Integrable modules over quantum symmetric pair coideal subalgebras

The quantum groups (a.k.a. quantized enveloping algebras) were introduced by Drinfeld [Dr85] and Jimbo [J85] independently, and their representation theory has been studied for many years and been applied to a number of areas of mathematics and mathematical physics such as orthogonal polynomials, combinatorics, knot theory, and integrable systems. One of the central objects in the representation theory of quantum groups is the integrable modules. The notion of integrable modules originates in the representation theory of semisimple Lie algebras. In this context, an integrable module is a module over a semisimple Lie algebra which can be integrated to a representation of a corresponding Lie group. The integrability can be described as the local nilpotency of the Chevalley generators of the Lie algebra. This description can be used to define the notion of integrable modules over quantum groups as it is.

The $\imath$quantum groups (a.k.a. quantum symmetric pair coideal subalgebras) of finite type were formulated by Letzter [Le99] in order to perform noncommutative harmonic analysis on quantum symmetric spaces based on earlier works such as [NS95], [Di96], [N96]. Letzter’s construction was further generalized to infinite type by Kolb [Ko14].

It has been pointed out that the representation theory of $\imath$quantum groups develops very slowly ([Le19], [Wa21]). In fact, we do not still know any general theory to classify the finite-dimensional simple modules over $\imath$quantum groups of finite type, although such $\imath$quantum groups are quantum deformations of the universal enveloping algebra of a finite-dimensional complex reductive Lie algebra. Several type-dependent classification can be found in e.g. [GK91], [IK05], [M06] [IT10], [Wa20], [We20], [Wa21]. Since the $\imath$quantum groups have close resemblance to the quantum groups, it seems to be essential for better understanding of their representation theory to formulate the notion of integrable modules over $\imath$quantum groups. This is the aim of the present paper.

Let us recall the definition of integrable modules over quantum groups. Let $\mathbf{U}$ be a quantum group, and $\{E_{i},F_{i}\mid i\in I\}$ its Chevalley generators. Then, a weight $\mathbf{U}$-module $V$ is said to be integrable if for each weight vector $v\in V$, there exist tuples $(a_{i})_{i\in I}$, $(b_{i})_{i\in I}$ of nonnegative integers such that

where the elements $E_{i}^{(a_{i}+1)}$, $F_{i}^{(b_{i}+1)}$ are divided powers.

Let us turn to the $\imath$quantum groups. An $\imath$quantum group is a certain subalgebra of a quantum group $\mathbf{U}$, and has a distinguished family $\{E_{j},F_{j},B_{k}\mid j\in I_{\bullet},\ k\in I_{\circ}\}$ of elements, which play similar roles to Chevalley generators of $\mathbf{U}$ in some sense. Here, the $I_{\bullet}$ is a subset of $I$, and $I_{\circ}$ is its complement $I\setminus I_{\bullet}$. An easy idea to define the integrable modules over the $\imath$quantum group is the local nilpotency with respect to the $E_{j}$, $F_{j}$, and $B_{k}$. However, this is not acceptable since the $B_{k}$ may acts semisimply on many modules which should be included in the class of integrable modules. Bao and Wang, in their theory of $\imath$canonical bases [BW18], [BW21], introduced the notion of $\imath$divided powers, which are analogues of the divided powers of quantum groups. Using the $\imath$divided powers, one may be able to resolve the semisimplicity issue of $B_{k}$ above. Instead, testing integrability would be too hard since the $\imath$divided powers are very complicated in general.

To overcome this difficulty, let us recall an alternative definition of the integrable $\mathbf{U}$-modules. Let $X^{+}$ denote the set of dominant weights. For each $\lambda\in X^{+}$, let $V(\lambda)$ denote the integrable highest weight module of highest weight $\lambda$ with highest weight vector $v_{\lambda}$. Also, let ${}^{\omega}V(\lambda)$ denote the integrable lowest weight module of lowest weight $-\lambda$ with lowest weight vector ${}^{\omega}v_{\lambda}$. Then, a weight $\mathbf{U}$-module $V$ is integrable if and only if for each weight vector $v\in V$, there exist $\lambda,\mu\in X^{+}$ and a $\mathbf{U}$-module homomorphism ${}^{\omega}V(\lambda)\otimes V(\mu)\rightarrow V$ which sends ${}^{\omega}v_{\lambda}\otimes v_{\mu}$ to $v$. Namely, integrable $\mathbf{U}$-modules are locally quotients of various ${}^{\omega}V(\lambda)\otimes V(\mu)$.

The $\mathbf{U}$-modules ${}^{\omega}V(\lambda)\otimes V(\mu)$ appear in Lusztig’s construction of the canonical basis of modified form of $\mathbf{U}$ ([Lu92]). They are replaced by certain $\mathbf{U}^{\imath}$-modules $L^{\imath}(\lambda,\mu)$ in Bao-Wang’s theory of $\imath$canonical bases [BW18], [BW21]. Hence, it is natural to define an integrable $\mathbf{U}^{\imath}$-module to be locally a quotient of $V^{\imath}(\lambda,\mu)$. This definition turns out to work well in the present paper as summarized below.

First, we determine a presentation of the $\mathbf{U}^{\imath}$-modules $V^{\imath}(\lambda,\mu)$. This enables one to test a given $\mathbf{U}^{\imath}$-module to be integrable in a systematical way. In many cases, one only needs to check the local nilpotency with respect to $E_{j}$, $F_{j}$, $B_{k}$ just like the quantum group case. In the remaining case, one needs to investigate local semisimplicity of some $B_{k}$. As an application, we show that each integrable $\mathbf{U}$-module is integrable as a $\mathbf{U}^{\imath}$-module.

Next, we concentrate on the $\imath$quantum groups of finite type. We also assume that the parameters of the $\mathbf{U}^{\imath}$ is chosen in a way such that the $\mathbf{U}^{\imath}$ is invariant under a certain anti-algebra involution $\rho$ on $\mathbf{U}$, and that the $\imath$canonical basis of the modified form of $\mathbf{U}^{\imath}$ is stable (or, strongly compatible). It turns out that each simple integrable $\mathbf{U}^{\imath}$-module is finite-dimensional and appears as submodule of a finite-dimensional $\mathbf{U}$-module.

Let us consider the space of matrix coefficients of all simple integrable $\mathbf{U}^{\imath}$-modules. As in the quantum group case ([Ka93, Section 7]), it admits a Peter-Weyl type decomposition. Furthermore, by using the stability of $\imath$canonical bases and the characterization of integrability, we show that this space has a basis which is dual to the $\imath$canonical basis of the modified form of $\mathbf{U}^{\imath}$. This shows that the space of matrix coefficients coincides with Bao-Song’s quantized coordinate ring [BS22], and hence, gives an intrinsic description of the latter.

The present paper is organized as follows. In Section 2, we review basic definitions and results concerning the structure and representation theory of quantum groups. Section 3 is devoted to formulating integrable modules over $\imath$quantum groups. In Section 4, we give an integrability criterion, and then prove that each integrable module over a quantum group is integrable when restricted to an $\imath$quantum group. As an application of this result, we investigate the space of matrix coefficients of all simple integrable modules over an $\imath$quantum group of finite type in Section 5.

The author thanks Paul Terwilliger for comments on the representation theory of $\imath$quantum groups, especially the $q$-Onsager algebra. He is also grateful to Stefan Kolb for fruitful discussion on a sufficient condition for a weight $\mathbf{U}^{\imath}$-module to be integrable, which stimulates him to formulate Proposition 4.3.1. This work was supported by JSPS KAKENHI Grant Number JP24K16903.

Throughout the present paper, we fix an indeterminate $q$, a Cartan datum $I=(I,\cdot)$ and a $Y$- and $X$-regular root datum $(Y,X,\langle,\rangle,\Pi^{\vee},\Pi)$ of type $I$. For each $i,j\in I$ and $n\in\mathbb{Z}_{\geq 0}$, set

For each $i\in I$, let $h_{i}\in\Pi^{\vee}$ and $\alpha_{i}\in\Pi$ denote the corresponding simple coroot and simple root, respectively; in particular, we have

Let $\mathrm{Br}$ and $W$ denote the braid group associated with $I$, and $s_{i}$ the simple reflection corresponding to $i\in I$. They act on $\mathbb{Z}[I]$, $Y$, and $X$ by

for each $i,j\in I,\ h\in Y,\ \lambda\in X$.

Let $Q:=\sum_{i\in I}\mathbb{Z}\alpha_{i}\subseteq X$ denote the root lattice, and $Q^{\pm}:=\pm\sum_{i\in I}\mathbb{Z}_{\geq 0}\alpha_{i}$ the positive and negative cones.

Let $\leq$ denote the dominance order on $X$, the partial ordering defined as follows: For each $\lambda,\mu\in X$, we have $\lambda\leq\mu$ if and only if $\mu-\lambda\in Q^{+}$.

Let $X^{+}$ denote the set of dominant weights:

Let $\mathbf{U}$ denote the quantum group associated with the root datum $(Y,X,\langle,\rangle,\Pi^{\vee},\Pi)$. Namely, the $\mathbf{U}$ is the unital associative $\mathbb{Q}(q)$-algebra with generators

subject to the following relations: For each $h,h_{1},h_{2}\in Y$ and $i,j\in I$,

where

The braid group $\mathrm{Br}$ acts on $\mathbf{U}$ by $s_{i}\mapsto T^{\prime\prime}_{i,1}$ ([Lu93, §37.1]). For each $w\in W$ with reduced expression $w=s_{i_{1}}\cdots s_{i_{l}}$, set

Let $\mathbf{U}^{-}$, $\mathbf{U}^{0}$, and $\mathbf{U}^{+}$ denote the negative, Cartan, and positive parts of $\mathbf{U}$, respectively. We have a triangular decomposition:

Let $\mathbf{B}(\pm\infty)$ denote the canonical basis of $\mathbf{U}^{\mp}$. They are graded by $Q^{\mp}$:

where

For each $b\in\mathbf{B}(\pm\infty)$, we set $\operatorname{wt}(b):=\lambda$.

Let $\omega$ denote the Chevalley involution on $\mathbf{U}$, that is, the algebra automorphism such that

Given a $\mathbf{U}$-module $V$, let ${}^{\omega}V=\{{}^{\omega}v\mid v\in V\}$ denote the $\mathbf{U}$-module $V$ twisted by $\omega$:

Let $\dot{\mathbf{U}}$ denote the modified form of $\mathbf{U}$ with idempotents $\{1_{\lambda}\mid\lambda\in X\}$, and $\dot{\mathbf{B}}$ the canonical basis of $\dot{\mathbf{U}}$.

A $\mathbf{U}$-module $V$ is said to be a weight module if it admits a linear space decomposition

such that

For each $v\in V_{\lambda}\setminus\{0\}$, we set $\operatorname{wt}(v):=\lambda$.

For each $\lambda\in X$, let $M(\lambda)$ denote the Verma module of highest weight $\lambda$ with highest weight vector $m_{\lambda}$.

For each $\lambda,\mu\in X$, set

and

The linear map

is a $\mathbf{U}$-module isomorphism ([Lu93, 23.3.1 (c)]). The $\mathbf{U}$-module $M(\lambda,\mu)$ (or rather $\dot{\mathbf{U}}1_{-\lambda+\mu}$) is a universal weight module of weight $-\lambda+\mu$ in the following sense: For each weight $\mathbf{U}$-module $V$ and a weight vector $v\in V_{-\lambda+\mu}$, there exists a unique $\mathbf{U}$-module homomorphism $M(\lambda,\mu)\rightarrow V$ which sends $m_{\lambda,\mu}$ to $v$.

A weight $\mathbf{U}$-module $V$ is said to be integrable if for each $\lambda\in X$ and $v\in V_{\lambda}$, there exist $(a_{i})_{i\in I},(b_{i})_{i\in I}\in\mathbb{Z}_{\geq 0}^{I}$ such that

For each $\lambda\in X^{+}$, let $V(\lambda)$ denote the integrable highest weight module of highest weight $\lambda$. It is the maximal integrable quotient of the Verma module $M(\lambda)$:

Twisting it by the Chevalley involution $\omega$, we obtain

Let $v_{\lambda}\in V(\lambda)$ denote the image of the highest weight vector $m_{\lambda}\in M(\lambda)$. Let $\mathbf{B}(\lambda)$ denote the canonical basis of $V(\lambda)$. The linear map

gives rise to a bijection

Combining this fact with the presentation (2.3.1), we obtain

For each $\lambda,\mu\in X^{+}$, set

and

By [Lu93, Proposition 23.3.6], we have

The $\mathbf{U}$-modules $V(\lambda,\mu)$ are universal integrable $\mathbf{U}$-modules in the following sense: Let $V$ be an integrable $\mathbf{U}$-module and $v\in V$ a weight vector. Then, there exist $\lambda,\mu\in X^{+}$ and a $\mathbf{U}$-module homomorphism $V(\lambda,\mu)\rightarrow V$ which sends $v_{\lambda,\mu}$ to $v$ ([Lu93, Proposition 23.3.10]).

In this subsection, we fix a subset $J\subseteq I$, and set $K:=I\setminus J$. We often regard $J$ as a Cartan datum and

as a root datum of type $J$.

Let $\mathbf{L}=\mathbf{L}_{J}$ denote the Levi subalgebra of the quantum group $\mathbf{U}$ associated with $J$, that is, the subalgebra generated by

The $\mathbf{L}$ itself is the quantum group associated with the root datum of type $J$ above. Let $\mathbf{L}^{-}$ and $\mathbf{L}^{+}$ denote the negative and the positive parts of $\mathbf{L}$, respectively. Noting that the Cartan part of $\mathbf{L}$ is the $\mathbf{U}^{0}$, we have a triangular decomposition

For each $\lambda\in X$, let $M_{\mathbf{L}}(\lambda)$ denote the Verma module over $\mathbf{L}$ of highest weight $\lambda$ with highest weight vector $m_{\mathbf{L},\lambda}$. Similarly, for each $\lambda,\mu\in X$, set $M_{\mathbf{L}}(\lambda,\mu):={}^{\omega}M_{\mathbf{L}}(\lambda)\otimes M_{\mathbf{L}}(\mu)$ and $m_{\mathbf{L};\lambda,\mu}:={}^{\omega}m_{\mathbf{L},\lambda}\otimes m_{\mathbf{L},\mu}\in M_{\mathbf{L}}(\lambda,\mu)$. Then, we have $M_{\mathbf{L}}(\lambda,\mu)\simeq\dot{\mathbf{L}}1_{-\lambda+\mu}$ as $\mathbf{L}$-modules, where $\dot{\mathbf{L}}$ denotes the modified form of $\mathbf{L}$.

Let $\mathbf{P}^{+}=\mathbf{P}^{+}_{J}$ and $\mathbf{P}^{-}=\mathbf{P}^{-}_{J}$ denote the parabolic and the opposite parabolic subalgebra of $\mathbf{U}$ associated with the subset $J$. Namely, the $\mathbf{P}^{+}$ (resp., $\mathbf{P}^{-}$) is the subalgebra of $\mathbf{U}$ generated by $\mathbf{L}$ and

We have triangular decompositions:

Although the algebras $\mathbf{P}^{\pm}$ are not quantum groups, we can construct their modified forms $\dot{\mathbf{P}}^{\pm}=\bigoplus_{\lambda\in X}\dot{\mathbf{P}}^{\pm}1_{\lambda}$ in a canonical way. Just like the modified forms of quantum groups, the $\dot{\mathbf{P}}^{\pm}$ have natural $\mathbf{P}^{\pm}$-bimodule structure. Furthermore, the $\mathbf{L}$-bimodule structure on $\mathbf{P}^{\pm}$ gives rise to an $\dot{\mathbf{L}}$-bimodule structure on $\dot{\mathbf{P}}^{\pm}$.

Let $\mathbf{R}^{+}=\mathbf{R}^{+}_{J}$ (resp., $\mathbf{R}^{-}=\mathbf{R}^{-}_{J}$) denote the nilradical part of $\mathbf{P}^{+}$ (resp., $\mathbf{P}^{-}$), that is, the two-sided ideal of $\mathbf{U}^{+}$ (resp., $\mathbf{U}^{-}$) generated by

Then, we have

Let $(I,I_{\bullet},\tau)$ be a generalized Satake diagram ([RV20, Definition 1]). From now on, we assume that the root datum $(Y,X,\langle,\rangle,\Pi^{\vee},\Pi)$ is a Satake datum of type $(I,I_{\bullet},\tau)$ ([Wa23c, Definition 3.1.3]). In particular, the $I_{\bullet}$ is a subdatum of $I$ which is supposed to be of finite type, the $\tau$ is an involutive automorphism on the Cartan datum $I$, and the lattices $Y$ and $X$ are supposed to be equipped with involutive automorphisms $\tau$ which are compatible with the one on $I$. Let $w_{\bullet}\in W$ denote the longest element of the Weyl group $W_{\bullet}\subseteq W$ associated with $I_{\bullet}$. Set

Also, let

denote the quotient map, and

the bilinear pairing induced from the one on $Y\times X$.

For later use, let us recall one of the axioms for generalized Satake diagrams from ([RV20, (2.19)]):

Let $\mathbf{U}^{\imath}$ denote the $\imath$quantum group associated with the Satake datum $(Y,X,\langle,\rangle,\Pi^{\vee},\Pi)$ and parameters ${\boldsymbol{\varsigma}}=(\varsigma_{k})_{k\in I_{\circ}}\in(\mathbb{Q}(q)^{\times})^{I_{\circ}}$, ${\boldsymbol{\kappa}}=(\kappa_{i})_{k\in I_{\circ}}\in\mathbb{Q}(q)^{I_{\circ}}$. It is the subalgebra of the quantum group $\mathbf{U}$ generated by

where

for each $k\in I_{\circ}$. The parameters are supposed to satisfy the following for all $k\in I_{\circ}$:

• •

  $\varsigma_{k}=\varsigma_{\tau(k)}$ if $k\cdot\theta(k)=0$.

• •

  $\kappa_{k}=0$ unless $\tau(k)=k$, $a_{k,j}=0$ for all $j\in I_{\bullet}$, and $a_{k,k^{\prime}}\in 2\mathbb{Z}$ for all $k^{\prime}\in I_{\circ}$ such that $\tau(k^{\prime})=k^{\prime}$ and $a_{k^{\prime},j}=0$ for all $j\in I_{\bullet}$.

They are as general as possible (see [Ko14, Definition 5.6]).

Let $\dot{\mathbf{U}}^{\imath}$ denote the modified form of $\mathbf{U}^{\imath}$ with idempotents $\{1_{\zeta}\mid\zeta\in X^{\imath}\}$.

Let $\mathbf{L}$, $\mathbf{P}^{\pm}$, and $\mathbf{R}^{\pm}$ denote the Levi, parabolic subalgebras, and the nilradical parts associated with $I_{\bullet}$, respectively.

A $\mathbf{U}^{\imath}$-module $V$ is said to be a weight module ([Wa23a, Definition 3.3.2]) if it admits a linear space decomposition

such that

• •

  $V_{\zeta}=\{v\in V\mid K_{h}v=q^{\langle h,\zeta\rangle}v\ \text{ for all }h\in Y^{\imath}\}$;

• •

  $E_{j}V_{\zeta}\subseteq V_{\zeta+\overline{\alpha_{j}}}$, $F_{j}V_{\zeta}\subseteq V_{\zeta-\overline{\alpha_{j}}}$ for all $j\in I_{\bullet}$;

• •

  $B_{k}V_{\zeta}\subseteq V_{\zeta-\overline{\alpha_{k}}}$ for all $k\in I_{\circ}$;

A weight $\mathbf{U}$-module $V=\bigoplus_{\lambda\in X}V_{\lambda}$ has a canonical weight $\mathbf{U}^{\imath}$-module structure:

For each $\lambda,\mu\in X$, set

and

Then, by Corollary 2.4.5, we have

as $\mathbf{U}$-modules. The $\mathbf{U}^{\imath}$-module $M^{\imath}(\lambda,\mu)$ is a universal weight module of weight $\overline{w_{\bullet}\lambda+\mu}$ in the following sense:

For each $\lambda,\mu\in X^{+}$, set $V^{\imath}(\lambda,\mu)$ to be the quotient $\mathbf{U}$-module of $M^{\imath}(\lambda,\mu)$ factored by the submodule generated by

Let $v^{\imath}_{\lambda,\mu}\in V^{\imath}(\lambda,\mu)$ denote the image of $m^{\imath}_{\lambda,\mu}\in M^{\imath}(\lambda,\mu)$ under the quotient map.