# Coordinate rings on symmetric spaces

Huanchen Bao  Department of Mathematics, National University of Singapore,
Singapore<EMAIL_ADDRESS>and  Jinfeng Song  Department of Mathematics,
National University of Singapore, Singapore<EMAIL_ADDRESS>
###### Abstract.

Let $G_{k}$ be a connected reductive group over an algebraically closed field
$k$ of char $\neq 2$. Let $\theta_{k}$ be an algebraic group involution of
$G_{k}$ and denote the fixed point subgroup by $K_{k}$. We construct an
integral model for the symmetric space $K_{k}\backslash G_{k}$ with a natural
action of the Chevalley group scheme over integers. We show the coordinate
ring $k[K_{k}\backslash G_{k}]$ admits a canonical basis, as well as a good
filtration as a $G_{k}$-module. We also construct a canonical basis and an
integral form for the space of $K_{k}$-biinvariant functions on $k[G_{k}]$.
Our results rely on the construction of quantized coordinate algebras of
symmetric spaces, using the theory of canonical bases on quantum symmetric
pairs.

##  1\. Introduction

### 1.1.

Let $G_{k}$ be a connected reductive group over an algebraically closed field
$k$. We denote the coordinate ring of $G_{k}$ by $k[G_{k}]$. It is known
[Lu07] that $k[G_{k}]$ admits an integral form ${}_{\mathbb{Z}}\mathbf{O}$
that defines the Chevalley group scheme $G_{\mathbb{Z}}$ over $\mathbb{Z}$
parameterizing split reductive groups of the given type. Lusztig [Lu07] gave a
construction of the ring ${}_{\mathbb{Z}}\mathbf{O}$ using his theory of
canonical bases on quantum groups.

Assume char $k\neq 2$ and let $\theta_{k}$ be an algebraic group involution of
the reductive group $G_{k}$. The classification of such involutions is
independent of the characteristic of $k$ (provided $\neq 2$) by Springer
[Spr87]. We denote the fixed point subgroup by $K_{k}$. In our previous work
[BS22], we have constructed the symmetric subgroup scheme over $\mathbb{Z}$
parameterizing such symmetric subgroups.

We study the symmetric space $K_{k}\backslash G_{k}$ in this paper. Let
$k[K_{k}\backslash G_{k}]$ be the coordinate ring. The first main result of
this paper is the construction of an integral form of $k[K_{k}\backslash
G_{k}]$, hence an integral model parameterizing the symmetric spaces
$K_{k}\backslash G_{k}$ for various algebraically closed field $k$.

######  Theorem 1 (Theorem 4.4 & Theorem 4.8).

There exists a commutative ring ${}_{\mathbb{Z}}\mathbf{O}(K\backslash
G)\subset{}_{\mathbb{Z}}\mathbf{O}$ over $\mathbb{Z}$, such that
${}_{k}\mathbf{O}(K\backslash
G)=k\otimes_{\mathbb{Z}}{}_{\mathbb{Z}}\mathbf{O}(K\backslash G)\cong
k[K_{k}\backslash G_{k}]$ for any algebraically closed field of char $\neq 2$.

There exists a natural basis $B(K\backslash G)$ on
${}_{\mathbb{Z}}\mathbf{O}(K\backslash G)$ which we call the canonical basis.

We actually construct a quantization ${}_{\mathcal{A}}\mathbf{O}(K\backslash
G)$ of the coordinate ring ${}_{\mathbb{Z}}\mathbf{O}(K\backslash G)$ over the
ring $\mathcal{A}=\mathbb{Z}[q,q^{-1}]$, where
${}_{\mathbb{Z}}\mathbf{O}(K\backslash G)$ is obtained via a specialization of
$q$ at $1$.

The desire to construct quantum homogeneous spaces was the original motivation
to study quantum symmetric pairs [Noumi, NS]. However, the existing
construction of quantum homogeneous spaces was entirely over the rational
field $\mathbb{Q}(q)$; cf. [Le03]. So it can not be specialized even to
recover the $q=1$ classical case. In order to construct the
$\mathcal{A}$-form, one is forced to utilize the theory of canonical bases on
quantum symmetric pairs developed by the first author and Wang in [BW18a].

### 1.2.

It is a classical result that the coordinate ring $k[G_{k}]$ admits a direct
sum decomposition $k[G_{k}]\cong\bigoplus_{\lambda\in
X^{+}}{}_{k}V(\lambda)^{*}\otimes{}_{k}V(-w_{0}\lambda)^{*}$ as $G_{k}\times
G_{k}$-modules if $k$ is of characteristic $0$. In positive characteristics,
the coordinate ring $k[G_{k}]$ is not semisimple anymore. In this case, we
instead have a filtration of $k[G_{k}]$ by $G_{k}\times G_{k}$-submodules
indexed by dominate weights in $X^{+}$, such that
$k[G_{k}]_{\leq\lambda}/k[G_{k}]_{<\lambda}\cong{}_{k}V(\lambda)^{*}\otimes{}_{k}V(-w_{0}\lambda)^{*}$
for any $\lambda\in X^{+}$. This is often referred as a good filtration
[Jan03]*§4.20 of $k[G_{k}]$.

In light of Lusztig’s construction of the coordinate ring $k[G_{k}]$ using
quantum groups, the good filtration on $k[G_{k}]$ can then be obtained via
dualizing the cell filtration [Lu94]*Chapter 29 on quantum groups.

Assume again char $k\neq 2$. The coordinate ring $k[K_{k}\backslash G_{k}]$ of
the symmetric space $K_{k}\backslash G_{k}$ is naturally a $G_{k}$-module. If
$k$ is of characteristic $0$, it follows from the direct sum decomposition of
$k[G_{k}]$ that $k[K_{k}\backslash G_{k}]\cong\bigoplus_{\lambda\in
X_{\imath}^{+}}{}_{k}V(-w_{0}\lambda)^{*}$ as a $G_{k}$-module. Here
$X_{\imath}^{+}$ is a subset of $X^{+}$ consisting of spherical dominant
weights (See §2.2.3). Little is known for positive characteristics.

We state the second main result of this paper.

######  Theorem 2 (Theorem 4.9).

The $G_{k}$-module $k[K_{k}\backslash G_{k}]$ admits a good filtration indexed
by $X_{\imath}^{+}$, such that

$k[K_{k}\backslash G_{k}]_{\leq\lambda}/k[K_{k}\backslash
G_{k}]_{<\lambda}\cong{}_{k}V(-w_{0}\lambda)^{*},\quad\text{for any
}\lambda\in X_{\imath}^{+}.$

### 1.3.

The construction of both the commutative ring
${}_{\mathbb{Z}}\mathbf{O}(K\backslash G)$ (or the quantization
${}_{\mathcal{A}}\mathbf{O}(K\backslash G)$) and the good filtration of
$k[K_{k}\backslash G_{k}]$ relys on the study of based modules of
$\imath$quantum groups.

Let $(\mathrm{U},\mathrm{U}^{\imath})$ be a quantum symmetric pair [BW18a]
associated with the symmetric pair $(G_{k},K_{k})$. The subalgebra
$\mathrm{U}^{\imath}\subset\mathrm{U}$ of the quantum group $\mathrm{U}$ is
refered as the $\imath$quantum group.

We now state the third main result of this paper on based
$\mathrm{U}^{\imath}$-modules, which is the technical foundation for both
Theorem 1 and Theorem 2.

######  Theorem 3 (Theorem 3.1).

Let $M$ be a finite-dimensional based $\mathrm{U}$-module. Let
$\mathrm{U}^{\imath,+}$ be the augmentation ideal of $\mathrm{U}^{\imath}$.
Then $M_{+}=\mathrm{U}^{\imath,+}M$ is a based $\mathrm{U}^{\imath}$-submodule
of $M$, and the coinvariant space $M/M_{+}$ is naturally a based
$\mathrm{U}^{\imath}$-module.

Based modules and based morphisms are well-behaved with respect to
specilizations. We are then able to deduce classical results on symmetric
spaces via specializations.

### 1.4.

In a subsequent paper, we shall apply results in this paper to construct the
Vinberg enveloping variety of the symmetric space $K_{k}\backslash G_{k}$ and
an integral model of the wonderful compactification of $K_{k}\backslash
G_{k}$.

Acknowledgement:  Both authors are supported by MOE grants A-0004586-00-00 and
A-0004586-01-00.

##  2\. Preliminaries

###  2.1. Quantum groups

We recall quantum groups following Lusztig’s book [Lu94].

####  2.1.1. Definitions

Let $(I,Y,X,A,(\alpha_{i})_{i\in I},(\alpha^{\vee}_{i})_{i\in I})$ be a root
datum of finite type; cf. [BS22]*§2.1. Let $D=diag(\varepsilon_{i})_{i\in I}$
be the diagonal matrix such that $DA$ is a symmetric matrix, with
$\epsilon_{i}\in\mathbb{\mathbb{Z}}_{>0}$ and $\\{\varepsilon_{i}\mid i\in
I\\}$ are relatively prime. Let $W=\langle s_{i}\mid i\in I\rangle$ be the
Weyl group associated with the root datum with the longest element denoted by
$w_{0}$. For $\lambda,\mu\in X$, we write $\lambda\leq\mu$ if
$\mu-\lambda\in\sum_{i\in I}\mathbb{Z}_{\geq 0}\alpha_{i}$. Let
$X^{+}=\\{\mu\in X\mid\langle\alpha_{i}^{\vee},\mu\rangle\geq 0,\forall i\in
I\\}$ be the set of _dominant weights_.

Let $q$ be an indeterminate. We write $\mathcal{A}=\mathbb{Z}[q,q^{-1}]$ to be
the subring of $\mathbb{Q}(q)$. Let $\mathrm{U}$ be the Drinfeld–Jimbo quantum
group associated with the root datum; cf. [Lu94]*§3.1. Recall that
$\mathrm{U}$ is an associated $\mathbb{Q}(q)$-algebra with 1. It has
generators $E_{i}$ $(i\in I)$, $F_{i}$ $(i\in I)$ and $K_{\mu}$ $(\mu\in Y)$.
The algebra $\mathrm{U}$ has a structure of a Hopf algebra. Let
$\varepsilon:\mathrm{U}\rightarrow\mathbb{Q}(q)$ be the counit, and
$\Delta:\mathrm{U}\rightarrow\mathrm{U}\otimes\mathrm{U}$ be the coproduct;
cf. [Lu94]*Lemma 3.1.4 & §3.1.11.

Following [Lu94]*§3.1.3, let $\sigma:\mathrm{U}\rightarrow\mathrm{U}$ be the
algebra anti-automorphism such that $\sigma(E_{i})=E_{i}$,
$\sigma(F_{i})=F_{i}$, for $i\in I$, and $\sigma(K_{\mu})=K_{-\mu}$, for
$\mu\in Y$, and let $\omega:\mathrm{U}\rightarrow\mathrm{U}$ be the algebra
automorphism such that $\omega(E_{i})=F_{i}$, $\omega(F_{i})=E_{i}$, for $i\in
I$, and $\omega(K_{\mu})=K_{-\mu}$, for $\mu\in Y$.

For any $\lambda\in X^{+}$, we denote by $V(\lambda)$ the highest weight
simple $\mathrm{U}$-module with highest weight $\lambda$. We denote the
highest weight vector by $v^{+}_{\lambda}$ and the canonical basis of
$V(\lambda)$ by $\mathrm{B}(\lambda)$.

####  2.1.2. Based modules

Let $\mathbf{A}=\mathbb{Q}(q)\cap\mathbb{Q}[[q^{-1}]]$ be the ring of rational
functions which are regular at $q=\infty$. Let $(M,B)$ be a finite-dimensional
based $\mathrm{U}$-module over $\mathbb{Q}(q)$; cf. [Lu94]*§27. We denote by
${}_{\mathcal{A}}M$ (resp., $L(M)$) the $\mathcal{A}$-submodule (resp.,
$\mathbf{A}$-submodule) of $M$ spanned by $B$. For any commutative
$\mathcal{A}$-algebra $R$, we write
${}_{R}M=R\otimes_{\mathcal{A}}{}_{\mathcal{A}}M$ and still denote the basis
by $B$. We often consider a commutative ring $R$ as an $\mathcal{A}$-algebra
via the ring homomorphism $\mathcal{A}\rightarrow R$, $q\mapsto 1$ unless
specified otherwise.

Following [Lu94]*Proposition 27.1.8, for any $\lambda\in X^{+}$, we have the
based submodule $M[\geq\lambda]\subset M$ (resp., $M[>\lambda]\subset M$) with
basis $B[\geq\lambda]=B\cap M[\geq\lambda]$ (resp., $B[>\lambda]=B\cap
M[>\lambda]$). We set $B[\lambda]=B[\geq\lambda]-B[>\lambda]$. Let
$B[\lambda]^{hi}$ be the subset $B[\lambda]$ defined in [Lu94]*§27.2.3.

For $\lambda\in X^{+}$, let $\pi_{\lambda}:M\rightarrow M/M[\geq\lambda]$ be
the canonical projection map. Then $M/M[\geq\lambda]$ is naturally a based
$\mathrm{U}$-module with a basis consisting of the image of $B-B[\geq\lambda]$
under the map $\pi_{\lambda}$; see [Lu94]*§27.1.4. In particular, the map
$\pi_{\lambda}:M\rightarrow M/M[\geq\lambda]$ is based in the sense of
[Lu94]*§27.1.3. We often abuse notations and denote by $B-B[\geq\lambda]$ the
basis of $M/M[\geq\lambda]$.

For any $\lambda\in X^{+}$, the subquotient $M[\geq\lambda]/M[>\lambda]$ is
isomorphic to the based module
$(\bigoplus_{i=1}^{n_{\lambda}}V(\lambda),B^{\prime})$, where $B^{\prime}$ is
given by the union of the canonical bases of the various summands $V(\lambda)$
and $n_{\lambda}$ is the cardinality of the subset $B[\lambda]^{hi}\subset B$.

We conclude that

(a) the based module $(M,B)$ admits a based filtration

$0=M[\geq\lambda_{0}]\subset M[\geq\lambda_{1}]\subset
M[\geq\lambda_{2}]\subset\cdots\subset M[\geq\lambda_{n}]=M,\quad\text{for
}\lambda_{i}\in X^{+},$

such that, for $1\leq i\leq n$, the map
$\phi_{\lambda_{i}}:M[\geq\lambda_{i}]/M[\geq\lambda_{i-1}]\overset{\sim}{\rightarrow}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}V(\lambda_{i})$,
$\bar{b}\mapsto(v_{\lambda_{i}}^{+})_{b}$, for $b\in B[\lambda_{i}]^{hi}$,
where $(v_{\lambda_{i}}^{+})_{b}$ is the highest vector of the copy
corresponding to $b$ of $V(\lambda_{i})$, defines an isomorphism of based
$\mathrm{U}$-modules.

The following lemma follows from [Ka91]*lemma 2.6.3.

######  Lemma 2.1.

Let $(M,B)$ be a finite-dimensional based $\mathrm{U}$-module. We have an
isomorphism of $\mathrm{U}$-modules

$\phi:M\longrightarrow\bigoplus_{i=1}^{n}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}V(\lambda_{i}),$

such that (for any $1\leq j\leq n$),

  1. (1)

$\phi$ restricts to an isomorphism
$M[\geq\lambda_{j}]\longrightarrow\bigoplus_{i=1}^{j}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}V(\lambda_{i})$;

  2. (2)

the induced morphism on $M[\geq\lambda_{j}]/M[\geq\lambda_{j-1}]$ agrees with
$\phi_{\lambda_{j}}$ in §2.1.2 (a);

  3. (3)

for any $b\in B[\lambda_{j}]$, we have
$\phi(b)\in\phi_{\lambda_{j}}(b)+q^{-1}\phi(L(M[\geq\lambda_{j-1}]))$;

  4. (4)

$\phi$ restricts to isomorphisms as $\mathbf{A}$-modules
$L(M[\geq\lambda_{j}])\longrightarrow\bigoplus_{i=1}^{j}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}L(V(\lambda_{i}))$.

####  2.1.3. The modified form

Let $\dot{\mathrm{U}}$ be the modified quantum group [Lu94]*§23.1 and
$\dot{\mathrm{B}}$ be its canonical basis ([Lu94]*§25.2.1). The algebra
(without unit) $\dot{\mathrm{U}}$ is a $(\mathrm{U},\mathrm{U})$-bimodule in a
natural way. Let ${{}_{\mathcal{A}}\dot{\mathrm{U}}}$ be the free
$\mathcal{A}$-submodule of $\dot{\mathrm{U}}$ with basis $\dot{\mathrm{B}}$. A
unital $\dot{\mathrm{U}}$-module is equivalent to a $\mathrm{U}$-module with a
direct sum decomposition into weight spaces by [Lu94]*§23.1.4. These are the
only modules we shall consider in this paper.

Lusztig [Lu94]*§29.1 defined a partition of the canonical basis
$\dot{\mathrm{B}}=\sqcup_{\lambda\in X^{+}}\dot{\mathrm{B}}[\lambda]$. For
$\lambda\in X^{+}$, let $\dot{\mathrm{U}}[\not\leq\lambda]$ (resp.,
${}_{\mathcal{A}}\dot{\mathrm{U}}[\not\leq\lambda]$) be the
$\mathbb{Q}(q)$-subspace (resp., $\mathcal{A}$-submodule) of
$\dot{\mathrm{U}}$ spanned by
$\sqcup_{\lambda^{\prime}\not\leq\lambda}\dot{\mathrm{B}}[\lambda^{\prime}]$.
Then $\dot{\mathrm{U}}[\not\leq\lambda]$ (resp.,
${}_{\mathcal{A}}\dot{\mathrm{U}}[\not\leq\lambda]$) is a two-sided ideal of
$\dot{\mathrm{U}}$ (resp., ${}_{\mathcal{A}}\dot{\mathrm{U}}$).

The quotient $\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]$ is a based
$\mathrm{U}$-module with the canonical basis consisting of the image of
$\dot{\mathrm{B}}[\leq\lambda]=\sqcup_{\lambda^{\prime}\leq\lambda}\dot{\mathrm{B}}[\lambda^{\prime}]$.
We abuse notations, and denote the quotient based module by
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda],\dot{\mathrm{B}}[\leq\lambda])$.
Similarly we define the subspace $\dot{\mathrm{U}}[<\lambda]$, the
$\mathcal{A}$-submodule ${}_{\mathcal{A}}\dot{\mathrm{U}}[<\lambda]$, and the
quotient based module
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not<\lambda],\dot{\mathrm{B}}[<\lambda])$.

####  2.1.4. Coordinate rings

Let $\mathbf{O}$ be the $\mathbb{Q}(q)$-vector space of all
$\mathbb{Q}(q)$-linear forms $f:\dot{\mathrm{U}}\rightarrow\mathbb{Q}(q)$ with
the following property: $f$ vanishes on $\dot{\mathrm{U}}[\not\leq\lambda]$
for some $\lambda\in X^{+}$. For any $a\in\dot{\mathrm{B}}$, we define the
linear form $\tilde{a}:\dot{\mathrm{U}}\rightarrow\mathbb{Q}(q)$, given by
$\tilde{a}(a^{\prime})=\delta_{a,a^{\prime}}$, for
$a^{\prime}\in\dot{\mathrm{B}}$. The set
$\\{\tilde{a}|a\in\dot{\mathrm{B}}\\}$ is a $\mathbb{Q}(q)$-basis of
$\mathbf{O}$, called the _canonical basis_ of $\mathbf{O}$. Note that elements
in $\mathbf{O}$ can be naturally viewed as linear forms of the completion
space $\widehat{\mathrm{U}}$, where $\widehat{\mathrm{U}}$ is the
$\mathbb{Q}(q)$-vector space consisting of formal sums of
$\mathbb{Q}(q)$-linear combinations of $\dot{\mathrm{B}}$.

Let ${}_{\mathcal{A}}\mathbf{O}$ be the $\mathcal{A}$-submodule of
$\mathbf{O}$, spanned by the canonical basis. Then by [Lu94]*§29.5.2,
${}_{\mathcal{A}}\mathbf{O}$ is a Hopf algebra over $\mathcal{A}$. For any
commutative $\mathcal{A}$-algebra $R$ with $1$, we write
${}_{R}\mathbf{O}=R\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}$. Unless
specified otherwise, we often consider a commutative ring $R$ as an
$\mathcal{A}$-algebra via the ring homomorphism $\mathcal{A}\rightarrow R$,
$q\mapsto 1$.

######  Theorem 2.2.

[Lu07] The group scheme $Spec\,{}_{\mathbb{Z}}\mathbf{O}$ is isomorphic to the
Chevalley group scheme $\mathbf{G}_{\mathbb{Z}}$ over $\mathbb{Z}$ associated
with the given root datum.

In particular, the ring ${}_{k}\mathbf{O}$ is canonically isomorphic to the
coordinate ring $k[G_{k}]$ of the connected reductive algebraic group $G_{k}$
associated with the given root datum, for any algebraically closed field $k$.

We next recall the Peter–Weyl filtration of $\mathbf{O}$ and
${}_{\mathcal{A}}\mathbf{O}$. For $\lambda\in X^{+}$, we define

$\mathbf{O}_{\leq\lambda}={\rm
Hom}_{\mathbb{Q}(q)}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda],\mathbb{Q}(q)),\quad{}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}=\mathbf{O}_{\leq\lambda}\cap{}_{\mathcal{A}}\mathbf{O}=\text{Hom}_{\mathcal{A}}\big{(}{}_{\mathcal{A}}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]),\mathcal{A}\big{)}.$

Then $\mathbf{O}_{\leq\lambda}$ (resp.,
${}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}$) is the $\mathbb{Q}(q)$-span
(resp., $\mathcal{A}$-span) of
$\\{\tilde{a}|a\in\dot{\mathrm{B}}[\leq\lambda]\\}$. We define
$\mathbf{O}_{<\lambda}$ and ${}_{\mathcal{A}}\mathbf{O}_{<\lambda}$ in the
similar way.

We then have

$\mathbf{O}=\cup_{\lambda\in
X^{+}}\mathbf{O}_{\leq\lambda},\quad{}_{\mathcal{A}}\mathbf{O}=\cup_{\lambda\in
X^{+}}{}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}.$

For an algebraically closed field $k$ and $\lambda\in X^{+}$, set
${}_{k}\mathbf{O}_{\leq\lambda}=k\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}$.
For any $\lambda\leq\lambda^{\prime}$, since the inclusion
${}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}\rightarrow{}_{\mathcal{A}}\mathbf{O}_{\leq\lambda^{\prime}}$
is compatible with bases, we have an inclusion
${}_{k}\mathbf{O}_{\leq\lambda}\rightarrow{}_{k}\mathbf{O}_{\leq\lambda^{\prime}}$
as well. Therefore we have
${}_{k}\mathbf{O}_{\leq\lambda}\subset{}_{k}\mathbf{O}$ for any $\lambda\in
X^{+}$ and ${}_{k}\mathbf{O}=\cup_{\lambda\in
X^{+}}{}_{k}\mathbf{O}_{\leq\lambda}$.

Let $G_{k}$ be the connected reductive group as in Theorem 2.2. We define
$G_{k}\times G_{k}$-module structure on ${}_{k}\mathbf{O}\cong k[G_{k}]$ as
follows

$((g_{1},g_{2})\cdot f)(x)=f(g_{1}^{-1}xg_{2}),\text{ for }(g_{1},g_{2})\in
G_{k}\times G_{k},f\in k[G_{k}],x\in G_{k}.$

Let ${}_{k}V(\lambda)^{*}$ be the dual Weyl module for any $\lambda\in X^{+}$.
Dualizing the construction in [Lu94]*§29.3, we obtain the following classical
Peter–Weyl filtration; cf. [Jan03]*Proposition 4.20.

######  Theorem 2.3.

For any $\lambda\in X^{+}$, the subspaces ${}_{k}\mathbf{O}_{\leq\lambda}$ and
${}_{k}\mathbf{O}_{<\lambda}$ are $G_{k}\times G_{k}$-submodules of
${}_{k}\mathbf{O}$ such that

${}_{k}\mathbf{O}_{\leq\lambda}/_{k}\mathbf{O}_{<\lambda}\cong{}_{k}V(\lambda)^{*}\otimes_{k}{}_{k}V(-w_{0}\lambda)^{*}.$

In particular, the $G_{k}\times G_{k}$-module ${}_{k}\mathbf{O}$ admits a good
filtration.

###  2.2. Quantum symmetric pairs

####  2.2.1. $\imath$Quantum groups

Let $(I,Y,X,A,(\alpha_{i})_{i\in I},(\alpha^{\vee}_{i})_{i\in
I}),I_{\bullet},\tau)$ be an $\imath$root datum; cf. [BS22]*§2.3.1. Let
$W_{I_{\bullet}}=\langle s_{i}\mid i\in I_{\bullet}\rangle$ be the parabolic
subgroup associated with $I_{\bullet}$, and let $w_{\bullet}$ be the longest
element of $W_{I_{\bullet}}$. Denote by $\theta=-w_{\bullet}\circ\tau$ the
involution on both $X$ and $Y$, by slightly abuse of notation. Set

$X_{\imath}=X/\langle\lambda-\theta\lambda\mid\lambda\in
X\rangle\qquad\text{and}\qquad Y^{\imath}=\\{\mu\in Y\mid\theta\mu=\mu\\}.$

We call $X_{\imath}$ the _$\imath$ weight lattice_, and call $Y^{\imath}$ the
_$\imath$ coweight lattice_. There is a natural pairing $Y^{\imath}\times
X_{\imath}\rightarrow\mathbb{Z}$ inherited from the pairing between $Y$ and
$X$. Note that this pairing is not necessarily perfect. For any $\lambda\in
X$, we write $\overline{\lambda}$ to denote its image in $X_{\imath}$.

For each $i\in I_{\circ}=I-I_{\bullet}$, we take a parameter
$\varsigma_{i}\in\pm q^{\mathbb{Z}}$. Let
$\mathrm{U}^{\imath}=\mathrm{U}^{\imath}_{\mathbf{\varsigma}}\subset\mathrm{U}$
be the $\imath$quantum group defined in [BW18a]*§3.3. By definition,
$\mathrm{U}^{\imath}$ is the $\mathbb{Q}(q)$-subalgebra of $\mathrm{U}$
generated by elements

$B_{i}=F_{i}+\varsigma_{i}T_{w_{\bullet}}(E_{\tau
i})\tilde{K}_{i}^{-1}\quad(i\in I_{\circ}),\qquad F_{i}\quad(i\in
I_{\bullet}),\qquad E_{i}\quad(i\in I_{\bullet}),\qquad K_{\mu}\quad(\mu\in
Y^{\imath}).$

We write $B_{i}=F_{i}$, for $i\in I_{\bullet}$. Here $T_{w_{\bullet}}$ denotes
Lusztig’s Braid group operator; cf. [BW18a]*§2.2.

The pair $(\mathrm{U},\mathrm{U}^{\imath})$ is called a quantum symmetric
pair. The subalgebra $\mathrm{U}^{\imath}$ is called an _$\imath$ quantum
group_. In what follows, we assume the parameters $(\varsigma_{i})_{i\in
I_{\circ}}$ are taken as the same as in [Wa23]*Lemma 4.2.1. The only reason we
pick these particular parameters is that the stability conjecture of
$\imath$canonical basis, or equivalently, Theorem 2.4, holds for these
parameters.

Let $\dot{\mathrm{U}}^{\imath}$ be the modified algebra of
$\mathrm{U}^{\imath}$ and let $\dot{\mathrm{B}}^{\imath}$ be the canonical
basis of $\dot{\mathrm{U}}^{\imath}$ defined in [BW18a]*§3.7 & §6.4. Let
${}_{\mathcal{A}}\dot{\mathrm{U}}^{\imath}$ be the $\mathcal{A}$-form of
$\dot{\mathrm{U}}^{\imath}$ [BW18a]*Definition 3.19. For any
$\mathcal{A}$-algebra $R$, set
${}_{R}\dot{\mathrm{U}}^{\imath}=R\otimes_{\mathcal{A}}{{}_{\mathcal{A}}\dot{\mathrm{U}}^{\imath}}$.
A unital $\dot{\mathrm{U}}^{\imath}$-module is equivalent to a
$\mathrm{U}^{\imath}$-module with a direct sum decomposition intro weight
spaces by [BW18a]*§3.7. These are the only modules we shall consider in this
paper.

####  2.2.2. Based $\mathrm{U}^{\imath}$-modules

Let $\psi_{\imath}$ be the anti-linear bar involution on $\mathrm{U}^{\imath}$
as in [BW18a]*Lemma 3.15. Let $(M,B)$ be a finite-dimensional based
$\mathrm{U}$-module. We consider $M$ naturally as a
$\mathrm{U}^{\imath}$-module by restriction. Then by [BW21]*Theorem 6.12, $M$
admits an anti-linear involution $\psi_{\imath}$ such that

$\psi_{\imath}(um)=\psi_{\imath}(u)\psi_{\imath}(m),\quad
u\in\mathrm{U}^{\imath},m\in M.$

Moreover, the $\mathrm{U}^{\imath}$-module $M$ admits a unique basis, called
the _$\imath$ canonical basis_, denoted by $B^{\imath}=\\{b^{\imath}|b\in
B\\}$, such that

  1. (1)

$\psi_{\imath}(b^{\imath})=b^{\imath}$ for any $b^{\imath}$;

  2. (2)

$b^{\imath}=b+\sum_{b^{\prime}\in B}t_{b;b^{\prime}}b^{\prime}$, for
$t_{b;b^{\prime}}\in q^{-1}\mathbb{Z}[q^{-1}]$;

  3. (3)

$B^{\imath}$ forms an $\mathcal{A}$-basis of the $\mathcal{A}$-lattice
${}_{\mathcal{A}}M$, and forms an $\mathbf{A}$-basis of the
$\mathbf{A}$-lattice $L(M)$;

  4. (4)

$(M,B^{\imath})$ is a based $\mathrm{U}^{\imath}$-module in the sense of
[BW21]*Definition 6.11.

In particular, we have the based $\mathrm{U}^{\imath}$-modules
$(V(\lambda),B(\lambda)^{\imath})$ for each $\lambda\in X^{+}$. For any based
$\mathrm{U}$\- (resp., $\mathrm{U}^{\imath}$-) module $M$, we write
${}_{\mathcal{A}}M$ to be the $\mathcal{A}$-submodule of $M$ spanned by the
canonical basis (resp., $\imath$canonical basis).

For any $\lambda\in X^{+}$, the based $\mathrm{U}$-submodule
$M[\geq\lambda]\subset M$ is also a based $\mathrm{U}^{\imath}$-submodule by
[BW21]*Theorem 6.12 (with respect to different bases). Let
$\phi:M\longrightarrow\bigoplus_{i=1}^{n}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}V(\lambda_{i})$ be the isomorphism in Lemma 2.1. We
conclude that

(a) the based $\mathrm{U}$-module $(M,B)$ admits a based filtration of based
$\mathrm{U}^{\imath}$-modules

$0=M[\geq\lambda_{0}]\subset M[\geq\lambda_{1}]\subset
M[\geq\lambda_{2}]\subset\cdots\subset M[\geq\lambda_{n}]=M,\quad\text{for
}\lambda_{i}\in X^{+},$

such that
$\phi_{\lambda_{i}}:M[\geq\lambda_{i}]/M[\geq\lambda_{i-1}]\overset{\sim}{\rightarrow}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}V(\lambda_{i})$ as based $\mathrm{U}^{\imath}$-modules for
$1\leq i\leq n$;

(b) for any $b\in B[\lambda_{j}]$, we have
$\phi(b^{\imath})=\phi_{\lambda_{j}}(b^{\imath})+q^{-1}\phi(L(M[\geq\lambda_{j-1}]))$.

####  2.2.3. Spherical modules

The following theorem was conjectured in [BW18a]*Remark 6.18 and proved in
[Wa23]*Theorem 4.3.1.

We define $X^{+}_{\imath}=\\{\lambda\in
X^{+}|\overline{\lambda}=\overline{0}\in X_{\imath}\\}$.

######  Theorem 2.4.

Let $\lambda\in X^{+}_{\imath}$. Then there exists a unique morphism as based
$\mathrm{U}^{\imath}$-modules

$V(\lambda)\rightarrow V(0),\qquad v^{+}_{\lambda}\mapsto v^{+}_{0}.$

Note that $V(\lambda)$ is generated by $v_{\lambda}^{+}$ as a
$\mathrm{U}^{\imath}$-module. Hence for any $\lambda\in X^{+}$, there exists a
non-zero map $V(\lambda)\rightarrow V(0)$ only if $\lambda\in X^{+}_{\imath}$.
We draw the following conclusions.

(a) For any $\lambda\in X^{+}$, there exists a unique morphism
$V(\lambda)\rightarrow V(0)$ as based $\mathrm{U}^{\imath}$-modules if and
only if $\lambda\in X^{+}_{\imath}$.

(b) For any $\lambda\in X^{+}$, we have

$\dim\text{\rm Hom}_{\mathrm{U}^{\imath}\text{\rm-
mod}}(V(\lambda),V(0))=\begin{cases}1,&\text{if }\lambda\in X^{+}_{\imath};\\\
0,&\text{otherwise}.\end{cases}$

####  2.2.4. Symmetric subgroups

We recall the construction of symmetric subgroups via $\imath$quantum groups
following [BS22]*§3.

Let $k$ be an algebraically closed field with characteristic not 2. Let
$G_{k}$ be a reductive linear algebraic group over $k$, and $\theta_{k}$ be an
algebraic group involution on $G_{k}$. The pair $(G_{k},\theta_{k})$ is then
called a _symmetric pair_. The closed subgroup $K_{k}=G_{k}^{\theta_{k}}$
consisting of $\theta_{k}$-fixed elements in $G_{k}$ is called a _symmetric
subgroup_. By [Spr87], the classification of symmetric pairs is independent of
the characteristic of $k$ (provided not $2$). By [BS22]*§3, one can associate
an $\imath$root datum to such a symmetric pair independent of the
characteristic of $k$ (provided not $2$).

On the other hand, starting with an $\imath$root datum. Let $\mathrm{U}$,
$\mathrm{U}^{\imath}$, $\dot{\mathrm{U}}$, $\dot{\mathrm{U}}^{\imath}$, etc.,
be the various quantum algebras associated with the given datum following §2.1
and §2.2.1. Let $G_{k}$ be the reductive group associated with the underlying
root datum defined over $k$. Set

${}_{k}\dot{\mathrm{U}}=k\otimes_{\mathcal{A}}{}_{\mathcal{A}}\dot{\mathrm{U}}\qquad\text{
and
}\qquad_{k}\dot{\mathrm{U}}^{\imath}=k\otimes_{\mathcal{A}}{}_{\mathcal{A}}\dot{\mathrm{U}}^{\imath}.$

Let ${}_{k}\widehat{\mathrm{U}}$ be the space consisting of formal sums of
$k$-linear combinations of canonical basis elements of
${}_{k}\dot{\mathrm{U}}$ defined in [Lu07]*§1.1. By [BS22]*Lemma 3.1, the
space ${}_{k}\dot{\mathrm{U}}^{\imath}$ is naturally a subspace of
${}_{k}\widehat{\mathrm{U}}$. By [Lu07], one can identify elements in the
coordinate ring $k[G_{k}]$ as $k$-linear forms on of a completion of the space
${}_{k}\dot{\mathrm{U}}$, that is,
$k[G_{k}]\hookrightarrow\text{Hom}_{k}({}_{k}\widehat{\mathrm{U}},k)$. We
define $\mathcal{I}=\\{f\in k[G_{k}]\mid
f({}_{k}\dot{\mathrm{U}}^{\imath})=0\\}$.

######  Theorem 2.5 ([BS22]*Theorem 3.8).

The subspace $\mathcal{I}$ is a reduced ideal of $k[G_{k}]$, defining a
symmetric subgroup $K_{k}=G_{k}^{\theta_{k}}$. The $\imath$root datum
associated with the pair $(G_{k},\theta_{k})$ is isomorphic to the one we
start with.

######  Remark 2.6.

We previously established Theorem 2.5 for quasi-split symmetric pairs in
[BS22], since Theorem 2.4 was only available for the quasi-split cases back
then. Theorem 2.4 was later established by Watanabe [Wa23] for all symmetric
pairs (of finite type). Hence results in [BS22]*§3 hold in this generality as
well; cf. [BS22]*Remark 3.3.

##  3\. Bases for coinvariants

###  3.1. Coinvariant spaces

Recall the counit $\varepsilon$ of $\mathrm{U}$ in §2.1.1. Set
$\mathrm{U}^{\imath,+}=\mathrm{U}^{\imath}\cap\text{ker }\varepsilon$ to be
the augmentation ideal of $\mathrm{U}^{\imath}$. Let $(M,B)$ be a finite-
dimensional based $\mathrm{U}$-module. We then obtain the based
$\mathrm{U}^{\imath}$-module $(M,B^{\imath})$. Let
$M_{+}=\mathrm{U}^{\imath,+}M$ be the $\mathrm{U}^{\imath}$-submodule of $M$.
We define

$B^{\imath}_{\ast}=\\{b^{\imath}\in B^{\imath}|b\in B[\lambda]^{hi}\text{ for
some }\lambda\in X_{\imath}^{+}\\}\subset B^{\imath}.$

######  Theorem 3.1.

The $\mathrm{U}^{\imath}$-submodule $M_{+}=\mathrm{U}^{\imath,+}M$ is a based
$\mathrm{U}^{\imath}$-submodule of $M$ with the basis $\\{b^{\imath}\in
B^{\imath}|b^{\imath}\not\in B^{\imath}_{\ast}\\}$.

In particular, the space of $\mathrm{U}^{\imath}$-coinvariants
$M^{\imath}_{\ast}=M/\mathrm{U}^{\imath,+}M$ is naturally a based
$\mathrm{U}^{\imath}$-module, with the basis given by the image of elements in
$B_{*}^{\imath}$. The natural projection $M\rightarrow M^{\imath}_{\ast}$ maps
${}_{\mathcal{A}}M$ to ${}_{\mathcal{A}}M^{\imath}_{\ast}$.

We abuse notations and denote the based quotient by
$(M^{\imath}_{\ast},B_{*}^{\imath})$, and write
$({}_{\mathcal{A}}M)^{\imath}_{\ast}={}_{\mathcal{A}}M^{\imath}_{\ast}$. We
also write
${}_{R}M^{\imath}_{\ast}=R\otimes_{\mathcal{A}}{}_{\mathcal{A}}M^{\imath}_{\ast}$
for any commutative $\mathcal{A}$-algebra $R$.

###### Proof.

Let $M[\geq\lambda_{1}]\subset M[\geq\lambda_{2}]\subset\cdots\subset
M[\geq\lambda_{n}]$ be the based filtration of $M$ as §2.2.2 (a). We define
$F(M)=\oplus_{b^{\imath}\in B^{\imath}_{\ast}}\mathbb{Q}(q)e_{b^{\imath}}$ to
be the trivial $\mathrm{U}^{\imath}$-module of dimension
$|B^{\imath}_{\ast}|$. By definition, we have $u\cdot x=\varepsilon(u)x$ for
any $u\in\mathrm{U}^{\imath}$ and $x\in F(M)$. It is a based
$\mathrm{U}^{\imath}$-module with basis $\\{e_{b^{\imath}}\mid b^{\imath}\in
B^{\imath}_{*}\\}$.

(a) There is a unique $\mathrm{U}^{\imath}$-module homomorphism
$f:M\rightarrow F(M)$ such that

$f:b^{\imath}\mapsto\begin{cases}e_{b^{\imath}},&\text{if }b\in
B[\lambda_{i}]^{hi}\text{ with }\lambda_{i}\in X^{+}_{\imath};\\\ 0,&\text{if
}b\in B[\lambda_{i}]^{hi}\text{ with }\lambda_{i}\notin
X^{+}_{\imath}.\end{cases}$

By [BW18a]*Lemma 6.2 and an straightforward induction on $n$, we see that $M$
is generated as an $\mathrm{U}^{\imath}$-module by $b^{\imath}$ for various
$b\in B[\lambda_{i}]^{hi}$ with $1\leq i\leq n$. This shows the uniqueness. It
also follows that the integral form ${}_{\mathcal{A}}M$ is generated by
$b^{\imath}$ for various $b\in B[\lambda_{i}]^{hi}$ with $1\leq i\leq n$ as an
${}_{\mathcal{A}}\dot{\mathrm{U}}^{\imath}$-module.

We next show the existence. We firstly define $\mathrm{U}^{\imath}$-module
homomorphisms $f_{i}:M\rightarrow F(M)$ for $1\leq i\leq n$ as follows: if
$\overline{\lambda_{i}}\neq\overline{0}\in X_{\imath}$, we define $f_{i}$ as
the zero map; if $\overline{\lambda_{i}}=\overline{0}\in X_{\imath}$, we
define $f_{i}$ as the composition

$f_{i}:M\xrightarrow{\phi}\bigoplus_{j=1}^{n}\bigoplus_{b\in
B[\lambda_{j}]^{hi}}V(\lambda_{j})\xrightarrow{p}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}V(\lambda_{i})\xrightarrow{\pi}\bigoplus_{b\in
B[\lambda_{i}]^{hi}}\mathbb{Q}(q)e_{b^{\imath}}\subset F(M),$ (3.1)

where $\phi$ is the isomorphism from Lemma 2.1, $p$ is the canonical
projection, and $\pi$ is direct sum of the based $\mathrm{U}$-module morphisms
defined in Theorem 2.4.

Let $1\leq i,j\leq n$ with $\overline{\lambda_{i}}=\overline{0}\in X_{\imath}$
and $b\in B[\lambda_{j}]$. By Lemma 2.1, we have

$f_{i}(b^{\imath})=\begin{cases}0,&\text{if }j<i;\\\ e_{b^{\imath}},&\text{if
}j=i\text{ and }b\in B[\lambda_{j}]^{hi};\\\ 0,&\text{if }j=i\text{ and
}b\not\in B[\lambda_{j}]^{hi};\\\ c_{b^{\imath}}e_{b^{\imath}},&\text{if
}i<j,\text{ for some }c_{b^{\imath}}\in q^{-1}\mathbb{Z}[q^{-1}].\end{cases}$
($\heartsuit$)

Hence a suitable $\mathbb{Z}[q^{-1}]$-linear combination of $f_{i}$’s will be
the desired map $f$.

(b) The map $f$ commutes with the bar involution $\psi_{\imath}$, that is,
$f(\psi_{\imath}(x))=\psi_{\imath}(f(x))$ for any $x\in M$.

Since $\\{b^{\imath}|b\in B[\lambda_{i}]^{hi}\\}$ is a set of generators of
$M$ as a $\mathrm{U}^{\imath}$-module, the claim follows immediately from the
fact that they are $\psi_{\imath}$-invariant.

(c) The map $f$ preserves the $\mathbf{A}$-lattice, that is,
$f(L(M))=L(f(M))$.

Since all three maps $\phi$, $p$, $\pi$ in (3.1) preserve the
$\mathbf{A}$-lattices, the map $f_{i}$ preverses the $\mathbf{A}$-lattices.
Since $f$ is a $\mathbb{Z}[q^{-1}]$-linear combination of $f_{i}$’s, the claim
follows.

(d) The map $f$ is a morphism of based $\mathrm{U}^{\imath}$-modules.

Thanks to $(b)$ and $(c)$, it suffices to consider the induced map
$f_{\infty}:L(M)/q^{-1}L(M)\rightarrow L(F(M))/q^{-1}L(F(M))$ at $q=\infty$;
cf. [Lu94]*§27.1.5.

Let $1\leq i,j\leq n$ with $\overline{\lambda_{i}}=\overline{0}\in X_{\imath}$
and $b\in B[\lambda_{j}]$. At $q=\infty$, we obtain from ($\heartsuit$ ‣ 3.1)
that

$f_{i,\infty}(b^{\imath})=\begin{cases}0,&\text{if }j\neq i;\\\
e_{b^{\imath}},&\text{if }j=i\text{ and }b\in B[\lambda_{j}]^{hi};\\\
0,&\text{if }j=i\text{ and }b\not\in B[\lambda_{j}]^{hi}.\\\ \end{cases}$

Since $f$ is a $\mathbb{Z}[q^{-1}]$-linear combination of $f_{i}$’s, claim (d)
follows.

(e) We have $\ker f=M_{+}$.

We have $M_{+}\subset\ker f$, since the action of $\mathrm{U}^{\imath}$ on
$F(M)$ is trivial, i.e., via the counit $\varepsilon$. Then it follows by
dimension counting that $M_{+}=\ker f$.

Now the theorem is proved. ∎

Retain the same notations from Theorem 3.1. Let ${}_{\mathcal{A}}M_{+}$ be the
$\mathcal{A}$-submodule of $M$ spanned by elements in
$B^{\imath}-B^{\imath}_{*}$. Then
${}_{\mathcal{A}}M_{+}=M_{+}\cap{}_{\mathcal{A}}M$ is an
${}_{\mathcal{A}}\dot{\mathrm{U}}^{\imath}$-module. For any commutative
$\mathcal{A}$-algebra $R$, set
${}_{R}M_{+}=R\otimes_{\mathcal{A}}{}_{\mathcal{A}}M_{+}$.

###  3.2. Coinvariant functors

For any based morphism $f:(M,B)\rightarrow(M^{\prime},B^{\prime})$ of finite-
dimensional based $\mathrm{U}$-modules, we naturally have the based morphism
$f:(M,B^{\imath})\rightarrow(M^{\prime},(B^{\prime})^{\imath})$ of based
$\mathrm{U}^{\imath}$-module. We then have the induced morphism
$f^{\imath}_{\ast}:M^{\imath}_{\ast}\rightarrow(M^{\prime})^{\imath}_{\ast}$
of $\mathrm{U}^{\imath}$-modules.

######  Proposition 3.2.

  1. (1)

The morphism
$f^{\imath}_{\ast}:M^{\imath}_{\ast}\rightarrow(M^{\prime})^{\imath}_{\ast}$
is a based morphism of $\mathrm{U}^{\imath}$-modules. Therefore
$(\cdot)^{\imath}_{\ast}$ is a functor from the category of finite-dimensional
based $\mathrm{U}$-modules to the category of finite-dimensional based
$\mathrm{U}^{\imath}$-modules.

  2. (2)

The functor $(\cdot)^{\imath}_{\ast}$ is an exact functor.

###### Proof.

We show part (1). Let $\lambda\in X^{+}$ and $b\in B[\lambda]^{hi}$. Since $f$
is a morphism of based $\mathrm{U}$-modules, we have
$f(b)\in(B^{\prime}[\lambda])^{hi}\sqcup\\{0\\}$ by investigating the induced
map at $q=\infty$. Therefore, $f$ maps $B^{\imath}_{\ast}$ to
$(B^{\prime})^{\imath}_{\ast}\sqcup\\{0\\}$. The claim now follows from
Theorem 3.1.

We show part (2). The right exactness is trivial. To show the left exactness,
suppose we have an injective based $\mathrm{U}$-module homomorphism
$f:(M,B)\rightarrow(M^{\prime},B^{\prime})$. Then $f$ is automatically an
injective based $\mathrm{U}^{\imath}$-module homomorphism
$f:(M,B^{\imath})\rightarrow(M^{\prime},(B^{\prime})^{\imath})$. Hence
$f(b)\neq 0$ for $b\in B^{\imath}$. In particular $f(b)\neq 0$ for $b\in
B_{*}^{\imath}$. Since $f$ maps $B^{hi}[\lambda]$ to
$B^{\prime}[\lambda]^{hi}$ for any $\lambda\in X^{+}$, we see that $f$ maps
$B_{*}^{\imath}$ to $(B^{\prime})_{*}^{\imath}$. We conclude that
$f^{\imath}_{\ast}$ is injective, since it is injective on the set of basis. ∎

Recall §2.1.3 that for $\lambda\in X^{+}$, we have the short exact sequence of
based $\mathrm{U}$-modules

$0\longrightarrow\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda]\longrightarrow\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]\longrightarrow\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not<\lambda]\longrightarrow
0.$

For any $\lambda\leq\lambda^{\prime}$ in $X^{+}$, we also have the short exact
sequence of based $\mathrm{U}$-modules

$0\longrightarrow\dot{\mathrm{U}}[\not\leq\lambda]/\dot{\mathrm{U}}[\not\leq\lambda^{\prime}]\longrightarrow\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda^{\prime}]\longrightarrow\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]\longrightarrow
0.$

######  Corollary 3.3.

  1. (1)

For any $\lambda\in X^{+}$, we have the following short exact sequence of
based $\mathrm{U}^{\imath}$-modules

$0\longrightarrow(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\longrightarrow(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\longrightarrow(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not<\lambda])^{\imath}_{\ast}\longrightarrow
0.$

  2. (2)

For any $\lambda\leq\lambda^{\prime}$ in $X^{+}$, we have the short exact
sequence of based $\mathrm{U}^{\imath}$-modules

$0\longrightarrow(\dot{\mathrm{U}}[\not\leq\lambda]/\dot{\mathrm{U}}[\not\leq\lambda^{\prime}])^{\imath}_{\ast}\longrightarrow(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda^{\prime}])^{\imath}_{\ast}\longrightarrow(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\longrightarrow
0.$

  3. (3)

For any $\lambda\in X^{+}$, we have
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\neq(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not<\lambda])^{\imath}_{\ast}$
if and only if $\lambda\in X^{+}_{\imath}$.

###### Proof.

The short exact sequences in part (1) and (2) follow from the exactness of the
functor $(\cdot)^{\imath}_{\ast}$ in Proposition 3.2.

If follows from §2.2.2 that
$\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda]\cong\bigoplus_{b\in\mathrm{B}[\lambda]^{hi}}V(\lambda)$.
Then by §2.2.3, we see
$(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\neq
0$ if and only if $\lambda\in X^{+}_{\imath}$. Then part (3) follows from the
short exact sequence in (1) now. ∎

##  4\. Coordinate rings of symmetric spaces

###  4.1. The quantum coordinate ring $\mathbf{O}(K\backslash G)$

Let $\mathbf{O}(K\backslash G)$ be the $\mathbb{Q}(q)$-subspace of
$\mathbf{O}$ consisting of linear forms on $\dot{\mathrm{U}}$ that vanish on
the subspace $\mathrm{U}^{\imath,+}\dot{\mathrm{U}}$.

######  Proposition 4.1.

  1. (1)

The subspace $\mathbf{O}(K\backslash G)$ is closed under multiplication.

  2. (2)

The coproduct $\Delta$ of $\mathbf{O}$ restricts to
$\Delta:\mathbf{O}(K\backslash G)\rightarrow\mathbf{O}(K\backslash
G)\otimes\mathbf{O}$.

###### Proof.

Part (1) follows if we can show
$\Delta(\mathrm{U}^{\imath,+})\subset\mathrm{U}^{\imath,+}\otimes\mathrm{U}+\mathrm{U}\otimes\mathrm{U}^{\imath,+}.$
Since the subspace $\mathrm{U}^{\imath,+}$ is the two-sided ideal of
$\mathrm{U}^{\imath}$, generated by $B_{i}$ ($i\in I$), $E_{i}$ ($i\in
I_{\bullet}$) and $K_{\mu}-1$ ($\mu\in Y^{\imath}$), it will suffice to check
$\Delta(x)\in\mathrm{U}^{\imath,+}\otimes\mathrm{U}+\mathrm{U}\otimes\mathrm{U}^{\imath,+}$,
when $x$ is a generator. This is direct and will be skipped. Part (2) follows
from the fact that $\mathrm{U}^{\imath,+}\dot{\mathrm{U}}$ is a right ideal of
$\dot{\mathrm{U}}$ ∎

Let ${}_{\mathcal{A}}\mathbf{O}(K\backslash G)=\mathbf{O}(K\backslash
G)\cap{}_{\mathcal{A}}\mathbf{O}$. Thanks to Proposition 4.1 (1), we see that
${}_{\mathcal{A}}\mathbf{O}(K\backslash G)$ is an $\mathcal{A}$-subalgebra of
${}_{\mathcal{A}}\mathbf{O}$. For any commutative $\mathcal{A}$-algebra $R$,
we write ${}_{R}\mathbf{O}(K\backslash
G)=R\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}(K\backslash G)$. For any
$\lambda\in X^{+}$, we define $\mathbf{O}(K\backslash
G)_{\leq\lambda}=\mathbf{O}(K\backslash G)\cap\mathbf{O}_{\leq\lambda}$ and
${}_{\mathcal{A}}\mathbf{O}(K\backslash
G)_{\leq\lambda}=\mathbf{O}(K\backslash
G)\cap{}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}$. We similarly define
$\mathbf{O}(K\backslash G)_{<\lambda}$ and
${}_{\mathcal{A}}\mathbf{O}(K\backslash G)_{<\lambda}$.

###  4.2. Canonical basis on $\mathbf{O}(K\backslash G)$

For any $\lambda\in X^{+}$, we have the finite-dimensional based
$\mathrm{U}$-module
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda],\dot{\mathrm{B}}[\leq\lambda])$
by §2.1.3. Then $\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]$ is
naturally a based $\mathrm{U}^{\imath}$-module with basis
$\dot{\mathrm{B}}[\leq\lambda]^{\imath}$.

We define $\dot{\mathrm{B}}[\leq\lambda]^{\imath}_{\ast}$ following §3.1. For
any $a\in\dot{\mathrm{B}}[\leq\lambda]$, and hence
$a^{\imath}\in\dot{\mathrm{B}}[\leq\lambda]^{\imath}$, we define the
$\mathbb{Q}(q)$-linear form

$\widetilde{a}^{\imath}:\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]\rightarrow\mathbb{Q}(q),\quad\text{such
that }\widetilde{a}^{\imath}(b^{\imath})=\delta_{a^{\imath},b^{\imath}}\text{
for all }b^{\imath}\in\dot{\mathrm{B}}[\leq\lambda]^{\imath}.$

Then we see that
$\\{\widetilde{a}^{\imath}|a^{\imath}\in\dot{\mathrm{B}}[\leq\lambda]^{\imath}\\}$
forms a $\mathbb{Q}(q)$-basis of $\mathbf{O}_{\leq\lambda}$ and an
$\mathcal{A}$-basis of ${}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}$.

Let $\lambda,\lambda^{\prime}\in X^{+}$ be such that
$\lambda^{\prime}\geq\lambda$. The natural projection
$\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda^{\prime}]\rightarrow\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]$
is a morphism of based $\mathrm{U}$-modules. It is then automatically a
morphism of based $\mathrm{U}^{\imath}$-modules by [BW18a]*§6

$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda^{\prime}],\dot{\mathrm{B}}[\leq\lambda^{\prime}]^{\imath})\rightarrow(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda],\dot{\mathrm{B}}[\leq\lambda]^{\imath}).$

Therefore

(a) the union $\cup_{\lambda\in
X^{+}}\\{\widetilde{a}^{\imath}|a^{\imath}\in\dot{\mathrm{B}}[\leq\lambda]^{\imath}\\}$
forms a $\mathbb{Q}(q)$-basis of $\mathbf{O}$ and an $\mathcal{A}$-basis of
${}_{\mathcal{A}}\mathbf{O}$.

######  Remark 4.2.

There is no natural $\imath$canonical basis $\dot{\mathrm{B}}^{\imath}$ on the
whole modified form $\dot{\mathrm{U}}$. One could consider a topological basis
via certain completion similar to [Lu94]*Chapter 30. However, due the
finiteness assumption on the elements in $\mathbf{O}$, we do naturally have
the dual basis $\cup_{\lambda\in
X^{+}}\\{\widetilde{a}^{\imath}|a^{\imath}\in\dot{\mathrm{B}}[\leq\lambda]^{\imath}\\}$.

######  Lemma 4.3.

The set $\mathrm{B}(K\backslash
G)_{\leq\lambda}=\\{\widetilde{a}^{\imath}|a^{\imath}\in\dot{\mathrm{B}}[\leq\lambda]^{\imath}_{\ast}\\}$
is a $\mathbb{Q}(q)$-basis of $\mathbf{O}(K\backslash G)_{\leq\lambda}$ and an
$\mathcal{A}$-basis of ${}_{\mathcal{A}}\mathbf{O}(K\backslash
G)_{\leq\lambda}$. In particular, ${}_{\mathcal{A}}\mathbf{O}(K\backslash
G)_{\leq\lambda}$ is a free $\mathcal{A}$-module.

Similar statements hold for $\mathbf{O}(K\backslash G)_{<\lambda}$ and
${}_{\mathcal{A}}\mathbf{O}(K\backslash G)_{<\lambda}$.

###### Proof.

Recall the exact functor $(\cdot)^{\imath}_{\ast}$ in §3.2. By definition, we
have

$\displaystyle\mathbf{O}(K\backslash G)_{\leq\lambda}$
$\displaystyle=\\{f\in{\rm
Hom}_{\mathbb{Q}(q)}\Big{(}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]),\mathbb{Q}(q)\Big{)}\mid
f\Big{(}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])_{+}\Big{)}=0\\}$
$\displaystyle={\rm
Hom}_{\mathbb{Q}(q)}\Big{(}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},\mathbb{Q}(q)\Big{)}.$

The lemma follows immediately by Theorem 3.1. ∎

By dualizing Corollary 3.3, we have the natural embeddings ($\text{for
}\lambda\leq\lambda^{\prime}$ in $X^{+}$)

$\displaystyle\mathbf{O}(K\backslash
G)_{\leq\lambda}\rightarrow\mathbf{O}(K\backslash G)_{\leq\lambda^{\prime}}$
$\displaystyle\text{ mapping }B(K\backslash G)_{\leq\lambda}\text{ into
}B(K\backslash G)_{\leq\lambda^{\prime}},$
$\displaystyle\mathbf{O}(K\backslash
G)_{<\lambda}\rightarrow\mathbf{O}(K\backslash G)_{\leq\lambda}$
$\displaystyle\text{ mapping }B(K\backslash G)_{<\lambda}\text{ into
}B(K\backslash G)_{\leq\lambda}.$

We set $\mathrm{B}(K\backslash G)_{\lambda}=\mathrm{B}(K\backslash
G)_{\leq\lambda}-\mathrm{B}(K\backslash G)_{<\lambda}$.

######  Theorem 4.4.

The set

$\mathrm{B}(K\backslash G)=\bigcup_{\lambda\in X^{+}}\mathrm{B}(K\backslash
G)_{\leq\lambda}=\bigsqcup_{\lambda\in X^{+}}\mathrm{B}(K\backslash
G)_{\lambda}=\bigsqcup_{\lambda\in X_{\imath}^{+}}\mathrm{B}(K\backslash
G)_{\lambda}$

is a $\mathbb{Q}(q)$-basis of $\mathbf{O}(K\backslash G)$ and an
$\mathcal{A}$-basis of ${}_{\mathcal{A}}\mathbf{O}(K\backslash G)$.

Moreover, for any commutative $\mathcal{A}$-algebra $R$ with $1$, the ring
${}_{R}\mathbf{O}(K\backslash
G)=R\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}(K\backslash G)$ is
naturally a free $R$-submodule of
${}_{R}\mathbf{O}=R\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}$.

###### Proof.

We have $\mathrm{B}(K\backslash G)_{\lambda}\neq\emptyset$ if and only if
$\lambda\in X^{+}_{\imath}$ by Corollary 3.3. The first claim follows
immediately.

We know ${}_{\mathcal{A}}\mathbf{O}(K\backslash
G)\subset{}_{\mathcal{A}}\mathbf{O}$ be definition. Since
${}_{\mathcal{A}}\mathbf{O}(K\backslash G)$ is spanned by the subset
$\mathrm{B}(K\backslash G)$ of the $\mathcal{A}$-basis $\cup_{\lambda\in
X^{+}}\\{\widetilde{a}^{\imath}|a^{\imath}\in\dot{\mathrm{B}}[\leq\lambda]^{\imath}\\}$
of ${}_{\mathcal{A}}\mathbf{O}$, the injectivity is clearly preserved after
the base change $R\otimes_{\mathcal{A}}-$. ∎

We call $\mathrm{B}(K\backslash G)$ the canonical basis of
$\mathbf{O}(K\backslash G)$.

######  Corollary 4.5.

The coproduct of ${}_{\mathcal{A}}\mathbf{O}$ induces the coproduct on the
$\mathcal{A}$-forms

${}_{\mathcal{A}}\mathbf{O}(K\backslash
G)\longrightarrow{{}_{\mathcal{A}}\mathbf{O}}(K\backslash
G)\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}.$ (4.1)

###### Proof.

It is known that the coproduct restricts to integral forms
$\Delta:{}_{\mathcal{A}}\mathbf{O}\rightarrow{}_{\mathcal{A}}\mathbf{O}\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}$.
The the corollory follows from Proposition 4.1 and the fact that
${}_{\mathcal{A}}\mathbf{O}(K\backslash G)$ is spanned by a subset the basis
of ${}_{\mathcal{A}}\mathbf{O}$ in §4.2 (a). ∎

###  4.3. The right action

For $\lambda\in X^{+}$, since $\dot{\mathrm{U}}[\not\leq\lambda]$ is a two-
sided ideal of $\dot{\mathrm{U}}$, the space
$\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]$ is a
$(\mathrm{U},\mathrm{U})$-bimodule. One can define the notion of based right
$\mathrm{U}$-modules in parallel with [Lu94]*§27. It follows by [Ka94] that
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda],\dot{\mathrm{B}}[\leq\lambda])$
is also a based right $\mathrm{U}$-module. By the construction of the functor
$(\cdot)^{\imath}_{\ast}$, the space
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}$ admits
a right $\mathrm{U}$-action. Recall that
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}$ is a
(left) based $\mathrm{U}^{\imath}$-module with $\imath$canonical basis
${\mathrm{B}}[\leq\lambda]^{\imath}_{\ast}$.

######  Lemma 4.6.

  1. (1)

$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda],\dot{\mathrm{B}}[\leq\lambda]^{\imath})$
is naturally a based right $\mathrm{U}$-module.

  2. (2)

The right $\mathrm{U}$-modules
$((\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},{\mathrm{B}}[\leq\lambda]^{\imath}_{\ast})$
and
$((\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not<\lambda])^{\imath}_{\ast},\dot{\mathrm{B}}[\not<\lambda]_{*}^{\imath})$
are based.

###### Proof.

We show part (1). We verify (the right module variant of the) conditions in
[Lu94]*§27.1.2. Conditions (a), (b) are trivial. Since the canonical basis and
$\imath$canonical basis coincide at $q=\infty$, condition (d) follows. We
define the bar involution on
$\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]$ by
$\psi_{\imath}=\Upsilon\psi$ following [BW18a]*Proposition 5.1, where $\psi$
is the bar involution fixing canonical basis, and $\Upsilon$ is an element in
a certain completion of $\mathrm{U}$. The $\imath$canonical basis
$\dot{\mathrm{B}}[\leq\lambda]^{\imath}$ is fixed by $\psi_{\imath}$ by
definition. For $u\in\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]$ and
$x\in\mathrm{U}$, we have

$\psi_{\imath}(ux)=\Upsilon\psi(ux)=\Upsilon\psi(u)\psi(x)=\psi_{\imath}(u)\psi(x).$

Hence the right $\mathrm{U}$-action on
$\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]$ is bar-equivariant with
respect to $\psi_{\imath}$. This verifies condition (c) in [Lu94]*§27.1.2.

Now it follows from Theorem 3.1 that
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])_{+}$ is a based right
$\mathrm{U}$-submodule, and
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}$ is a
based quotient of right $\mathrm{U}$-submodule. ∎

The following short exact sequence of based $\mathrm{U}^{\imath}$-modules in
Corollary 3.3

$0\longrightarrow(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\longrightarrow(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\longrightarrow(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not<\lambda])^{\imath}_{\ast}\longrightarrow
0$

is also an exact sequence as based right $\mathrm{U}$-modules, thanks to Lemma
4.6.

Recall the anti-algebra automorphism $\sigma$ and the algebra homomorphism
$\omega$ in §2.1.1. For $\lambda\in X^{+}$, let ${}^{\sigma\omega}V(\lambda)$
be the right $\mathrm{U}$-module with the same underlying vector space as
$V(\lambda)$, and with the $\mathrm{U}$-action twisted by
${\sigma\circ\omega}$. It was proved in [Ka94] the canonical basis
$\dot{\mathrm{B}}$ is $\sigma$-stable; it was proved in [Lu22] that the
canonical basis $\dot{\mathrm{B}}$ is $\omega$-stable. Then we can conclude
that $({}^{\sigma\omega}V(\lambda),B(\lambda))$ is a based right
$\mathrm{U}$-module.

######  Proposition 4.7.

For each $\lambda\in X^{+}$, we have an isomorphism of based right
$\mathrm{U}$-modules,

$\left((\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},\dot{\mathrm{B}}[\lambda]_{*}^{\imath}\right)\cong\begin{cases}\left({}^{\sigma\omega}V(\lambda),{}B(\lambda)\right),&\text{if
}\lambda\in X^{+}_{\imath};\\\ 0,&\text{if }\lambda\notin
X^{+}_{\imath}.\end{cases}$

###### Proof.

By [Lu94]*Proposition 29.2.2 & Proposition 29.3.1, we have the
$(\mathrm{U},\mathrm{U})$-bimodule isomorphism

$\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda]\cong
V(\lambda)\boxtimes{}^{\sigma\omega}V(\lambda),$

such that we have the induced bijection of bases $\mathrm{B}[\lambda]\cong
B(\lambda)\times B(\lambda)$. Here $\boxtimes$ denotes the external tensor
product emphasizing the bimodule structure.

Note that $V(\lambda)^{\imath}_{\ast}\cong\mathbb{Q}(q)$ if $\lambda\in
X^{+}_{\imath}$; and $V(\lambda)^{\imath}_{\ast}$ is the zero module if
$\lambda\notin X^{+}_{\imath}$ by §2.2.3. Then we have the following right
$\mathrm{U}$-module isomorphisms

$\big{(}\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda]\big{)}^{\imath}_{\ast}\cong
V(\lambda)^{\imath}_{\ast}\boxtimes{}^{\sigma\omega}V(\lambda)\cong\begin{cases}{}^{\sigma\omega}V(\lambda),&\text{if
}\lambda\in X^{+}_{\imath};\\\ 0,&\text{if }\lambda\notin
X^{+}_{\imath}.\end{cases}$

The compactibility on bases is clear. We finish the proof. ∎

###  4.4. Symmetric spaces

Let $k$ be an algebraically closed field of characteristic $\neq 2$. Let
$\mathcal{A}\rightarrow k$ be the ring homomorphism given by $q\mapsto 1$. Set

${}_{k}\mathbf{O}=k\otimes_{\mathcal{A}}{{}_{\mathcal{A}}\mathbf{O}}\qquad\text{
and }\qquad_{k}\mathbf{O}(K\backslash
G)=k\otimes_{\mathcal{A}}{{}_{\mathcal{A}}\mathbf{O}(K\backslash G)}.$

The $k$-algebra ${}_{k}\mathbf{O}(K\backslash G)$ is naturally a subalgebra of
${}_{k}\mathbf{O}$ by Theorem 4.4. For $\lambda\in X^{+}$, set
${}_{k}\mathbf{O}(K\backslash
G)_{\leq\lambda}=k\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}(K\backslash
G)_{\leq\lambda}$. Still by Theorem 4.4, the $k$-algebra
${}_{k}\mathbf{O}(K\backslash G)_{\leq\lambda}$ is a $k$-subalgebra of
${}_{k}\mathbf{O}$, and moreover we have ${}_{k}\mathbf{O}(K\backslash
G)_{\leq\lambda}={}_{k}\mathbf{O}(K\backslash
G)\cap{}_{k}\mathbf{O}_{\leq\lambda}$. As a result, we have

$_{k}\mathbf{O}(K\backslash G)=\cup_{\lambda\in
X^{+}}{}_{k}\mathbf{O}(K\backslash G)_{\leq\lambda}.$ (4.2)

Let $(G_{k},\theta_{k})$ be the symmetric pair associated with our
$\imath$root datum, as constructed in §2.2.4. Recall that by Theorem 2.2, the
$k$-algebra ${}_{k}\mathbf{O}$ is canonically isomorphic to the algebra
$k[G_{k}]$ of regular functions on $G_{k}$. Let $K_{k}=G_{k}^{\theta_{k}}$ be
the symmetric subgroup.

######  Theorem 4.8.

As regular functions on $G_{k}$, we have

$_{k}\mathbf{O}(K\backslash G)=\\{f\in{}_{k}\mathbf{O}\mid f(hg)=f(g),\text{
for any }h\in K_{k},\;g\in G_{k}\\}.$ (4.3)

Therefore ${}_{k}\mathbf{O}(K\backslash G)$ is isomorphic to the coordinate
ring of the symmetric space $K_{k}\backslash G_{k}$. Moreover, the basis
$B(K\backslash G)$ of ${}_{\mathcal{A}}\mathbf{O}(K\backslash G)$ specializes
to a basis of ${}_{k}\mathbf{O}(K\backslash G)$.

###### Proof.

For $\lambda\in X^{+}$, let us write

${}^{K_{k}}k[G_{k}]_{\leq\lambda}=\\{f\in k[G_{k}]_{\leq\lambda}\mid
f(hg)=f(g),\text{ for any }h\in K_{k},\;g\in G_{k}\\}.$

It will suffice to prove that

${}^{K_{k}}k[G_{k}]_{\leq\lambda}={}_{k}\mathbf{O}(K\backslash
G)_{\leq\lambda},\qquad\text{for any }\lambda\in X^{+}.$

Let $\mathcal{I}\subset k[G_{k}]$ be the defining ideal of the closed subgroup
$K_{k}$, and $\delta:k[G_{k}]\rightarrow k[G_{k}]\otimes k[G_{k}]$ be the
coproduct of $k[G_{k}]$. Then we have

${}^{K_{k}}k[G_{k}]_{\leq\lambda}=\\{f\in
k[G_{k}]_{\leq\lambda}\mid\delta(f)-1\otimes f\in\mathcal{I}\otimes
k[G_{k}]\\}.$

By [Lu07]*§3.1, under the isomorphism $k[G_{k}]\cong{}_{k}\mathbf{O}$, the
constant function 1 is the dual canonical basis element $\widetilde{1_{0}}$.
Recall from Theorem 2.5 that

$\mathcal{I}=\\{f\in k[G_{k}]\mid f({}_{k}\dot{\mathrm{U}}^{\imath})=0\\}.$

Therefore we have

${}^{K_{k}}k[G_{k}]_{\leq\lambda}$ $\displaystyle=\\{f\in
k[G_{k}]_{\leq\lambda}\mid f(yx)=\widetilde{1_{0}}(y)f(x),\;\text{for
}y\in{}_{k}\dot{\mathrm{U}}^{\imath},\;x\in{}_{k}\dot{\mathrm{U}}\\}$
$\displaystyle=\\{f\in\text{Hom}_{k}\big{(}{}_{k}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]),k\big{)}\mid
f\big{(}{}_{k}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])_{+}\big{)}=0\\}.$

Here the last equality follows from the fact that, the integral form of the
based $\mathrm{U}^{\imath}$-module
${}_{\mathcal{A}}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])_{+}$ is
spanned by $yx-\widetilde{1_{0}}(y)x$, for
$y\in{}_{\mathcal{A}}\dot{\mathrm{U}}^{\imath}$,
$x\in{}_{\mathcal{A}}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])$, as
an $\mathcal{A}$-module.

On the other hand, recall that

${}_{\mathcal{A}}\mathbf{O}(K\backslash
G)_{\leq\lambda}=\\{f\in\text{Hom}_{\mathcal{A}}\big{(}{}_{\mathcal{A}}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]),\mathcal{A}\big{)}\mid
f\big{(}{}_{\mathcal{A}}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])_{+}\big{)}=0\\}.$

By Theorem 3.1, we have

${}_{k}\mathbf{O}(K\backslash
G)_{\leq\lambda}=\\{f\in\text{Hom}_{k}\big{(}{}_{k}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]),k\big{)}\mid
f\big{(}{}_{k}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])_{+}\big{)}=0\\}.$

This completes the proof of the equality (4.3).

For the last assertion, it is known that the symmetric space $K_{k}\backslash
G_{k}$ is affine by [Sp83]*Proposition 2.2. Therefore the coordinate ring
$k[K_{k}\backslash G_{k}]$ is isomorphic to the subalgebra of
$K_{k}$-invariant functions in $k[G_{k}]$ (cf. [Spr09]*Exercise 5.5.9 (8)). We
complete the proof of the theorem. ∎

Thanks to Theorem 4.8, the base change of the coproduct in Corollary 4.5 gives
the map

$k[K_{k}\backslash G_{k}]\longrightarrow k[K_{k}\backslash
G_{k}]\otimes_{k}k[G_{k}],$

which is the comorphism of the action map $K_{k}\backslash
G_{k}\times_{k}G_{k}\longrightarrow K_{k}\backslash G_{k}$.

We hence have the following geometric reformulation of Theorem 4.8.

(a) The space $Spec\,{}_{\mathbb{Z}}\mathbf{O}(K\backslash G)$ admits a
natural right action of the Chevalley group scheme $G_{\mathbb{Z}}$, whose
geometric fiber over any algebraically closed field $k$ of characteristic not
$2$ is isomorphic to the symmetric space $K_{k}\backslash G_{k}$ with the
natural right action of $G_{k}$.

###  4.5. Filtrations

The coordinate ring ${}_{k}\mathbf{O}(K\backslash G)\cong k[K_{k}\backslash
G_{k}]$ admits a left $G_{k}$-action by

$(g\cdot f)(x)=f(xg),\quad\text{ for }x\in K_{k}\backslash G_{k},f\in
k[K_{k}\backslash G_{k}],g\in G_{k}.$

Recall the Peter–Weyl filtration
$\\{{}_{k}\mathbf{O}_{\leq\lambda}\mid\lambda\in X^{+}\\}$ of the coordinate
ring ${}_{k}\mathbf{O}\cong k[G_{k}]$ in Theorem 2.3. For $\lambda\in X^{+}$,
recall

${}_{k}\mathbf{O}(K\backslash G)_{\leq\lambda}={}_{k}\mathbf{O}(K\backslash
G)\cap{}_{k}\mathbf{O}_{\leq\lambda}.$

Since each subspace ${}_{k}\mathbf{O}_{\leq\lambda}$ is a
($G_{k},G_{k}$)-subbimodule of ${}_{k}\mathbf{O}$, we deduce that
${}_{k}\mathbf{O}(K\backslash G)_{\leq\lambda}$ is a $G_{k}$-submodule.
Similarly we have the $G_{k}$-submodule ${}_{k}\mathbf{O}(K\backslash
G)_{<\lambda}$.

######  Theorem 4.9.

For $\lambda\in X^{+}$. We have the $G_{k}$-module isomorphisms

${}_{k}\mathbf{O}(K\backslash G)_{\leq\lambda}/{}_{k}\mathbf{O}(K\backslash
G)_{<\lambda}\cong\begin{cases}{}_{k}V(-w_{0}\lambda)^{*},&\text{if
}\lambda\in X^{+}_{\imath};\\\ 0,&\text{if }\lambda\notin
X^{+}_{\imath}.\end{cases}$

###### Proof.

By the construction of canonical basis of
${}_{\mathcal{A}}\mathbf{O}(K\backslash G)$, we have the following canonical
isomorphism as vector spaces,

${}_{k}\mathbf{O}(K\backslash G)_{\leq\lambda}/{}_{k}\mathbf{O}(K\backslash
G)_{<\lambda}$ $\displaystyle\cong
k\otimes_{\mathcal{A}}\big{(}{}_{\mathcal{A}}\mathbf{O}(K\backslash
G)_{\leq\lambda}/{}_{\mathcal{A}}\mathbf{O}(K\backslash G)_{<\lambda}\big{)}$
$\displaystyle\cong
k\otimes_{\mathcal{A}}\left(\text{Hom}_{\mathcal{A}}\big{(}{}_{\mathcal{A}}(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},\mathcal{A}\big{)}\right)$
$\displaystyle\cong\text{Hom}_{k}\big{(}{}_{k}(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},k\big{)}.$

The second identity follows from the exact sequence in Corollary 3.3. The left
$G_{k}$-action is given by

$(g\cdot f)(x)=f(xg),\quad g\in
G_{K},x\in\big{(}{}_{k}(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},f\in\text{Hom}_{k}\big{(}{}_{k}(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},k\big{)}.$

By Proposition 4.7, we have the right $G_{k}$-module isomorphism

${}_{k}(\dot{\mathrm{U}}[\not<\lambda]/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\cong\begin{cases}{}^{\sigma\omega}_{k}V(\lambda),&\text{if
}\lambda\in X^{+}_{\imath};\\\ 0,&\text{if }\lambda\notin
X^{+}_{\imath}.\end{cases}$

Note that $\sigma\circ\omega(g)=\text{Ad}_{h}\circ\omega(g^{-1})$ for any
$g\in G_{k}$ and some $h$ in a maximal torus of $G_{k}$. Therefore as left
$G_{k}$-modules, we have

${}_{k}\mathbf{O}(K\backslash G)_{\leq\lambda}/{}_{k}\mathbf{O}(K\backslash
G)_{<\lambda}\cong\begin{cases}{}_{k}V(-w_{0}\lambda)^{*},&\text{if
}\lambda\in X^{+}_{\imath};\\\ 0,&\text{if }\lambda\notin
X^{+}_{\imath}.\end{cases}\qed$

For a rational $G_{k}$-module $V$, an ascending chain $F^{0}=(0)\subset
F^{1}\subset F^{2}\subset\cdots$ of $G_{k}$-submodules is called a _good
filtration_ if $\cup_{j}F^{j}=V$, and for each $j\geq 1$, the quotient
$F^{j}/F^{j-1}$ is isomorphic to ${}_{k}V(\lambda_{j})^{*}$ for some
$\lambda_{j}\in X^{+}$.

By Theorem 2.3, $k[G_{k}]$ has a good filtration as a $G_{k}\times
G_{k}$-module. Theorem 4.9 generalises this result to arbitrary symmetric
pairs, which we restate as follows.

######  Corollary 4.10.

The coordinate ring of the symmetric space $K_{k}\backslash G_{k}$ has a good
filtration as a $G_{k}$-module.

######  Remark 4.11.

Let $B_{k}\subset G_{k}$ be a Borel subgroup with the unipotent radical
$N_{k}\subset B_{k}$. Then the similar arguments in the proof of Theorem 4.8
and Theorem 4.9 can be applied to give a construction of an integral form and
a good filtration of the coordinate ring $k[N_{k}\backslash G_{k}]$.

The construction of the (integral form of) ring $k[N_{k}\backslash G_{k}]$
will actually be easier, since one only need to consider Lusztig’s (dual)
canonical bases. The good filtration of the $G_{k}$-module $k[N_{k}\backslash
G_{k}]$ actually splits, thanks to the left multiplication by the maximal
torus.

###  4.6. Spherical functions

Let $\mathbf{O}(K\backslash G/K)$ be the $\mathbb{Q}(q)$-subspace of
$\mathbf{O}$ consisting of linear forms on $\dot{\mathrm{U}}$ that vanish on
the subspace
$\mathrm{U}^{\imath,+}\dot{\mathrm{U}}+\dot{\mathrm{U}}\mathrm{U}^{\imath,+}$.
The following proposition can be proved similarly to Proposition 4.1, whose
proof we omit.

######  Proposition 4.12.

The subspace $\mathbf{O}(K\backslash G/K)$ is closed under multiplication.

We define ${}_{\mathcal{A}}\mathbf{O}(K\backslash G/K)=\mathbf{O}(K\backslash
G/K)\cap{}_{\mathcal{A}}\mathbf{O}$, and $\mathbf{O}(K\backslash
G/K)_{\leq\lambda}=\mathbf{O}(K\backslash G/K)\cap\mathbf{O}_{\leq\lambda}$
for $\lambda\in X^{+}$. We similarly define $\mathbf{O}(K\backslash
G/K)_{<\lambda}$, ${}_{\mathcal{A}}\mathbf{O}(K\backslash G/K)_{\leq\lambda}$,
and ${}_{\mathcal{A}}\mathbf{O}(K\backslash G/K)_{<\lambda}$.

By Lemma 4.6 (1), for $\lambda\in X^{+}$,
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda],\mathrm{B}[\leq\lambda]^{\imath})$
is a based right $\mathrm{U}$-module, hence a based right
$\mathrm{U}^{\imath}$-module. We denote its $\imath$canonical basis by
${}^{\imath}\mathrm{B}[\leq\lambda]^{\imath}=\\{{}^{\imath}b^{\imath}|b\in\mathrm{B}[\leq\lambda]\\}$
as a right $\mathrm{U}^{\imath}$-module. Following §4.2, we define the
$\mathbb{Q}(q)$-linear form

${}^{\imath}\widetilde{a}^{\imath}:\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda]\rightarrow\mathbb{Q}(q),\quad{}^{\imath}\widetilde{a}^{\imath}({}^{\imath}b^{\imath})=\delta_{{}^{\imath}a^{\imath},{}^{\imath}b^{\imath}}\text{
for all
}{}^{\imath}b^{\imath}\in{}^{\imath}\dot{\mathrm{B}}[\leq\lambda]^{\imath}.$

Then
$\\{{}^{\imath}\widetilde{a}^{\imath}|{}^{\imath}a^{\imath}\in{}^{\imath}\dot{\mathrm{B}}[\leq\lambda]^{\imath}\\}$
forms a $\mathbb{Q}(q)$-basis of $\mathbf{O}_{\leq\lambda}$ and an
$\mathcal{A}$-basis of ${}_{\mathcal{A}}\mathbf{O}_{\leq\lambda}$.

By Lemma 4.6 (2),
$\big{(}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},\mathrm{B}[\leq\lambda]^{\imath}_{\ast}\big{)}$
is a based right $\mathrm{U}$-module. We denote its $\imath$canonical basis by
${}^{\imath}\mathrm{B}[\leq\lambda]^{\imath}_{\ast}$ as a right
$\mathrm{U}^{\imath}$-module. By the right variant of Theorem 3.1, we see that
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}\mathrm{U}^{\imath,+}$
is a based right $\mathrm{U}^{\imath}$-submodule of
$(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast}$. By
applying the right variant of the construction in §3.1, we get a subset
${}^{\imath}_{\ast}\mathrm{B}[\leq\lambda]^{\imath}_{\ast}$ of the basis
${}^{\imath}\mathrm{B}[\leq\lambda]^{\imath}$, such that the based quotient
$\big{(}{}^{\imath}_{\ast}(\dot{\mathrm{U}}/\dot{\mathrm{U}}[\not\leq\lambda])^{\imath}_{\ast},{}^{\imath}_{\ast}\mathrm{B}[\leq\lambda]^{\imath}_{\ast}\big{)}$
is a based right $\mathrm{U}^{\imath}$-module. Similar to §4.2, we see that
$\mathrm{B}(K\backslash
G/K)_{\leq\lambda}=\\{{}^{\imath}\tilde{a}^{\imath}|{}^{\imath}\tilde{a}^{\imath}\in{}^{\imath}_{\ast}\mathrm{B}[\leq\lambda]^{\imath}_{\ast}\\}$
is a $\mathbb{Q}(q)$-basis of $\mathbf{O}(K\backslash G/K)_{\leq\lambda}$ and
an $\mathcal{A}$-basis of ${}_{\mathcal{A}}\mathbf{O}(K\backslash
G/K)_{\leq\lambda}$. For any $\lambda\in X^{+}$, the natural embedding

$\mathbf{O}(K\backslash G/K)_{<\lambda}\rightarrow\mathbf{O}(K\backslash
G/K)_{\leq\lambda}\text{ maps }\mathrm{B}(K\backslash G/K)_{<\lambda}\text{ to
}\mathrm{B}(K\backslash G/K)_{\leq\lambda}.$

We define

$\mathrm{B}(K\backslash G/K)_{\lambda}=\mathrm{B}(K\backslash
G/K)_{\leq\lambda}-\mathrm{B}(K\backslash G/K)_{<\lambda}\quad\text{ and
}\quad\mathrm{B}(K\backslash G/K)=\bigsqcup_{\lambda\in
X^{+}}\mathrm{B}(K\backslash G/K)_{\lambda}.$

The following theorem is analogous to Theorem 4.4 and Theorem 4.8. The proof
will be omitted.

######  Theorem 4.13.

  1. (1)

The set $\mathrm{B}(K\backslash G/K)$ is a $\mathbb{Q}(q)$-basis of
$\mathbf{O}(K\backslash G/K)$ and an $\mathcal{A}$-basis of
${}_{\mathcal{A}}\mathbf{O}(K\backslash G/K)$.

  2. (2)

For any algebraically closed field $k$ of char $\neq 2$, we have

${}_{k}\mathbf{O}(K\backslash
G/K)=\\{f\in{}_{k}\mathbf{O}|f(hg)=f(gh)=f(g),\forall g\in G_{k},h\in
K_{k}\\}.$

Here ${}_{k}\mathbf{O}(K\backslash
G/K)=k\otimes_{\mathcal{A}}{}_{\mathcal{A}}\mathbf{O}(K\backslash G/K)$.

We call $\mathrm{B}(K\backslash G/K)$ the canonical basis of
$\mathbf{O}(K\backslash G/K)$.

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