}^{n}$ $\displaystyle
x_{a,2n}d_{a,2n}x_{n,2n}^{s_{n}}x_{n-1,2n}^{s_{n-1}}\cdots
x_{1,2n}^{s_{1}}x_{e_{1},f_{1}+n}\cdots x_{e_{r},f_{r}+n}$
$\displaystyle={{(q^{2s_{n}+2s_{n-1}+\cdots
2s_{1}}-1)}\over{(q^{2}-1)}}x_{n,2n}^{s_{n}}x_{n-1,2n}^{s_{n-1}}\cdots
x_{1,2n}^{s_{1}}x_{e_{1},f_{1}+n}\cdots
x_{e_{r},f_{r}+n}+\sum_{a}\mathscr{P}_{\theta}d_{a,2n}$

and so

$\displaystyle\left(\sum_{a=1}^{n}x_{a,2n}d_{a,2n}\right)$ $\displaystyle\cdot
x_{n,2n}^{s_{n}}x_{n-1,2n}^{s_{n-1}}\cdots
x_{1,2n}^{s_{1}}x_{e_{1},f_{1}+n}\cdots x_{e_{r},f_{r}+n}$
$\displaystyle={{(q^{2s_{n}+2s_{n-1}+\cdots+2s_{1}}-1)}\over{(q^{2}-1)}}x_{n,2n}^{s_{n}}x_{n-1,2n}^{s_{n-1}}\cdots
x_{1,2n}^{s_{1}}x_{e_{1},f_{1}+n}\cdots x_{e_{r},f_{r}+n}$

for any sequence $(e_{1},f_{1}),\dots,(e_{r},f_{r})$ with $f_{i}<n$ for each
$i$. Thus $\sum_{a=1}^{n}x_{a,2n}d_{a,2n}$ acts on $\mathscr{P}_{\theta}$ the
same way as $(K_{2\epsilon_{n}}-1)/(q^{2}-1)$. This proves the first assertion
in the diagonal case. The proof for the second case is the same with the order
of the subscripts switched. ∎

###  6.2. Relationships between Cartan elements

In the next lemma, we write $K_{2\epsilon_{r}+\cdots+2\epsilon_{N}}$ in terms
of elements in the $U_{q}(\mathfrak{g})$-module generated by
$K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}$ and $K_{2\epsilon_{N}}$ where $N$
is either $n$ or $2n$ depending on $\mathfrak{g}$. This is the crucial step in
showing that $K_{2\epsilon_{r}+\cdots+2\epsilon_{N}}$ acts the same as an
element in $\mathscr{PD}_{\theta}$ on $\mathscr{P}_{\theta}$.

The proof of this lemma is very similar to the proof of [18], Proposition
4.2.1. However, in [18], the coproduct used is the opposite of the one used
here which in turn impacts the formulas for the adjoint action. We include a
proof here so that the reader will not need to adjust the proof of [18] with
respect to subtle differences between the setups in the two papers.

######  Lemma 6.2.

The element $K_{2\epsilon_{r}+\cdots+2\epsilon_{N}}$ can be expressed as a
linear combination of the following three terms, each with nonzero coefficient

  * •

$\left(({\rm ad}\ E_{r}E_{r+1}\cdots E_{N-1})\cdot
K_{2\epsilon_{N}}\right)\left(({\rm ad}\ F_{N-1}\cdots F_{r})\cdot
K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}\right)$

  * •

$\left(({\rm ad}\ F_{N-1}\cdots F_{r})\cdot
K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}\right)\left(({\rm ad}\
E_{r}E_{r+1}\cdots E_{N-1})\cdot K_{2\epsilon_{N}}\right)$

  * •

$K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}K_{2\epsilon_{N}}$

where $N=n$, $r=1,\dots,n$ in Type AI, $N=2n$ and $r=1,\dots,2n$ for Type AII,
and $N=n$, $r=1,\dots,n$ and $N=2n$, $r=n+1,\dots,2n$ in the diagonal case.

###### Proof.

Set $\beta_{r}=\alpha_{r}+\cdots+\alpha_{N-1}$. Straightforward calculations
show that

$\displaystyle({\rm ad}\ E_{N-1})\cdot
K_{2\epsilon_{N}}=(1-q^{-2})E_{N-1}K_{2\epsilon_{N}}$

and

$\displaystyle({\rm ad}\ E_{j})\cdot
X_{\beta_{j-1}}K_{2\epsilon_{N}}=(E_{j}X_{\beta_{j-1}}-qX_{\beta_{j-1}}E_{j})K_{2\epsilon_{N}}$

where $X$ is a sum of monomials in the $E_{i}$ of weight $\beta_{j-1}$. It
follows that

$\displaystyle({\rm ad}\ E_{r}E_{r+1}\cdots E_{N-2}E_{N-1})\cdot
K_{2\epsilon_{N}}=(1-q^{-2})E_{\beta_{r}}K_{2\epsilon_{N}}$

where $E_{\beta_{r}}$ is the root vector associated to the root $\beta_{r}$
(see [10], Section 7.3.1 or [19], Section 3.3).

We have a similar result for the $F_{j}^{\prime}s$. In particular, we have

$\displaystyle({\rm ad}\ F_{r})\cdot
K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}=(1-q^{2})F_{r}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}$

and

$\displaystyle({\rm ad}\ F_{j})\cdot
Y_{\beta^{\prime}_{j}}K_{\beta^{\prime}_{j}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}=(F_{j}Y_{\beta^{\prime}_{j}}-q^{-1}Y_{\beta^{\prime}_{j}}F_{j})K_{\beta^{\prime}_{j}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}$

where $Y$ is a sum of monomials in the $F_{i}$ of weight
$\alpha_{N-1}+\cdots+\alpha_{j-1}$. By induction, we produce the $q^{-1}$
commutator
$[F_{N-1},[F_{N-2},\cdots,[F_{r+1},F_{r}]_{q^{-1}}]_{q^{-1}}]\cdots]_{q^{-1}}$
which is just a (nonzero) scalar multiple of the root vector $F_{\beta_{r}}$
([10], Section 7.3.1). Hence

$\displaystyle({\rm ad}\ F_{N-1}\cdots F_{r})\cdot
K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}=(1-q^{2})\gamma
F_{\beta_{r}}K_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}$

for some nonzero scalar $\gamma$.

Since $E_{\beta_{r}}$ and $F_{\beta_{r}}$ are root vectors under one of
Lusztig’s automorphisms ([10], Section 7.3.1), it follows that
$[E_{\beta_{r}},F_{\beta_{r}}]=(K_{\beta_{r}}-K_{\beta_{r}}^{-1})(q-q^{-1})$.
Hence

$\displaystyle q^{-2}$
$\displaystyle(E_{\beta_{r}}K_{2\epsilon_{N}})(F_{\beta_{r}}K_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}})-(F_{\beta_{r}}K_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}})(E_{\beta_{r}}K_{2\epsilon_{N}})$
$\displaystyle=(E_{\beta_{r}}F_{\beta_{r}}-F_{\beta_{r}}E_{\beta_{r}})K_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}K_{2\epsilon_{N}}$
$\displaystyle=(q-q^{-1})^{-1}(K_{\beta_{r}}^{2}-1)K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}K_{2\epsilon_{N}}.$

Note that $2\beta_{r}=2\epsilon_{r}-2\epsilon_{N}.$ Thus the above simplifies
to

$\displaystyle(q-q^{-1})^{-1}(K_{2\epsilon_{r}+2\epsilon_{r+1}+\cdots+2\epsilon_{N}}-K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}K_{2\epsilon_{N}}).$

Therefore, $K_{2\epsilon_{r}+\cdots+2\epsilon_{N}}$ can be written as a linear
combination of the elements
$K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}K_{2\epsilon_{N}}$,
$(E_{\beta_{r}}K_{2\epsilon_{N}})(F_{\beta_{r}}K_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}})$,
and
$(F_{\beta_{r}}K_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}})(E_{\beta_{r}}K_{2\epsilon_{N}})$
with each term appearing with nonzero coefficient. ∎

###  6.3. Acting as quantum Weyl Algebra elements

Let $\psi$ denote the map from $U_{q}(\mathfrak{g})$ to ${\rm End}\
\mathscr{P}_{\theta}$ that agrees with the action of $U_{q}(\mathfrak{g})$ on
$\mathscr{P}_{\theta}$. We show that
$\psi(K_{2\epsilon_{r}+\dots+2\epsilon_{m}})$ agrees with the image under
$\phi$ of an element of $\mathscr{PD}_{\theta}$. Moreover, we determine a
bound on the degree of these elements using the degree function defined in
Section 5.1.

######  Proposition 6.3.

The image $\psi(K_{2\epsilon_{r}+\dots+2\epsilon_{N}})$ of
$K_{2\epsilon_{r}+\dots+2\epsilon_{N}}$ inside ${\rm End}\
\mathscr{P}_{\theta}$ is equal to the image $\phi_{a_{r}}$ for some
$a_{r}\in\mathscr{PD}_{\theta}$ where

  * •

$N=n$, $r=1,\dots,n$ in Type AI

  * •

$N=2n$ and $r=1,\dots,2n$ for Type AII

  * •

$N=n$, $r=1,\dots,n$ and $N=2n$, $r=n+1,\dots,2n$ in the diagonal case.

Moreover, $\deg a_{r}\leq 2(N-r+1)$.

###### Proof.

By Lemma 6.1, $\psi(K_{2\epsilon_{N}})=\phi_{a_{N}}$ for an appropriate
element $a_{N}\in\mathscr{PD}_{\theta}$. Moreover, one sees from the formulas
for $\psi(K_{2\epsilon_{N}})$ given in this lemma that $\deg
a_{N}=2=2(N-N+1)$. Now assume that for some $r\leq N$,
$\psi(K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}})=\phi_{a_{r+1}}$ for an element
$a_{r+1}$ of degree at most ${2(N-r)}$ where $N=n$ in Type AI, $N=2n$ in Type
AII, and $N=n$ or $N=2n$ in the diagonal case with $N-n\leq r\leq N-1$.

By Lemma 6.2, $K_{2\epsilon_{r}+\cdots+2\epsilon_{N}}$ is a linear combination
of products of the form $xy$ where $x\in({\rm ad}\ U_{q}(\mathfrak{g}))\cdot
K_{2\epsilon_{N}}$ and $y\in({\rm ad}\ U_{q}(\mathfrak{g}))\cdot
K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}$. Since $\phi(\mathscr{PD}_{\theta})$
is a subalgebra and, by Proposition 5.4, an $({\rm ad}\
U_{q}(\mathfrak{g}))$-submodule of ${\rm End}\ \mathscr{P}_{\theta}$, it
follows that
$\psi(K_{2\epsilon_{r}+\cdots+2\epsilon_{N}})\in\phi(\mathscr{PD}_{\theta})$,
and so the main assertion of the lemma follows by induction.

We now turn our attention to understanding the degree assertion at the
conclusion of the lemma. As explained above, $\deg a_{N}=2$. Now assume that
$\deg a_{r+1}=2(N-(r+1)+1)$. It follows that
$\deg(a_{r+1}a_{N})\leq\deg(a_{r+1})+\deg(a_{N})=2(N-(r+1)+1)+2=2(N-r+1).$
Since the filtration $\mathcal{J}$ is preserved by the action of
$U_{q}(\mathfrak{g})$, we also have $\deg(E_{\beta_{r}}K_{2\epsilon_{N}})\leq
2$ and $\deg(F_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}})\leq
2(N-r).$ Therefore

$\deg(E_{\beta_{r}}F_{\beta_{r}}-F_{\beta_{r}}E_{\beta_{r}})K_{\beta_{r}}K_{2\epsilon_{r+1}+\cdots+2\epsilon_{N}}K_{2\epsilon_{N}}\leq
2(N-r+1)$

and so $\deg a_{r}\leq 2(N-r+1).$. The final assertion of the proposition now
follows by induction. ∎

##  7\. The locally finite subalgebra

###  7.1. The locally finite subalgebra for $U_{q}(\mathfrak{gl}_{N})$

In [6] and [7], a complete description of the locally finite subalgebra of
$U_{q}(\mathfrak{sl}_{N})$ as a direct sum of $({\rm ad}\
U_{q}(\mathfrak{sl}_{N}))$-modules is given. This result is then generalized
to the simply connected quantized enveloping algebra (see Section 2.2) in [7]
(see also [5], 7.1). In particular, we have

(25)
$\displaystyle\mathcal{F}(\check{U}_{q}(\mathfrak{sl}_{N}))=\bigoplus_{\lambda\in-P^{+}}({\rm
ad}\ U_{q}(\mathfrak{sl}_{N}))\cdot K_{2\lambda}.$

Note that $-P^{+}=\\{-\lambda|\ \lambda\in P^{+}\\}=\\{w_{0}\lambda|\
\lambda\in P^{+}\\}$. Thus, $-P^{+}$ is the $\mathbb{N}$ linear combination of
the $w_{0}\omega_{i},i=1,\dots,N-1$. As explained in Section 2.1, the
fundamental weight $\omega_{i}$ is equal to
$\hat{\omega}_{i}=\epsilon_{1}+\cdots+\epsilon_{i}$ plus a scalar multiple of
$\hat{\omega}_{N}=\epsilon_{1}+\cdots+\epsilon_{N}$. We see from (1) that this
scalar is $i/N$ which is not an integer and so, the simply connected quantized
enveloping algebra $\check{U}_{q}(\mathfrak{sl}_{N})$ is not a subalgebra of
$U_{q}(\mathfrak{gl}_{N})$. However, the two algebras are closely related. For
instance, we can extend $U_{q}(\mathfrak{gl}_{N})$ in a similar manner to the
construction of $\check{U}_{q}(\mathfrak{sl}_{N})$ so that the resulting
algebra contains both $\check{U}_{q}(\mathfrak{sl}_{N})$ and
$U_{q}(\mathfrak{gl}_{N})$. To do this,define an algebra
$\check{U}_{q}(\mathfrak{gl}_{N})$ generated by $U_{q}(\mathfrak{gl}_{N})$ and
the Laurent polynomial ring $\mathbb{C}(q)[K_{\hat{\omega}_{N}/N}^{\pm 1}]$
subject to the additional relations

  * •

$(K_{\hat{\omega}_{N}/N})^{N}=K_{\hat{\omega}_{N}}$

  * •

$K_{\hat{\omega}_{N}/N}$ commutes with all elements of
$U_{q}(\mathfrak{gl}_{N})$ and thus is in the center of
$\check{U}_{q}(\mathfrak{gl}_{N})$.

The algebra $\check{U}_{q}(\mathfrak{gl}_{N})$ can be given a Hopf structure
by insisting that $K_{\hat{\omega}_{N}/N}$ satisfies the same formulas for
coproduct, counit, and antipode as an element $K\in U_{q}(\mathfrak{gl}_{N})$
(as given in Section 2.2). By (1), it follows that
$\check{U}_{q}(\mathfrak{sl}_{N})$ can be identified with a subalgebra of
$\check{U}_{q}(\mathfrak{gl}_{N})$. Though much of the discussion below
focuses on either $\check{U}_{q}(\mathfrak{sl}_{N})$ or
$U_{q}(\mathfrak{gl}_{N})$, it will sometimes be useful to pass to the larger
algebra $\check{U}_{q}(\mathfrak{gl}_{N})$.

Since $K_{\hat{\omega}_{N}}$ is in the center of $U_{q}(\mathfrak{gl}_{N})$,
and, similarly, $K_{\hat{\omega}_{N}/N}$ is in the center of
$\check{U}_{q}(\mathfrak{gl}_{N})$, we have

(26) $\displaystyle({\rm ad}\ U_{q}(\mathfrak{sl}_{N}))\cdot
uK_{\hat{\omega}_{N}}^{s}=[({\rm ad}\ U_{q}(\mathfrak{sl}_{N}))\cdot
u]K_{\hat{\omega}_{N}}^{s}$

for any $s$ with $Ns\in\mathbb{Z}$ and any $u\in U_{q}(\mathfrak{gl}_{N})$.
Consider $\lambda=-\sum_{i}\lambda_{i}\omega_{i}$ and
$\lambda^{\prime}=-\sum_{i}\lambda_{i}\hat{\omega}_{i}$, we have the following
$U_{q}(\mathfrak{sl}_{N})$-module isomorphism

(27) $\displaystyle({\rm ad}\ U_{q}(\mathfrak{sl}_{N}))\cdot
K_{2\lambda}\cong({\rm ad}\ U_{q}(\mathfrak{sl}_{N}))\cdot
K_{2\lambda^{\prime}}$

via the map sending $({\rm ad}\ a)\cdot K_{2\lambda}$ to $({\rm ad}\ a)\cdot
K_{2\lambda^{\prime}}$ for all $a\in U_{q}(\mathfrak{sl}_{N})$.

Note that $w_{0}\Lambda^{+}_{N}$ is equal to the $\mathbb{N}$ linear
combination of the weights
$w_{0}\hat{\omega}_{i}=\epsilon_{N-i+1}+\cdots+\epsilon_{N}$, $i=1,\dots,N-1$.
It follows from (25) and (27) that

$\displaystyle\mathcal{F}(\check{U}_{q}(\mathfrak{sl}_{N}))\cong\bigoplus_{\lambda\in
w_{0}\Lambda^{+}_{N}}({\rm ad}\ U_{q}(\mathfrak{sl}_{N}))\cdot
K_{2\lambda}\subseteq\bigoplus_{\lambda\in
w_{0}\Lambda^{+}_{N},r\in\mathbb{Z}}({\rm ad}\ U_{q}(\mathfrak{sl}_{N}))\cdot
K_{2\lambda+r\hat{\omega}_{N}}\subseteq\mathcal{F}(U_{q}(\mathfrak{gl}_{N})).$

Note that the above equalities, isomorphisms, and inequalities all hold when
$({\rm ad}\ U_{q}(\mathfrak{sl}_{N}))$ is replaced by $({\rm ad}\
U_{q}(\mathfrak{gl}_{N}))$, $({\rm ad}\ \check{U}_{q}(\mathfrak{sl}_{N}))$ or
$({\rm ad}\ \check{U}_{q}(\mathfrak{gl}_{N}))$. This is because the adjoint
action of additional elements of the form $K_{\mu}$ in these Hopf algebras is
semisimple with the same eigenspaces as that of the original Cartan subalgebra
of $U_{q}(\mathfrak{sl}_{N})$. Thus the action of these extra elements
preserve the decomposition into $({\rm ad}\
U_{q}(\mathfrak{sl}_{N}))$-modules.

Recall [7] that $U_{q}(\mathfrak{sl}_{N})$ admits an $({\rm ad}\
U_{q}(\mathfrak{sl}_{N}))$ filtration $\mathcal{G}$ defined by the degree
function $\deg_{\mathcal{G}}(K_{i})=-1$, $\deg_{\mathcal{G}}(F_{i})=1$, and
$\deg_{\mathcal{G}}(E_{i})=0$ for $i=1,\dots,N-1$. Moreover this extends to a
filtration on $\check{U}_{q}(\mathfrak{sl}_{N})$ by setting
$\deg_{\mathcal{G}}K_{\beta}=m$ where
$\beta=m_{1}\alpha_{1}+\cdots+m_{N-1}\alpha_{N-1}$ and
$m=m_{1}+\cdots+m_{N-1}$. This filtration can also be extended to
$U_{q}(\mathfrak{gl}_{N})$ by setting
$\deg_{\mathcal{G}}K_{\epsilon_{i}}=N-i$. Indeed, this extension restricts to
the desired degree function on $U_{q}(\mathfrak{sl}_{N})$ since
$K_{i}=K_{\epsilon_{i}}K_{\epsilon_{i+1}}^{-1}$ and
$\deg_{\mathcal{G}}K_{\epsilon_{i}}-\deg_{\mathcal{G}}K_{\epsilon_{i+1}}=N-i-(N-i+1)=-1$.
Note that on passage to the graded setting, the relations stay the same except
for the ones relating $E_{i}$ and $F_{i}$. Indeed, we have

$\displaystyle E_{i}F_{i}-F_{i}E_{i}=(q-q^{-1})^{-1}K_{i}^{-1}+{\rm lower\
degree\ terms}.$

Let $G^{-}$ be the subalgebra of $U_{q}(\mathfrak{sl}_{N})$ generated by the
$F_{i}K_{i},i=1,\dots,N-1$. Note that all elements of $U^{+}$ and all elements
of $G^{-}$ have degree $0$ with respect to $\mathcal{G}$. As in [7] and
Chapter 7 of [5] ${\rm gr}_{\mathcal{G}}U^{+}G^{-}K_{\beta}$ is an $({\rm ad}\
U_{q}(\mathfrak{sl}_{N}))$-submodule of ${\rm
gr}_{\mathcal{G}}(\check{U}_{q}(\mathfrak{sl}_{N}))$ (resp. ${\rm
gr}_{\mathcal{G}}U_{q}(\mathfrak{gl}_{N})$) for all $\beta\in P^{+}$ (resp.
$\beta\in\sum_{i=1}^{N}\mathbb{Z}\epsilon_{i}$). Moreover,

${\rm gr}_{\mathcal{G}}U^{+}G^{-}K_{\beta}\ \cap\ {\rm
gr}_{\mathcal{G}}U^{+}G^{-}K_{\beta^{\prime}}=0$

for $\beta\neq\beta^{\prime}$. Hence both ${\rm
gr}_{\mathcal{G}}(\check{U}_{q}(\mathfrak{sl}_{N}))$ and ${\rm
gr}_{\mathcal{G}}U_{q}(\mathfrak{gl}_{N})$ can be written as a direct sum of
modules with respect to the adjoint action of the form ${\rm
gr}_{\mathcal{G}}U^{+}G^{-}K_{\beta}$.

Write $\mathbb{Z}_{odd}$ for the set of odd integers.

######  Theorem 7.1.

The locally finite subalgebra of $U_{q}(\mathfrak{gl}_{N})$ admits the
following decomposition into a direct sum of $({\rm ad}\