Factorisation of quasi K-matrices for quantum symmetric pairs

Abstract

The theory of quantum symmetric pairs provides a universal K-matrix which is an analog of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this paper we restrict to the cases where the underlying Lie algebra is $\mathfrak {sl}_n$ or the Satake diagram has no black dots. In these cases we give an explicit formula for the quasi K-matrix as a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the underlying symmetric Lie algebra in the same way as the factorisation of the quasi R-matrix depends on the Weyl group of the Lie algebra. We conjecture that our formula holds in general.

1 Introduction

1.1 Background

Let $\mathfrak {g}$ be a complex semisimple Lie algebra and $U_q(\mathfrak {g})$ the corresponding Drinfeld-Jimbo quantised enveloping algebra with positive and negative parts $U^+$ and $U^-$, respectively. The quasi R-matrix for $U_q(\mathfrak {g})$ is a canonical element in a completion of $U^-\otimes U^+$ which plays a pivotal role in many applications of quantum groups. In the theory of canonical or crystal bases for $U_q(\mathfrak {g})$ developed by G. Lusztig and M. Kashiwara, the quasi R-matrix appears as an intertwiner of two bar involutions on $\Delta (U_q(\mathfrak {g}))$, where $\Delta$ denotes the coproduct of $U_q(\mathfrak {g})$. The quasi R-matrix is used to define canonical bases of tensor products of $U_q(\mathfrak {g})$-modules and of the modified quantised enveloping algebra $\dot{\mathbf {U}}$, see [23, Part IV].

Moreover, the quasi R-matrix for $U_q(\mathfrak {g})$ is used in [23, Chapter 32] to construct a family of commutativity isomorphisms. These maps turn suitable categories of $U_q(\mathfrak {g})$-modules into braided monoidal categories and hence have applications in low-dimensional topology, in particular the construction of invariants of knots and links, see [26]. Up to completion, the commutativity isomorphisms come from a universal R-matrix for $U_q(\mathfrak {g})$.

As in [3] we denote the quasi R-matrix of $U_q(\mathfrak {g})$ by R. One of the key properties of R is that it admits a factorisation as a product of quasi R-matrices for $U_q(\mathfrak {sl}_2)$. Let $\{\alpha _i\,|\,i\in I\}$ be the set of simple roots for $\mathfrak {g}$. The quasi R-matrix corresponding to $i\in I$ is given by

$$\begin{aligned} R_{i} = \sum _{r \ge 0} (-1)^r q_{i}^{-r(r-1)/2} \dfrac{(q_{i} - q_{i}^{-1})^r}{ [r]_{q_{i}}!} F_{i}^r \otimes E_{i}^r \end{aligned}$$

(1.1)

where $E_i, F_i\in U_q(\mathfrak {g})$ are generators of the copy of $U_q(\mathfrak {sl}_2)$ labelled by i, and $q_i=q^{(\alpha _i,\alpha _i)/2}$. Let $\sigma _i$ for $i\in I$ be the generators of the Weyl group W of $\mathfrak {g}$, and let $T_i: U_q(\mathfrak {g})\rightarrow U_q(\mathfrak {g})$ for $i\in I$ be the corresponding Lusztig automorphisms. For any reduced expression $w_0 = \sigma _{i_1} \cdots \sigma _{i_t}$ of the longest element $w_0 \in W$ define

$$\begin{aligned} R^{[j]}=\big (T_{i_1}\dots T_{i_{j-1}}\otimes T_{i_1}\dots T_{i_{j-1}}\big )(R_{i_j}) \qquad \text{ for } j=1, \dots ,t. \end{aligned}$$

With this notation the quasi R-matrix of $U_q(\mathfrak {g})$ can be written as

$$\begin{aligned} R = R^{[t]} \cdot R^{[t-1]} \cdots R^{[2]} \cdot R^{[1]}, \end{aligned}$$

(1.2)

see [14, 15, 21, 11, 8.30].

In the present paper we aim to find a similar factorisation in the theory of quantum symmetric pairs. Let $\theta :\mathfrak {g}\rightarrow \mathfrak {g}$ be an involutive Lie algebra automorphism and let $\mathfrak {k}=\{x\in \mathfrak {g}\,|\,\theta (x)=x\}$ be the corresponding fixed Lie subalgebra. We refer to $(\mathfrak {g},\mathfrak {k})$ as a symmetric pair. A comprehensive theory of quantum symmetric pairs was developed by Letzter [16, 17]. This theory provides families of subalgebras $B_{{{\mathbf {c}}},{{\mathbf {s}}}}\subset U_q(\mathfrak {g})$ with parameters ${{\mathbf {c}}}$ and ${{\mathbf {s}}}$, which are quantum group analogs of $U(\mathfrak {k})$. Crucially, $B_{{{\mathbf {c}}},{{\mathbf {s}}}}$ is a right coideal subalgebra of $U_q(\mathfrak {g})$, that is

$$\begin{aligned} \Delta (B_{{{\mathbf {c}}},{{\mathbf {s}}}})\subseteq B_{{{\mathbf {c}}},{{\mathbf {s}}}}\otimes U_q(\mathfrak {g}). \end{aligned}$$

Initially, the theory of quantum symmetric pairs was used to perform harmonic analysis on quantum group analogs of symmetric spaces, see [19, 25]. Recent pioneering work by Bao and Wang [4] and by Ehrig and Stroppel [7] has placed quantum symmetric pairs in a much broader representation theoretic context. Both papers consider a bar involution for specific quantum symmetric pair coideal subalgebras of type $\textit{AIII}/\textit{AIV}$. Moreover, Bao and Wang construct an intertwiner $\mathfrak {X}$ (denoted by $\Upsilon$ in [4]) between the bar involutions on $B_{{{\mathbf {c}}},{{\mathbf {s}}}}$ and on $U_q(\mathfrak {g})$. The intertwiner $\mathfrak {X}$ is an analog for quantum symmetric pairs of the quasi R-matrix for $U_q(\mathfrak {g})$. Using the intertwiner, Bao and Wang show that large parts of Lusztig’s theory of canonical bases [23, Part IV] extend to the setting of quantum symmetric pairs. While [4] restricts to the specific quantum symmetric pairs of type $\textit{AIII}/\textit{AIV}$, the more recent work [5] develops a theory of canonical bases for all quantum symmetric pairs of finite type.

Following the program outlined in [4], the existence of the bar involution and the intertwiner $\mathfrak {X}$ was proved for general quantum symmetric pairs in [2] and [3], respectively. The intertwiner $\mathfrak {X}$ was then used in [3] to construct a universal K-matrix for $B_{{{\mathbf {c}}},{{\mathbf {s}}}}$ which is an analog of the universal R-matrix for $U_q(\mathfrak {g})$. For this reason we call the intertwiner $\mathfrak {X}$ the quasi K-matrix for $B_{{{\mathbf {c}}},{{\mathbf {s}}}}$. The universal K-matrix gives suitable categories of $B_{{{\mathbf {c}}},{{\mathbf {s}}}}$-modules the structure of a braided module category and allows similar applications in low-dimensional topology as braided monoidal categories, see [13].

In [4] and [3] the quasi K-matrix is defined recursively by the intertwiner property for the two bar involutions. It is noted at the end of [5, 4.4] that this recursion can be solved in principle. However, to this date no closed formula for $\mathfrak {X}$ is known.

1.2 Results

In the present paper we provide a general closed formula for the quasi K-matrix $\mathfrak {X}$ in many cases, and we conjecture that our formula holds for all quantum symmetric pairs of finite type. In doing so, we take guidance from the known factorisation (1.2) of the quasi R-matrix of $U_q(\mathfrak {g})$.

Recall that the involutive automorphism $\theta$ is determined up to conjugation by a Satake diagram $(I,X,\tau )$. Here, X is a subset of I and $\tau :I\rightarrow I$ is a diagram automorphism. The $\tau$-orbits in $I{{\setminus }}X$ correspond to rank one subdiagrams of the Satake diagram $(I,X,\tau )$. Associated to the symmetric Lie algebra $(\mathfrak {g},\theta )$ is a restricted root system $\Sigma$ with Weyl group $\widetilde{W}=W(\Sigma )$ which can be considered as a subgroup of W. The Coxeter generators $\widetilde{\sigma }_i$ of $\widetilde{W}$ are parametrised by the $\tau$-orbits in $I{{\setminus }}X$. We introduce the notion of a partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}}$ for any $\widetilde{w}\in \widetilde{W}$ with a reduced expression $\widetilde{w}=\widetilde{\sigma }_{i_1}\dots \widetilde{\sigma }_{i_t}$. More precisely, for $j=1, \dots , t$ let $\mathfrak {X}_i$ denote the quasi K-matrix corresponding to the rank one Satake subdiagram $(\{i,\tau (i)\}\cup X, X, \tau |_{\{i,\tau (i)\}\cup X})$ of $(I,X,\tau )$. The Lusztig automorphisms $T_w: U_q(\mathfrak {g})\rightarrow U_q(\mathfrak {g})$ for $w\in W$ allow us to define automorphisms $\widetilde{T}_i:=T_{\widetilde{\sigma }_i}$ for all $i\in I{{\setminus }}X$. For $j=1,\dots ,t$ we set

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}}^{[j]} = \Psi \circ \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{j-1}} \circ \Psi ^{-1} (\mathfrak {X}_{i_j}) \end{aligned}$$

where $\Psi$ denotes an algebra automorphism of an extension of $\widetilde{U}^+=\bigoplus _{\mu \in Q^+(2\Sigma )} U^+_\mu$ defined in Eq. (3.45). In analogy to (1.2) we now define the partial quasi K-matrix corresponding to $\widetilde{w}\in \widetilde{W}$ by

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}} = \mathfrak {X}_{\widetilde{w}}^{[t]} \cdot \mathfrak {X}_{\widetilde{w}}^{[t-1]} \cdots \mathfrak {X}_{\widetilde{w}}^{[2]} \cdot \mathfrak {X}_{\widetilde{w}}^{[1]}. \end{aligned}$$

(1.3)

The following theorem is the main result of the present paper. It gives an explicit formula for $\mathfrak {X}$ in the case that ${{\mathbf {s}}}={{\mathbf {0}}}=(0,0,\dots ,0)$ for certain Satake diagrams. Let $\widetilde{w}_0\in \widetilde{W}$ denote the longest element.

Theorem A

(Corollary 3.21) Let $\mathfrak {g}$ be of type $A_n$ or $X = \emptyset$. Then the quasi K-matrix $\mathfrak {X}$ for $B_{{{\mathbf {c}}},{{\mathbf {0}}}}$ is given by $\mathfrak {X}= \mathfrak {X}_{\widetilde{w}_0}$ for any reduced expression of the longest element $\widetilde{w}_0 \in \widetilde{W}$.

We conjecture that Theorem A holds for all Satake diagrams of finite type, see Conjecture 3.22. The proof of Theorem A proceeds in three steps and resembles the construction of the quasi R-matrix in [15]. First we construct the quasi K-matrices corresponding to all rank one Satake diagrams of type $A_n$ in the case where ${{\mathbf {s}}}={{\mathbf {0}}}$. The difficulty here is that there are many rank one cases, see Table 1. Secondly, we prove Theorem A in rank two by direct calculation. In rank two the longest element $\widetilde{w}_0\in \widetilde{W}$ has two reduced expressions. We show for each reduced expression that the element $\mathfrak {X}_{\widetilde{w}_0}$ defined by (1.3) satisfies the defining recursive relations for the quasi K-matrix. This involves tedious calculations which we have banned to Appendix A. The calculations require explicit knowledge of the rank one quasi K-matrices. The restriction to type $A_n$ or $X=\emptyset$ in Theorem A hence stems from the fact that the rank one quasi K-matrices are only known in type $A_n$.

In the cases considered in Appendix A, the calculations imply in particular that $\mathfrak {X}_{\widetilde{w}_0}$ is independent of the chosen reduced expression for $\widetilde{w}_0$. We conjecture that this is true in general.

Conjecture B

(Conjecture 3.14) Assume that $(I,X,\tau )$ is a Satake diagram of rank two. Then the element $\mathfrak {X}_{\widetilde{w}}$ defined by (1.3) depends only on $\widetilde{w} \in \widetilde{W}$ and not on the chosen reduced expression.

Conjecture B is all that is needed to prove Theorem A for all Satake diagrams of finite type. Indeed, assume that Conjecture B holds for all rank two Satake subdiagrams of the given Satake diagram. Then we can use braid relations for the operators $\widetilde{T}_i$ to show that $\mathfrak {X}_{\widetilde{w}}$ is independent of the chosen reduced expression for $\widetilde{w}\in \widetilde{W}$. In the case of the longest element $\widetilde{w}_0\in \widetilde{W}$ we choose different reduced expressions for $\widetilde{w}_0$ to show that $\mathfrak {X}_{\widetilde{w}_0}$ satisfies the defining recursive relations for the quasi K-matrix for $B_{{{\mathbf {c}}},{{\mathbf {0}}}}$. In summary, we obtain the following result in the case ${{\mathbf {s}}}={{\mathbf {0}}}$.

Theorem C

(Theorems 3.17, 3.20) Let $(I,X,\tau )$ be a Satake diagram such that all rank two Satake subdiagrams satisfy Conjecture B. Then the following hold:

• (1) The partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}}$ depends only on $\widetilde{w} \in \widetilde{W}$ and not on the chosen reduced expression.

• (2) The quasi K-matrix $\mathfrak {X}$ for $B_{{{\mathbf {c}}},{{\mathbf {0}}}}$ is given by $\mathfrak {X}=\mathfrak {X}_{\widetilde{w}_0}$ where $\widetilde{w}_0\in \widetilde{W}$ denotes the longest element.

In the case ${{\mathbf {s}}}\ne {{\mathbf {0}}}$ it is harder to give an explicit formula for the quasi K-matrix $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$ of $B_{{{\mathbf {c}}},{{\mathbf {s}}}}$. However, we can make use of the fact that $B_{\mathbf {c},\mathbf {s}}$ is obtained from $B_{{{\mathbf {c}}},{{\mathbf {0}}}}$ via a twist by a character $\chi _{{\mathbf {s}}}$ of $B_{{{\mathbf {c}}},{{\mathbf {0}}}}$. We consider the element $R^\theta _{{{\mathbf {c}}},{{\mathbf {s}}}}=\Delta (\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}})R(\mathfrak {X}^{-1}_{{{\mathbf {c}}},{{\mathbf {s}}}}\otimes 1)$ which was introduced in [4] under the name quasi R-matrix for $B_{\mathbf {c},\mathbf {s}}$, and which lives in a completion of $B_{\mathbf {c},\mathbf {s}}\otimes U^+$, see also [13, 3.4]. We show that the quasi K-matrix $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$ for $B_{\mathbf {c},\mathbf {s}}$ satisfies the relation

$$\begin{aligned} \mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}=(\chi _{{{\mathbf {s}}}}\otimes \mathrm {id})(R^\theta _{{{\mathbf {c}}},{{\mathbf {0}}}}). \end{aligned}$$

(1.4)

Hence the explicit formulas (1.2) and (1.3) for R and $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {0}}}}$, respectively, provide a formula for the quasi K-matrix of $B_{\mathbf {c},\mathbf {s}}$ also in the case ${{\mathbf {s}}}\ne {{\mathbf {0}}}$. However, in this case we do not obtain a factorisation as in Eq. (1.3).

1.3 Organisation

In Sect. 2 we recall background material on the restricted root system $\Sigma$ and its Weyl group $W(\Sigma )$. We show in particular how $W(\Sigma )$ can be identified with a subgroup $\widetilde{W}$ of the Weyl group W of $\mathfrak {g}$, see Proposition 2.7. Section 3 forms the heart of the paper. In Sects. 3.1 and 3.2 we fix notation for quantised enveloping algebras and quantum symmetric pairs, respectively. In Sect. 3.3 we determine explicit closed formulas for the quasi K-matrices $\mathfrak {X}$ for $B_{{{\mathbf {c}}},{{\mathbf {0}}}}$ in all cases where the Satake diagram is of rank one and of type $A_n$. We plan to return to the remaining rank one cases in the future. The theory of partial quasi K-matrices $\mathfrak {X}_{\widetilde{w}}$ for $\widetilde{w}\in \widetilde{W}$ is developed in Sect. 3.4. Here the rank two case is crucial. The explicit calculations which prove Theorem A in rank two are left for Appendix A. Based on the rank two results we prove in Theorem 3.17 that $\mathfrak {X}_{\widetilde{w}}$ is independent of the chosen reduced expression for $\widetilde{w}$. This gives the first part of Theorem C, and the second part as well as Theorem A follow, see Theorem 3.20. The case ${{\mathbf {s}}}\ne {{\mathbf {0}}}$ is treated in Sect. 3.5 where Formula (1.4) is proved in Proposition 3.25.

2 The restricted Weyl group

In this section we recall the construction of involutive automorphisms $\theta : \mathfrak {g} \rightarrow \mathfrak {g}$ of a semisimple Lie algebra $\mathfrak {g}$ following [12] and [19]. This allows us to construct a subgroup $W^{\Theta }$ of the Weyl group W consisting of elements fixed under the corresponding group automorphism of W. Of particular importance is a subgroup $\widetilde{W}$ of $W^{\Theta }$ which has an interpretation as the Weyl group of the corresponding restricted root system. The results in this section do not claim originality, but are all in some form contained in [22, 24] and also [8].

2.1 Involutive automorphisms of semisimple Lie algebras

Let $\mathfrak {g}$ be a finite dimensional complex semisimple Lie algebra. Let $\mathfrak {h} \subset \mathfrak {g}$ be a Cartan subalgebra and $\Phi \subset \mathfrak {h}^{*}$ the corresponding root system. Choose a set of simple roots $\Pi = \{ \alpha _i \mid i \in I\}$ where I is an index set labelling the nodes of the Dynkin diagram of $\mathfrak {g}$. Let $\Phi ^{+}$ be the corresponding set of positive roots and set $V = \mathbb {R}\Phi$. For $i \in I$, let $\sigma _i : V \rightarrow V$ denote the reflection at the hyperplane orthogonal to $\alpha _i$. We write W to denote the Weyl group generated by the simple reflections $\sigma _i$. We fix the W-invariant non-degenerate bilinear form $(-,-)$ on V such that $(\alpha ,\alpha )= 2$ for all short roots $\alpha \in \Phi$ in each component. Let $\{e_i, f_i, h_i \mid i \in I\}$ be the Chevalley generators for $\mathfrak {g}$.

Involutive automorphisms of $\mathfrak {g}$ are classified up to conjugation by pairs $(X, \tau )$ where $X \subset I$ and $\tau : I \rightarrow I$ is a diagram automorphism. More precisely, for any subset $X \subset I$ let $\mathfrak {g}_X$ denote the Lie subalgebra of $\mathfrak {g}$ generated by $\{e_i, f_i, h_i \mid i \in X\}$. Let $Q_X$ denote the sublattice of the root lattice $Q = \mathbb {Z}\Pi$ generated by $\{ \alpha _i \mid i \in X\}$. This is the root lattice for $\mathfrak {g}_X$. Let $\rho _X \in V$ and $\rho _X^{\vee } \in V^{*}$ denote the half sum of positive roots and positive coroots for $\mathfrak {g}_X$. The Weyl group $W_X$ of $\mathfrak {g}_X$ is the parabolic subgroup of W generated by $\{\sigma _i \mid i \in X\}$. Let $w_X \in W_X$ denote the longest element of $W_X$.

Definition 2.1

([27, p. 109], see also [12, Definition 2.3]) Let $X \subset I$ and $\tau :I \rightarrow I$ a diagram automorphism such that $\tau (X) = X$. The pair $(X, \tau )$ is called a Satake diagram if it satisfies the following properties:

• (1) $\tau ^2 = \text {id}_{I}$.

• (2) The action of $\tau$ on X coincides with the action of $-w_X$, that is

  $$\begin{aligned} -w_X(\alpha _i) = \alpha _{\tau (i)}\quad \hbox { for all}\ i\in X. \end{aligned}$$

• (3) If $j \in I {{\setminus }} X$ and $\tau (j) = j$, then $\alpha _j(\rho _X^{\vee }) \in \mathbb {Z}$.

Remark 2.2

When we need to identify the underlying Lie algebra, we write $(I,X,\tau )$ to indicate the Satake diagram. With this convention, if $(I,X,\tau )$ is a Satake diagram and $i \in I {{\setminus }} X$ then $(\{i, \tau (i)\} \cup X, X, \tau |_{\{i,\tau (i)\}\cup X})$ is also a Satake diagram.

Graphically, the ingredients of a Satake diagram are recorded in the Dynkin diagram of $\mathfrak {g}$. The nodes labelled by X are colored black and a double sided arrow is used to indicate the diagram automorphism. See [1, p. 32/33] for a complete list of Satake diagrams for simple $\mathfrak {g}$.

We can now construct an involution $\theta$ corresponding to the Satake diagram $(X, \tau )$ as in [12, 2.4]. Let $\omega : \mathfrak {g} \rightarrow \mathfrak {g}$ be the Chevalley involution given by

$$\begin{aligned} \omega (e_i) = -f_i, \qquad \omega (h_i) = -h_i, \qquad \omega (f_i) = -e_i. \end{aligned}$$

(2.1)

The diagram automorphism $\tau$ can be lifted to a Lie algebra automorphism $\tau : \mathfrak {g} \rightarrow \mathfrak {g}$ denoted by the same symbol. Recall from [28, Lemma 56] that the action of the Weyl group W on $\mathfrak {h}$ can be lifted to an action of the corresponding braid group Br(W) on $\mathfrak {g}$. We denote the action of $w \in Br(W)$ on $\mathfrak {g}$ by $\text {Ad}(w)$. Let $s: I \rightarrow \mathbb {C}^{\times }$ be a function such that

$$\begin{aligned} s(i)&= 1&\text{ if } i \in X \text{ or } \tau (i) = i, \end{aligned}$$

(2.2)

$$\begin{aligned} \dfrac{s(i)}{s(\tau (i))}&= (-1)^{\alpha _i(2\rho _X^{\vee })}&\text{ if } i \not \in X \text{ and } \tau (i) \ne i. \end{aligned}$$

(2.3)

This map extends to a group homomorphism $s_Q: Q \rightarrow \mathbb {C}^{\times }$ such that $s_Q(\alpha _i) = s(i)$ for each simple root $\alpha _i$. Let $\text {Ad}(s): \mathfrak {g} \rightarrow \mathfrak {g}$ denote the Lie algebra automorphism such that the restriction to any root space $\mathfrak {g}_{\alpha }$ is given by multiplication by $s_Q(\alpha )$.

Given a Satake diagram $(X, \tau )$ we define a Lie algebra automorphism $\theta =\theta (X,\tau )$ of $\mathfrak {g}$ by

$$\begin{aligned} \theta (X, \tau ) = \text {Ad}(s) \circ \text {Ad}(w_X) \circ \tau \circ \omega : \mathfrak {g} \rightarrow \mathfrak {g}. \end{aligned}$$

(2.4)

Here, the longest element $w_X \in W_X$ is considered as an element in the braid group Br(W). The Lie algebra automorphism $\theta =\theta (X,\tau )$ is involutive, that is $\theta ^2=\mathrm {id}$. Any involutive Lie algebra automorphism of $\mathfrak {g}$ is $\text {Aut}(\mathfrak {g})$-conjugate to an automorphism of the form $\theta (X,\tau )$, see for example [12, Theorems 2.5, 2.7].

2.2 The subgroup $W^{\Theta }$

For any Satake diagram $(X, \tau )$, the automorphism $\theta = \theta (X, \tau )$ satisfies $\theta (\mathfrak {h}) = \mathfrak {h}$. More explicitly, the definition (2.4) implies that $\theta |_{\mathfrak {h}} = -w_X \circ \tau$. The dual map $\Theta : \mathfrak {h}^{*} \rightarrow \mathfrak {h}^{*}$ is given by the same expression

$$\begin{aligned} \Theta = -w_X \circ \tau \end{aligned}$$

(2.5)

where now $w_X$ and $\tau$ act on $\mathfrak {h}^{*}$. We obtain a group automorphism

$$\begin{aligned} \Theta _W :W \rightarrow W, \qquad w \mapsto \Theta \circ w \circ \Theta . \end{aligned}$$

(2.6)

Let $W^{\Theta } = \{ w \in W \mid \Theta _W(w) = w\}$ denote the subgroup of elements fixed by $\Theta _W$. In this section we recall the structure of the subgroup $W^\Theta$ following [8] and [22]. For any $i \in I$ one has

$$\begin{aligned} \Theta _W(\sigma _i)=(-w_X\circ \tau ) \circ \sigma _i \circ (-w_X\circ \tau ) =w_X\sigma _{\tau (i)}w_X. \end{aligned}$$

(2.7)

This formula and property (2) in Definition 2.1 imply that $W_X$ is a subgroup of $W^{\Theta }$. Recall that for any subset $J \subset I$ we write $w_J$ to denote the longest element in the parabolic subgroup $W_J$. We use this notation in particular for $J=\{i,\tau (i)\}\cup X$. For all $i \in I {{\setminus }} X$ define

$$\begin{aligned} \widetilde{\sigma }_i = w_{\{i, \tau (i)\} \cup X} w_X^{-1}. \end{aligned}$$

(2.8)

Recall that there exists a diagram automorphism $\tau _0:I \rightarrow I$ such that the longest element $w_0 \in W$ satisfies

$$\begin{aligned} w_0(\alpha _i) = -\alpha _{\tau _0(i)} \end{aligned}$$

(2.9)

for all $i \in I$. By inspection of the list of Satake diagrams in [1, p. 32/33] one sees that the set X is $\tau _0$-invariant.

By Remark 2.2 the triple $(\{i, \tau (i)\} \cup X, X, \tau |_{\{i,\tau (i)\}\cup X})$ is a Satake diagram for any $i \in I {{\setminus }} X$. Hence, for any $i \in I {{\setminus }} X$ there exists a diagram automorphism $\tau _{0,i} : \{i, \tau (i)\} \cup X \rightarrow \{i, \tau (i)\} \cup X$ such that X is $\tau _{0,i}$-invariant and

$$\begin{aligned} w_{\{i, \tau (i)\} \cup X} (\alpha _j) = -\alpha _{\tau _{0,i}(j)} \end{aligned}$$

(2.10)

for all $j \in \{i,\tau (i)\} \cup X$. With this notation we get

$$\begin{aligned} w_{\{i, \tau (i)\} \cup X} \sigma _j = \sigma _{\tau _{0,i}(j)}w_{\{i,\tau (i)\} \cup X} \end{aligned}$$

(2.11)

and hence

$$\begin{aligned} w_{\{i,\tau (i)\} \cup X} w_X = \tau _{0,i}(w_X) w_{\{i,\tau (i)\} \cup X} = w_X w_{\{i,\tau (i)\} \cup X}. \end{aligned}$$

This proves that $w_X$ and $w_{\{i,\tau (i)\} \cup X}$ commute in W for any $i \in I {{\setminus }} X$. Hence Eq. (2.7) implies that

$$\begin{aligned} \widetilde{\sigma }_i \in W^{\Theta } \qquad \text{ for } \text{ all } i \in I {{\setminus }} X. \end{aligned}$$

(2.12)

Let $\widetilde{W} \subset W^{\Theta }$ denote the subgroup of W generated by all $\widetilde{\sigma }_i$ for $i \in I {{\setminus }} X$. Let $l : W \rightarrow \mathbb {N}_0$ denote the length function with respect to W. Let

$$\begin{aligned} W^X = \{w \in W \mid l(\sigma _iw) > l(w) \, \hbox { for all}\ i \in X\} \end{aligned}$$

(2.13)

be the set of minimal length left coset representatives of $W_X$. By [24, 25.1] the subset

$$\begin{aligned} \mathcal {W} = \{ w \in W^X \mid wW_X = W_Xw\} \end{aligned}$$

(2.14)

is a subgroup of W. By (2.8) and (2.11) we have $\widetilde{\sigma }_i \in \mathcal {W}$ for all $i \in I {{\setminus }} X$. Let $\mathcal {W}^{\tau } = \{ w \in \mathcal {W} \mid \tau (w) = w\}$. As $\tau (X) = X$ the definition of $\widetilde{\sigma }_i$ implies that $\widetilde{W} \subseteq \mathcal {W}^{\tau }$. The following lemma is a generalisation of [24, A.1(a)] and [8, Lemma 2] in the spirit of [8, Remark 8].

Lemma 2.3

Any $w \in \mathcal {W}^{\tau }$ may be written as $w = \widetilde{\sigma }_{i_1} \cdots \widetilde{\sigma }_{i_k}$ such that $\widetilde{\sigma }_{i_1}, \dotsc , \widetilde{\sigma }_{i_k} \in \widetilde{W}$ and $l(w) = l(\widetilde{\sigma }_{i_1}) + \cdots + l(\widetilde{\sigma }_{i_k})$.

Lemma 2.3 implies that $\widetilde{W} = \mathcal {W}^{\tau }$. Set $\widetilde{S} = \{\widetilde{\sigma }_i \mid i \in I {{\setminus }} X\}$. The following theorem is stated in [24, 25.1] and proved in [22, Theorem 5.9(i)]. A proof using only Weyl group combinatorics is indicated in [8, Remark 8].

Theorem 2.4

The pair $(\widetilde{W}, \widetilde{S})$ is a Coxeter system.

Let $\lambda : \widetilde{W} \rightarrow \mathbb {N}_0$ denote the length function with respect to $\widetilde{W}$. The following proposition, proved in [22, Theorem 5.9(iii)] and indicated in [8, Corollary 6] implies that reduced expressions in $\widetilde{W}$ are also reduced in W.

Proposition 2.5

Let $w, w^{\prime } \in \widetilde{W}$. Then $l(ww^{\prime }) = l(w) + l(w^{\prime })$ if and only if $\lambda (ww^{\prime }) = \lambda (w) + \lambda (w^{\prime })$.

Remark 2.6

By (2.11) the group $\widetilde{W}$ acts on $W_X$ by conjugation. Moreover $W_X\cap \widetilde{W}=\{\mathrm {id}\}$. One can show that $W^{\Theta } = W_X \rtimes \widetilde{W}$.

2.3 The restricted root system

The group $\widetilde{W}$ has an interpretation as the Weyl group of the restricted root system of the symmetric Lie algebra $(\mathfrak {g},\theta )$. This fact is implicit in [22] but we feel that it is beneficial to explain this connection in some detail. As $\theta (\mathfrak {h}) = \mathfrak {h}$, we obtain a direct sum decomposition

$$\begin{aligned} \mathfrak {h} = \mathfrak {h}_1 \oplus \mathfrak {a} \end{aligned}$$

(2.15)

where $\mathfrak {h}_1 = \{x\in \mathfrak {h}\,|\,\theta (x)=x\}$ and $\mathfrak {a} = \{x\in \mathfrak {h}\,|\,\theta (x)=-x\}$. The restricted root system $\Sigma \subset \mathfrak {a}^*$ is obtained by restriction of all roots in $\Phi$ to $\mathfrak {a}$, that is

$$\begin{aligned} \Sigma = \Phi |_{\mathfrak {a}} {{\setminus }} \{0\}. \end{aligned}$$

(2.16)

As $\Theta (\Phi )=\Phi$ we have $\Theta (V)=V$. Moreover, the inner product $(-,-)$ is $\Theta$-invariant. Hence we obtain an orthogonal direct sum decomposition

$$\begin{aligned} V = V_{+1} \oplus V_{-1} \end{aligned}$$

(2.17)

where $V_{\lambda } = \{ \alpha \in V \mid \Theta (\alpha ) = \lambda \alpha \}$ for $\lambda = \pm 1$. Indeed, any $\alpha \in V$ can be written as

$$\begin{aligned} \alpha = \dfrac{\alpha + \Theta (\alpha )}{2} + \dfrac{\alpha - \Theta (\alpha )}{2} \end{aligned}$$

(2.18)

where $(\alpha + \lambda \Theta (\alpha ))/2 \in V_\lambda$ for $\lambda = \pm \, 1$. For any $\beta \in V_{-1}$ and $h \in \mathfrak {h}_1$, we have $\beta (h) = 0$ as $\beta (h) = \beta (\Theta (h)) = \Theta (\beta )(h)= -\beta (h)$. Hence we may consider $V_{-1}$ as a subspace of $\mathfrak {a}^{*}$ and $V_{-1}=\mathbb {R}\Sigma$. For any $\beta \in V$ we define

$$\begin{aligned} \widetilde{\beta } = \dfrac{\beta - \Theta (\beta )}{2}. \end{aligned}$$

(2.19)

Equation (2.18) implies that $\Sigma = \{ \widetilde{\beta } \mid \beta \in \Phi , \widetilde{\beta } \ne 0\}$. We write $\widetilde{\Pi }=\{\widetilde{\alpha }_i\,|\, i\in I{{\setminus }} X\}$ and define $Q(\Sigma )=\mathbb {Z}\Sigma =\mathbb {Z}\widetilde{\Pi }$ and $Q^+(\Sigma )=\mathbb {N}_0\widetilde{\Pi }$. For any $w \in W^{\Theta }$, we have

$$\begin{aligned} \begin{aligned} w(\widetilde{\beta })&= w\bigg ( \dfrac{\beta - \Theta (\beta )}{2} \bigg ) = \dfrac{w(\beta ) - w(\Theta (\beta ))}{2}\\&= \dfrac{w(\beta ) - \Theta (w(\beta ))}{2} = \widetilde{w(\beta )}. \end{aligned} \end{aligned}$$

(2.20)

Hence the group $W^{\Theta }$ acts on $\Sigma$.

We have an inner product on $V_{-1}$ by restriction of the inner product on V. As the inner product on V is W-invariant and $V_{-1}$ is a $W^{\Theta }$-invariant subspace, the restriction of the inner product on $V_{-1}$ is $W^{\Theta }$-invariant. For any $i \in X$, we have $\alpha _i \in V_{+1}$. Hence the direct sum decomposition (2.17) implies that $\sigma _i(\widetilde{\beta }) = \widetilde{\beta }$ for all $\widetilde{\beta } \in \Sigma , i \in X$. The group $\widetilde{W}$ on the other hand can be interpreted in terms of the restricted root system $\Sigma$. We include a pedestrian proof, avoiding the more sophisticated setting in [22].

Proposition 2.7

• (1) The reflections at the hyperplanes perpendicular to elements of $\Sigma$ generate a finite reflection group $W(\Sigma )$.

• (2) There is an isomorphism of groups $\rho :\widetilde{W}\rightarrow W(\Sigma )$ which sends $\widetilde{\sigma }_i$ to the reflection at the hyperplane orthogonal to $\widetilde{\alpha }_i$ for any $i\in I{{\setminus }} X$.

Proof

For any $i \in I {{\setminus }} X$ we have

$$\begin{aligned} \widetilde{\sigma }_i(\widetilde{\alpha }_i)&= \big (w_X^{-1}w_{\{i,\tau (i)\} \cup X} \big )(\widetilde{\alpha }_i)\\&= w_{\{i, \tau (i)\} \cup X} (\widetilde{\alpha }_i)\\&= \big ( w_{\{i,\tau (i)\} \cup X}(\alpha _i) \big )|_{\mathfrak {a}}. \end{aligned}$$

It hence follows from (2.10) that

$$\begin{aligned} \widetilde{\sigma }_i(\widetilde{\alpha }_i) = -\,\widetilde{\alpha }_i. \end{aligned}$$

(2.21)

Now suppose $\widetilde{\beta } \in V_{-1}$ such that $(\widetilde{\beta }, \widetilde{\alpha }_i) = 0$. Using the $W^{\Theta }$-invariance of the bilinear form on $V_{-1}$, we obtain

$$\begin{aligned} (\widetilde{\sigma }_i(\widetilde{\beta }), \widetilde{\alpha }_i)&=(\widetilde{\beta }, \widetilde{\sigma }_i(\widetilde{\alpha }_i) )\\&=-(\widetilde{\beta }, \widetilde{\alpha }_i)\\&=0. \end{aligned}$$

On the other hand, by the definition of $\widetilde{\sigma }_i$ we have

$$\begin{aligned} \widetilde{\sigma }_i(\beta ) = \beta + n_i\alpha _i + n_{\tau (i)}\alpha _{\tau (i)} + \sum _{j \in X}n_j\alpha _j \end{aligned}$$

for some $n_j \in \mathbb {Q}$. From this we obtain

$$\begin{aligned} \widetilde{\sigma }_i(\widetilde{\beta }) = \widetilde{\beta } + m_i\widetilde{\alpha }_i \end{aligned}$$

where $m_i = n_i + n_{\tau (i)}$. Hence,

$$\begin{aligned} 0&= (\widetilde{\sigma }_i(\widetilde{\beta }), \widetilde{\alpha }_i) = (\widetilde{\beta } + m_i\widetilde{\alpha }_i, \widetilde{\alpha }_i)\\&= (\widetilde{\beta }, \widetilde{\alpha }_i) + m_i(\widetilde{\alpha }_i, \widetilde{\alpha }_i)\\&= m_i(\widetilde{\alpha }_i, \widetilde{\alpha }_i). \end{aligned}$$

The inner product is positive definite so it follows that $m_i = 0$. Hence $\widetilde{\sigma }_i(\widetilde{\beta }) = \widetilde{\beta }$. This together with (2.21) implies that $\widetilde{\sigma }_i$ is the reflection at the hyperplane orthogonal to $\widetilde{\alpha }_i$. As seen above, the action of $\widetilde{W}$ on $V_{-1}$ gives a group homomorphism

$$\begin{aligned} \rho : \widetilde{W} \rightarrow W(\Sigma ). \end{aligned}$$

Adapting the proof of [10, Theorem 1.5] one shows that $\rho$ is surjective which implies part 1. To prove part 2 it remains to show that $\rho$ is injective. This is a consequence of Lemma 2.8 below. $\square$

Lemma 2.8

The action of $\widetilde{W}$ on $\Sigma$ is faithful.

Proof

Assume that there exists $w \in \widetilde{W}$ such that $w \ne 1_{\widetilde{W}}$ and

$$\begin{aligned} w (\widetilde{\alpha }_i) = \widetilde{\alpha }_i \quad \text{ for } \text{ all } i \in I. \end{aligned}$$

We can rewrite this formula as

$$\begin{aligned} w(\alpha _i) - w(\Theta (\alpha _i)) = \alpha _i - \Theta (\alpha _i). \end{aligned}$$

(2.22)

For all $i \in X$ we have $w(\alpha _i) > 0$ as $l(w\sigma _i) = l(w)+1$. Hence there exists $i \in I {{\setminus }} X$ such that $w(\alpha _i) < 0$. In this case also $w(\alpha _{\tau (i)}) < 0$ since elements of $\widetilde{W}$ are fixed under $\tau$. Consider Eq. (2.22) for this i: The right hand side lies in $Q^+$ and is of the form

$$\begin{aligned} \alpha _i + \alpha _{\tau (i)} + \sum _{j \in X} n_j\alpha _j \end{aligned}$$

(2.23)

where $n_j \in \mathbb {N}_0$ for each $j \in X$. We can write the left hand side as

$$\begin{aligned} w(\alpha _i) - w(\Theta (\alpha _i)) = w(\alpha _i) + w(\alpha _{\tau (i)}) + \sum _{j \in X} m_j w(\alpha _j) \end{aligned}$$

(2.24)

where $m_j \in \mathbb {N}_0$ for each $j \in X$. Hence inserting (2.23) and (2.24) into (2.22), we get

$$\begin{aligned} w(\alpha _i) + w(\alpha _{\tau (i)}) + \sum _{j \in X}m_jw(\alpha _j) = \alpha _i + \alpha _{\tau (i)} + \sum _{j \in X}n_j\alpha _j. \end{aligned}$$

Now we apply the tilde map to the above equation. The terms involving $\alpha _j$ for $j \in X$ vanish, because the tilde map is zero on $Q_X$ and w commutes with $\Theta$. We get

$$\begin{aligned} \widetilde{w(\alpha _i)} + \widetilde{w(\alpha _{\tau (i)})} = \widetilde{\alpha }_i + \widetilde{\alpha }_{\tau (i)}. \end{aligned}$$

The right hand side lies in $Q^+(\Sigma )$. The left hand side lies in $-Q^+(\Sigma )$ because $w(\alpha _i)$ and $w(\alpha _{\tau (i)})$ lie in $-Q^+$. Hence both sides of the equation must vanish. However, this is not possible, in particular for the right hand side which is $2\widetilde{\alpha }_i$. We have a contradiction. $\square$

Proposition 2.7 has the following consequence which we note for later reference.

Corollary 2.9

For any $i\in I{{\setminus }} X$ and $\mu \in Q(\Sigma )$ the relation

$$\begin{aligned} \widetilde{\sigma }_i(\mu ) = \mu - 2\frac{(\mu ,\widetilde{\alpha }_i)}{(\widetilde{\alpha }_i,\widetilde{\alpha }_i)}\widetilde{\alpha }_i \end{aligned}$$

holds and $\displaystyle 2\frac{(\mu ,\widetilde{\alpha }_i)}{(\widetilde{\alpha }_i,\widetilde{\alpha }_i)}\in \mathbb {Z}$.

3 Factorisation of quasi K-matrices

As explained in the introduction, the quasi R-matrix for $U_q(\mathfrak {g})$ [23, Chapter 4] has a deep connection to the Weyl group W. This was first observed by Levendorskiĭ and Soibelman [21], and independently by Kirillov and Reshetikhin [14]. In this section, we establish a similar connection between the quasi K-matrix $\mathfrak {X}$ for a quantum symmetric pair and the restricted Weyl group $\widetilde{W}$. In particular, we will see in many cases that the quasi K-matrix $\mathfrak {X}$ factorises into a product of quasi K-matrices for Satake diagrams of rank one.

We first fix notation for quantised enveloping algebras and quantum symmetric pairs in Sects. 3.1 and 3.2. Recall that the construction of quantum symmetric pairs depends on an additional choice of parameters $\mathbf {c} \in \mathcal {C}$ and $\mathbf {s} \in \mathcal {S}$, see Definition 3.1. In Sects. 3.3 and 3.4 we find explicit formulas for $\mathfrak {X}$ in the case $\mathbf {s} = (0, \dotsc , 0)$. Section 3.5 then deals with the case of general parameters $\mathbf {s}$.

3.1 Quantised enveloping algebras

In this section we fix notation for quantum groups mostly following the conventions in [23] and [11]. Let q be an indeterminate and $\mathbb {K}$ a field of characteristic zero. Denote by $\mathbb {K}(q)$ the field of rational functions in q with coefficients in $\mathbb {K}$. The quantised enveloping algebra $U_q(\mathfrak {g})$ is the associative $\mathbb {K}(q)$-algebra generated by elements $E_i, F_i, K_i^{\pm 1}$ for all $i \in I$ satisfying the relations given in [23, 3.1.1] or [11, 4.3]. The algebra $U_q(\mathfrak {g})$ has the structure of a Hopf algebra with coproduct $\Delta$, counit $\varepsilon$ and antipode S given by

$$\begin{aligned} \Delta (E_i)&= E_i \otimes 1 + K_i\otimes E_i,&\varepsilon (E_i)&= 0,&S(E_i)&= -K_i^{-1}E_i, \end{aligned}$$

(3.1)

$$\begin{aligned} \Delta (F_i)&= F_i \otimes K_i^{-1} + 1 \otimes F_i,&\varepsilon (F_i)&= 0,&S(F_i)&= -F_iK_i, \end{aligned}$$

(3.2)

$$\begin{aligned} \Delta (K_i)&= K_i \otimes K_i,&\varepsilon (K_i)&= 1,&S(K_i)&= K_i^{-1}, \end{aligned}$$

(3.3)

for all $i \in I$. Let $U^{+}, U^{-}$ and $U^0$ denote the subalgebras of $U_q(\mathfrak {g})$ generated by $\{E_i \mid i \in I\}$, $\{F_i \mid i \in I\}$ and $\{K_i^{\pm 1} \mid i \in I\}$, respectively. For $\lambda = \sum _{i \in I} n_i\alpha _i \in Q$ we also write

$$\begin{aligned} K_\lambda = \prod _{i \in I}K_i^{n_i}. \end{aligned}$$

(3.4)

The elements $K_\lambda$ for $\lambda \in Q$ form a vector space basis of $U^0$. For any $U^0$-module M and any $\mu \in Q$ let

$$\begin{aligned} M_{\mu } = \{m \in M \mid K_im = q^{(\mu , \alpha _i)}m \,\, \hbox { for all}\ i \in I\} \end{aligned}$$

(3.5)

denote the corresponding weight space. In particular, both $U^+$ and $U^-$ are $U^0$-modules via the left adjoint action so we may apply the above notation. This gives algebra gradings

$$\begin{aligned} U^+&= \bigoplus _{\mu \in Q^+} U_{\mu }^+,&U^-&= \bigoplus _{\mu \in Q^+} U_{-\mu }^-. \end{aligned}$$

(3.6)

By [23, 39.4.3] the braid group Br(W) acts on $U_q(\mathfrak {g})$ by algebra automorphisms analogously to the action of Br(W) on $\mathfrak {g}$. For any $i \in I$ let $T_i$ be the algebra automorphism of $U_q(\mathfrak {g})$ denoted by $T_{i,1}^{\prime \prime }$ in [23, 37.1]. The automorphisms $T_i$ for $i \in I$ satisfy braid relations

$$\begin{aligned} \underbrace{T_iT_jT_i \cdots }_{\small {m_{ij} \text{ factors }}} = \underbrace{T_jT_iT_j \cdots }_{\small {m_{ij} \text{ factors }}} \end{aligned}$$

(3.7)

where $m_{ij}$ denotes the order of $\sigma _i\sigma _j \in W$. Hence for any $w \in W$, there is a well-defined algebra automorphism $T_w: U_q(\mathfrak {g}) \rightarrow U_q(\mathfrak {g})$ defined by

$$\begin{aligned} T_w = T_{i_1}T_{i_2} \cdots T_{i_k} \end{aligned}$$

(3.8)

where $w = \sigma _{i_1}\sigma _{i_2}\dots \sigma _{i_k}$ is a reduced expression.

In Sects. 3.2 and 3.4 it is necessary to consider a completion $\mathscr {U}$ of $U_q(\mathfrak {g})$. We recall the construction of $\mathscr {U}$ following [3, 3.1]. Let $\mathcal {O}_{int}$ denote the category of finite dimensional $U_q(\mathfrak {g})$-modules of type 1, and let $\mathcal {V}ect$ denote the category of vector spaces over $\mathbb {K}(q)$. Both categories are tensor categories and the forgetful functor $\mathcal {F}or: \mathcal {O}_{int} \rightarrow \mathcal {V}ect$ is a tensor functor. We let $\mathscr {U} = \mathrm {End}(\mathcal {F}or)$ be the $\mathbb {K}(q)$-algebra of all natural transformations from the functor $\mathcal {F}or$ to itself. Observe that $U_q(\mathfrak {g})$ and $\widehat{U^+} = \prod _{\mu \in Q^+} U_{\mu }^+$ may be considered as subalgebras of $\mathscr {U}$, see [3, 3.1]. We usually write elements in $\widehat{U^+}$ additively as infinite sums $\sum _{\mu \in Q^+} u_\mu$ with $u_\mu \in U^+_\mu$.

By [23, 1.2.13] for any $i\in I$ there exist uniquely determined linear maps ${}_ir$, $r_i: U^{+} \rightarrow U^{+}$ satisfying

$$\begin{aligned} {}_{i}r(E_j)&= \delta _{ij},&{}_{i}r(xy)&= {}_{i}r(x)y + q^{(\alpha _i,\mu )}x{}_{i}r(y) \end{aligned}$$

(3.9)

$$\begin{aligned} r_{i}(E_j)&= \delta _{ij},&r_{i}(xy)&= q^{(\alpha _i, \nu )}r_{i}(x)y + xr_{i}(y), \end{aligned}$$

(3.10)

for all $j \in I$, $x \in U^{+}_{\mu }$ and $y \in U^{+}_{\nu }$ where $\mu ,\nu \in Q^+$. We may extend the skew derivation $_{i}r: U^+ \rightarrow U^+$ to a linear map

$$\begin{aligned} {}_{i}r: \widehat{U^+} \rightarrow \widehat{U^+}, \quad \sum _{\mu \in Q^+} u_{\mu } \mapsto \sum _{\mu \in Q^+} {}_{i}r(u_{\mu }) \end{aligned}$$

(3.11)

where ${}_ir(u_\mu )$ is the component in $U^+_{\mu -\alpha _i}$ for all $\mu \in Q^+$ with $\mu \ge \alpha _i$. Similarly we may extend the skew derivation $r_i: U^+ \rightarrow U^+$ to a linear map $r_i: \widehat{U^+} \rightarrow \widehat{U^+}$.

Finally, recall that the bar involution for $U_q(\mathfrak {g})$ is the $\mathbb {K}$-algebra automorphism

(3.12)

defined by $\overline{q}^U = q^{-1}$ and $\overline{E_i}^U = E_i$, $\overline{F_i}^U = F_i$, $\overline{K_i}^U = K_i^{-1}$ for all $i \in I$.

3.2 Quantum symmetric pairs

We recall the definition of quantum symmetric pair coideal subalgebras $B_{\mathbf {c},\mathbf {s}}$ as introduced by Letzter [16]. Here we follow the conventions in [12]. Let $(X, \tau )$ be a Satake diagram and let $s:I \rightarrow \mathbb {K}^{\times }$ be a function satisfying equations (2.2) and (2.3). Let $\mathcal {M}_X~=~U_q(\mathfrak {g}_X)$ be the subalgebra of $U_q(\mathfrak {g})$ generated by $\{E_i, F_i, K_i^{\pm 1} \mid i \in X \}$, and let $U_\Theta ^0$ be the subalgebra of $U^0$ generated by $\{K_\lambda \,|\,\lambda \in Q, \Theta (\lambda )=\lambda \}$. Quantum symmetric pair coideal subalgebras depend on a choice of parameters $\mathbf {c} = (c_i)_{i \in I {{\setminus }} X} \in (\mathbb {K}(q)^{\times })^{I {{\setminus }} X}$ and $\mathbf {s} = (s_i)_{i \in I {{\setminus }} X} \in (\mathbb {K}(q)^{\times })^{I {{\setminus }} X}$ with added constraints. Define

$$\begin{aligned} I_{ns} = \{i \in I {{\setminus }} X \mid \tau (i) = i \text{ and } a_{ij} = 0 \text{ for } \text{ all } j \in X \} \end{aligned}$$

(3.13)

where $a_{ij} = 2\frac{(\alpha _i, \alpha _j)}{(\alpha _j,\alpha _j)}$ for $i,j \in I$ are the entries of the Cartan matrix of $\mathfrak {g}$. Define the parameter sets

$$\begin{aligned} \mathcal {C}&= \{ \mathbf {c} \in (\mathbb {K}(q)^{\times })^{I {{\setminus }} X} \mid c_i = c_{\tau (i)} \text{ if } \tau (i) \ne i \text{ and } (\alpha _i, \Theta (\alpha _i)) = 0 \}, \end{aligned}$$

(3.14)

$$\begin{aligned} \mathcal {S}&= \{ \mathbf {s} \in (\mathbb {K}(q)^{\times })^{I {{\setminus }} X} \mid s_j \ne 0 \Rightarrow (j \in I_{ns} \text{ and } a_{ij} \in -2\mathbb {N}_0 \; \forall i \in I_{ns} {{\setminus }} \{j\}) \}. \end{aligned}$$

(3.15)

Definition 3.1

Let $(X, \tau )$ be a Satake diagram, $\mathbf {c} = (c_i)_{i \in I {{\setminus }} X} \in \mathcal {C}$ and $\mathbf {s} = (s_i)_{i \in I {{\setminus }} X} \in \mathcal {S}$. The quantum symmetric pair coideal subalgebra $B_{\mathbf {c},\mathbf {s}}= B_{\mathbf {c},\mathbf {s}}(X, \tau )$ is the subalgebra of $U_q(\mathfrak {g})$ generated by $\mathcal {M}_X, U_{\Theta }^0$ and the elements

$$\begin{aligned} B_i = F_i - c_is(\tau (i))T_{w_X}(E_{\tau (i)})K_i^{-1} + s_iK_i^{-1} \end{aligned}$$

(3.16)

for all $i\in I{{\setminus }} X$.

By [12, Proposition 5.2] the algebra $B_{\mathbf {c},\mathbf {s}}$ is a right coideal subalgebra of $U_q(\mathfrak {g})$, that is

$$\begin{aligned} \Delta (B_{\mathbf {c},\mathbf {s}}) \subseteq B_{\mathbf {c},\mathbf {s}}\otimes U_q(\mathfrak {g}). \end{aligned}$$

(3.17)

All through this paper we assume that the parameters $\mathbf {c} = (c_i)_{i \in I {{\setminus }} X} \in \mathcal {C}$ and $\mathbf {s} = (s_i)_{i \in I {{\setminus }} X} \in \mathcal {S}$ satisfy the additional relations

$$\begin{aligned} c_{\tau (i)}&= q^{(\alpha _i, \Theta (\alpha _i) - 2\rho _X)} \overline{c_i}^U, \end{aligned}$$

(3.18)

$$\begin{aligned} \overline{s_i}^U&= s_i \end{aligned}$$

(3.19)

for all $i \in I {{\setminus }} X$. By [2, Theorem 3.11], if condition (3.18) holds, then there exists a $\mathbb {K}$-algebra automorphism

(3.20)

such that

(3.21)

The map is called the bar involution for $B_{\mathbf {c},\mathbf {s}}$ and plays a similar role as the bar involution (3.12) for $U_q(\mathfrak {g})$. In particular there exists a quasi K-matrix $\mathfrak {X}\in \mathscr {U}$ which resembles the quasi R-matrix for $U_q(\mathfrak {g})$. More explicitly, following a program outlined by H. Bao and W. Wang in [4], it was proved in [3, Theorem 6.10] that there exists a uniquely determined element $\mathfrak {X}= \sum _{\mu \in Q^+} \mathfrak {X}_{\mu } \in \prod _{\mu \in Q^+} U_{\mu }^+$ with $\mathfrak {X}_0 = 1$ and $\mathfrak {X}_{\mu } \in U_{\mu }^+$ such that the equation

$$\begin{aligned} \overline{b}^B \mathfrak {X}= \mathfrak {X}\overline{b}^U \end{aligned}$$

(3.22)

holds in $\mathscr {U}$ for all $b \in B_{\mathbf {c},\mathbf {s}}$. For symmetric pairs of type AIII/IV with $X=\emptyset$ the existence of the quasi K-matrix satisfying (3.22) was first observed in [4]. The quasi K-matrix is an essential building block for the construction of the universal K-matrix in [3]. To unify notation we define $c_i = s_i = 0$ and $B_i = F_i$ for $i\in X$. Moreover, we write

$$\begin{aligned} X_i&= -s(\tau (i))T_{w_X}(E_{\tau (i)})&\text{ for } \text{ all } i \in I {{\setminus }} X, \end{aligned}$$

(3.23)

$$\begin{aligned} X_j&= 0&\text{ for } \text{ all } j\in X. \end{aligned}$$

(3.24)

By [3, Proposition 6.1] the quasi K-matrix $\mathfrak {X}= \sum _{\mu \in Q^+}\mathfrak {X}_{\mu }$ is the unique solution to the recursive system of equations

$$\begin{aligned} {}_{i}r(\mathfrak {X}_\mu ) = -(q_i - q_i^{-1}) \big (q^{-(\Theta (\alpha _i),\alpha _i)}c_iX_i \mathfrak {X}_{\mu + \Theta (\alpha _i)-\alpha _i} + s_i\mathfrak {X}_{\mu -\alpha _i} \big ) \quad \text{ for } \text{ all } i\in I\nonumber \\ \end{aligned}$$

(3.25)

with the normalisation $\mathfrak {X}_0 = 1$. Using the extension of the skew derivation ${}_{i}r$ to $\widehat{U^+}$ given by (3.11) we can rewrite (3.25) in the compact form

$$\begin{aligned} {}_{i}r(\mathfrak {X}) = -(q_i-q_i^{-1}) \big (q^{-(\Theta (\alpha _i),\alpha _i)}c_iX_i \mathfrak {X}+ s_i\mathfrak {X}\big ) \qquad \text{ for } \text{ all } i\in I. \end{aligned}$$

(3.26)

In Sect. 3.3 and in Appendix A we will use the above formula to perform uniform calculations with $\mathfrak {X}$. Equation (3.25) implies in particular that

$$\begin{aligned} {}_{j}r(\mathfrak {X}_\mu )=0 \qquad \text{ for } \text{ all } j\in X \end{aligned}$$

(3.27)

as $c_j = s_j = 0$ for all $j \in X$. This property has the following consequence which was already observed in [5, Proposition 4.15]. Recall that $w_0\in W$ denotes the longest element. Moreover, for any $w\in W$ recall the definition of the subalgebra $U^+[w]$ of $U^+$ given in [11, 8.21, 8.24].

Lemma 3.2

For any $\mu \in Q^+$ the relation $\mathfrak {X}_\mu \in U^+[w_Xw_0]$ holds.

Proof

In view of (3.27), Equation (4) of [11, 8.26] implies that $\mathfrak {X}_\mu \in U^+[\sigma _jw_0]$ for all $j \in X$. By [9, Theorem 7.3] we have

$$\begin{aligned} \bigcap _{j \in X} U^+[\sigma _jw_0] = U^+[w_Xw_0] \end{aligned}$$

which completes the proof of the Lemma. $\square$

We write $\mathfrak {X}_{\mathbf {c},\mathbf {s}}$ for $\mathfrak {X}$ if we need to specify the dependence on the parameters. Any diagram automorphism $\eta : I \rightarrow I$ induces a map $\eta : \mathbb {K}(q)^{I {{\setminus }} X} \rightarrow \mathbb {K}(q)^{I {{\setminus }} X}$ by

$$\begin{aligned} \eta ({{\mathbf {c}}}) = {{\mathbf {d}}} \quad \text{ with } d_i = c_{\eta ^{-1}(i)} \text{ where } {{\mathbf {c}}}=(c_i)_{i\in I{{\setminus }} X} \text{ and } {{\mathbf {d}}}=(d_i)_{i\in I{{\setminus }} X}. \end{aligned}$$

(3.28)

This notation allows us to record the effect of diagram automorphisms on the quasi K-matrix $\mathfrak {X}$.

Lemma 3.3

Let $\eta : I \rightarrow I$ be a diagram automorphism and $\mathbf {c} \in \mathcal {C}$, $\mathbf {s} \in \mathcal {S}$. Then $\eta (\mathbf {c}) \in \mathcal {C}$, $\eta (\mathbf {s}) \in \mathcal {S}$ and

$$\begin{aligned} \eta (\mathfrak {X}_{\mathbf {c},\mathbf {s}}) = \mathfrak {X}_{\eta (\mathbf {c}),\eta (\mathbf {s})}. \end{aligned}$$

(3.29)

Proof

The relations $\eta (\mathbf {c}) \in \mathcal {C}$, $\eta (\mathbf {s}) \in \mathcal {S}$ follow from the definitions (3.14) and (3.15) of $\mathcal {C}$ and $\mathcal {S}$. By [3, Proposition 6.1], property (3.22) is equivalent to the relation

$$\begin{aligned} B_i^{\mathbf {c},\mathbf {s}} \mathfrak {X}_{\mathbf {c},\mathbf {s}} = \mathfrak {X}_{\mathbf {c},\mathbf {s}} \overline{B_i^{\mathbf {c},\mathbf {s}}}^U \quad \text{ for } \text{ all } i \in I \end{aligned}$$

(3.30)

where we write $B_i^{\mathbf {c},\mathbf {s}}$ instead of $B_i$ to denote the dependence on $\mathbf {c}$ and $\mathbf {s}$.

By construction, $\eta (B_i^{\mathbf {c},\mathbf {s}}) = B_{\eta (i)}^{\eta (\mathbf {c}),\eta (\mathbf {s})}$ and $\eta :U_q(\mathfrak {g}) \rightarrow U_q(\mathfrak {g})$ commutes with the bar involution on $U_q(\mathfrak {g})$. Hence applying $\eta$ to relation (3.30) we obtain

$$\begin{aligned} B_{\eta (i)}^{\eta (\mathbf {c}),\eta (\mathbf {s})} \eta (\mathfrak {X}_{\mathbf {c},\mathbf {s}}) = \eta (\mathfrak {X}_{\mathbf {c},\mathbf {s}}) \overline{B_{\eta (i)}^{\eta (\mathbf {c}),\eta (\mathbf {s})}}^U\quad \text{ for } \text{ all } i \in I \end{aligned}$$

which proves (3.29). $\square$

3.3 Quasi K-matrices for Satake diagrams of rank one

For the remainder of this paper, following Remark 2.2, we denote Satake diagrams as triples $(I, X, \tau )$ to also indicate the underlying Lie algebra.

Definition 3.4

A subdiagram of a Satake diagram $(I,X,\tau )$ is a triple $(J, X \cap J, \tau |_J)$ such that $J \subset I$ and $(J,X \cap J, \tau |_J)$ is a Satake diagram for the subdiagram of the Dynkin diagram of $\mathfrak {g}$ indexed by J.

Adding a connected component consisting entirely of black dots to a Satake diagram does not change the corresponding quasi K-matrix. For this reason we only consider subdiagrams $(J,X \cap J, \tau |_J)$ with the property that any connected component of $X \cap J$ is connected to a white node of J. Let $\widetilde{I}$ be the set of $\tau$-orbits of $I {{\setminus }} X$. There is a projection map

$$\begin{aligned} \pi :I {{\setminus }} X \longrightarrow \widetilde{I} \end{aligned}$$

(3.31)

that takes any white node to the $\tau$-orbit it belongs to.

Definition 3.5

The rank of a Satake diagram $(I,X,\tau )$ is  defined by  $\mathrm {rank}(I,X,\tau )=|\pi (I {{\setminus }} X ) |$.

In other words, a Satake diagram has rank n if there are n distinct orbits of white nodes. By Proposition 2.7 the rank of a Satake diagram coincides with the rank of the corresponding restricted root system $\Sigma$.

Given a Satake diagram $(I,X,\tau )$, any $i\in I{{\setminus }} X$ determines a subdiagram $(\{i,\tau (i)\}\cup X, X, \tau |_{\{i,\tau (i)\}\cup X})$ of rank one. Let $\mathfrak {X}_i$ be the quasi K-matrix corresponding to this rank one subdiagram. For any $w\in W$ we define $\widehat{U^+[w]} = \prod _{\mu \in Q^+} U^+[w]_{\mu }$. As $U[w]^+$ is a subalgebra of $U^+$ we obtain that $\widehat{U^+[w]}$ is a subalgebra of $\widehat{U^+}$ and hence of $\mathscr {U}$. Formulating Lemma 3.2 in the present setting we obtain

$$\begin{aligned} \mathfrak {X}_i \in \widehat{U^+[\widetilde{\sigma }_i]}. \end{aligned}$$

(3.32)

In the following lemma we consider the case $\tau (i)=i$ and make the dependence of $\mathfrak {X}_i$ on the parameter $c_i$ more explicit.

Lemma 3.6

Assume that $\mathbf {s} = (0, \dotsc ,0)$ and $i \in I {{\setminus }} X$ satisfies $\tau (i) = i$. Then

$$\begin{aligned} \mathfrak {X}_i = \sum _{n \in \mathbb {N}_0} c_i^n \mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))} \end{aligned}$$

(3.33)

where $\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))} \in U^+_{n(\alpha _i - \Theta (\alpha _i))}$ is independent of $\mathbf {c}$.

Proof

It follows from the recursion (3.25) and the assumption $s_i = 0$ that $\mathfrak {X}_i = \sum _{n \in \mathbb {N}_0} \mathfrak {X}_{n(\alpha _i - \Theta (\alpha _i))}$ with $\mathfrak {X}_{n(\alpha _i - \Theta (\alpha _i))} \in U^+_{n(\alpha _i - \Theta (\alpha _i))}$. Again by (3.25), the elements $\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))} = c_i^{-n}\mathfrak {X}_{n(\alpha _i - \Theta (\alpha _i))}$ for $n \in \mathbb {N}$ satisfy the relations

$$\begin{aligned} {}_{i}r(\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))}) = -(q-q^{-1})q^{-(\Theta (\alpha _i),\alpha _i)}X_i\mathcal {E}_{(n-1)(\alpha _i - \Theta (\alpha _i))} \end{aligned}$$

(3.34)

and

$$\begin{aligned} {}_{j}r(\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))}) = 0 \quad \text{ for } j \in X. \end{aligned}$$

(3.35)

The relations (3.34) and (3.35) are independent of $\mathbf {c}$ and determine $\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))}$ uniquely if we additionally impose $\mathcal {E}_0 = 1$. $\square$

The quasi K-matrices of rank one are the building blocks for quasi K-matrices of higher rank. In the following we give explicit formulas for rank one quasi K-matrices of type A shown on the left hand side of Table 1 in the case $\mathbf {s} = (0, \dotsc ,0)$. The additional rank one Satake diagrams on the right hand side of Table 1 are not tackled in this paper. However, the rank one quasi K-matrices of type BII and DII have been determined in [6, 5.3].

Recall the definition of the q-number

$$\begin{aligned}{}[n]_{q_i} = [n]_i = \dfrac{q_i^n - q_i^{-n}}{q_i - q_i^{-1}} \qquad \text{ for } \text{ any } n \in \mathbb {Z} \text{ and } i \in I \end{aligned}$$

(3.36)

and of the q-factorial $[n]_i!=[1]_i [2]_i \dots [n]_i$ for $n\ge 0$, see for example [11, Chapter 0]. If all roots $\alpha \in \Phi$ are of the same length, then we simply write [n] and [n]!.

Throughout the following calculations in the present section and in Appendix A, we use a modified form of the q-number $[n]_i$. For $n\ge 1$ define

$$\begin{aligned} \{ n \}_i = q_i^{n-1}[n]_i = 1 + q_i^2 + \dots + q_i^{2(n-1)}. \end{aligned}$$

(3.37)

Define $\{n\}_i! = \prod _{k=1}^{n}\{k\}_i$, and let $\{n\}_i!!$ be the double factorial of $\{n\}_i$ defined by

$$\begin{aligned} \{n\}_i!! = \prod _{k=0}^{\lceil \frac{n}{2}\rceil -1} \{n - 2k\}_i. \end{aligned}$$

(3.38)

For $n=0$ we set $\{0\}_i! = 1$ and $\{0\}_i!!=1$. Again, we omit the index i if all roots are of the same length. Further we use the following conventions. For any $x,y \in U_q(\mathfrak {g})$, $a \in \mathbb {K}(q)$ we denote by $[x,y]_a$ the element $xy - ayx$. For any $i,j \in I$ we write $T_{ij} = T_i \circ T_j:U_q(\mathfrak {g})\rightarrow U_q(\mathfrak {g})$ and we extend this definition recursively.

Table 1 Satake diagrams of symmetric pairs of rank one

3.3.1 Type $AI_1$

Consider the Satake diagram of type $AI_1$.

Lemma 3.7

The quasi K-matrix $\mathfrak {X}$ of type $AI_1$ is given by

$$\begin{aligned} \mathfrak {X}= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{2n\}!! } (q^2c_1)^n E_1^{2n}. \end{aligned}$$

(3.39)

Proof

By Eq. (3.26), we need to show that

$$\begin{aligned} {}_{1}r(\mathfrak {X}) = (q-q^{-1})(q^2c_1)E_1\mathfrak {X}. \end{aligned}$$

Using the recursive formula (3.9) for ${}_1r$, we see that

$$\begin{aligned} {}_{1}r(E_1^n) = \{n\}E_1^{n-1} \qquad \text{ for } \text{ all } n\in \mathbb {N}. \end{aligned}$$

(3.40)

Hence

$$\begin{aligned} {}_{1}r(\mathfrak {X})&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}!!} (q^2c_1)^n {}_{1}r(E_1^{2n})\\&= \sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{\{2n\}!!} (q^2c_1)^n \{2n\} E_1^{2n-1}\\&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^{n+1}}{\{2n\}!!} (q^2c_1)^{n+1} E_1^{2n+1}\\&= (q-q^{-1})(q^2c_1)E_1\mathfrak {X}\end{aligned}$$

as required. $\square$

3.3.2 Type $AII_3$

Consider the Satake diagram of type $AII_3$.

Lemma 3.8

The quasi K-matrix $\mathfrak {X}$ of type $AII_3$ is given by

$$\begin{aligned} \mathfrak {X}= \sum _{n \ge 0} \dfrac{(qc_2)^n}{\{n\}!} [E_2, T_{13}(E_2)]_{q^{-2}}^n. \end{aligned}$$

(3.41)

Proof

Since $T_{13}(E_2) = [E_1, T_3(E_2)]_{q^{-1}}$, property (3.9) of the skew derivative ${}_1r$ implies that ${}_{1}r(T_{13}(E_2)) = (1-q^{-2})T_3(E_2)$. Again by property (3.9), it follows that ${}_{1}r([E_2,T_{13}(E_2)]_{q^{-2}}) = 0$. Hence ${}_{1}r(\mathfrak {X}) = 0$. By symmetry, we also have ${}_{3}r(\mathfrak {X}) = 0$.

We want to show that

$$\begin{aligned} {}_{2}r(\mathfrak {X}) = (q-q^{-1})c_2T_{13}(E_2)\mathfrak {X}. \end{aligned}$$

Since ${}_{2}r(T_{13}(E_2)) = 0$ by [11, 8.26, (4)], the relation

$$\begin{aligned} {}_{2}r([E_2,T_{13}(E_2)]_{q^{-1}}) = (1-q^{-2})T_{13}(E_2) \end{aligned}$$

holds in $U_q(\mathfrak {sl}_4)$.

Moreover, the element $T_{13}(E_2)$ commutes with the element $[E_2, T_{13}(E_2)]_{q^{-2}}$. Indeed, this follows from the fact that $E_2$ commutes with $[T_{213}(E_2), E_2]_{q^{-2}}$ by applying the automorphism $T_{13}$. This implies that the relation

$$\begin{aligned} {}_{2}r([E_2, T_{13}(E_2)]_{q^{-2}}^n) = (1-q^{-2})\{n\}T_{13}(E_2)[E_2, T_{13}(E_2)]_{q^{-2}}^{n-1} \end{aligned}$$

holds in $U_q(\mathfrak {sl}_4)$. Using this, we obtain

$$\begin{aligned} {}_{2}r(\mathfrak {X})&= \sum _{n \ge 0} \dfrac{(qc_2)^n}{\{n\}!} {}_{2}r([E_2,T_{13}(E_2)]_{q^{-2}}^n) \\&= (1-q^{-2})T_{13}(E_2) \sum _{n \ge 1} \dfrac{(qc_2)^n}{\{n-1\}!} [E_2, T_{13}(E_2)]_{q^{-2}}^{n-1}\\&= (q-q^{-1})c_2T_{13}(E_2)\mathfrak {X}\end{aligned}$$

as required. $\square$

3.3.3 Type $AIII_{11}$

Consider the Satake diagram of type $AIII_{11}$.

Note that $s(1) = s(2) = 1$ by (2.3) and $c_1 = c_2$ by (3.14) and (3.18).

Lemma 3.9

The quasi K-matrix $\mathfrak {X}$ of type $AIII_{11}$ is given by

$$\begin{aligned} \mathfrak {X}= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{n\}!}c_1^n (E_1E_2)^n. \end{aligned}$$

(3.42)

Proof

By symmetry, we only need to show that

$$\begin{aligned} {}_{1}r(\mathfrak {X}) = (q-q^{-1})c_1E_2\mathfrak {X}. \end{aligned}$$

By Eq. (3.40), we have

$$\begin{aligned} {}_{1}r(\mathfrak {X})&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{n\}!}c_1^n {}_{1}r((E_1E_2)^n)\\&= \sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{\{n\}!} \{n\} c_1^nE_1^{n-1}E_2^n\\&= (q-q^{-1})c_1 E_2\mathfrak {X}\end{aligned}$$

as required. $\square$

3.3.4 Type AIV for $n \ge 2$

Consider the Satake diagram of type AIV for $n \ge 2$.

By (2.3), we have $s(1) = -s(n)$ and by (3.18), we have $c_1 = q^{-2}\overline{c_n}$.

Lemma 3.10

The quasi K-matrix $\mathfrak {X}$ of type AIV is given by

$$\begin{aligned} \mathfrak {X}= \bigg ( \sum _{k \ge 0} \dfrac{ (c_1s(n) )^k}{\{k\}!} T_1T_{w_X}(E_n)^k \bigg ) \bigg ( \sum _{k \ge 0} \dfrac{ (c_ns(1) )^k}{\{k\}!} T_nT_{w_X}(E_1)^k \bigg ). \end{aligned}$$

(3.43)

Proof

We have ${}_{i}r(\mathfrak {X}) = 0$ for $i \in X$. Hence by symmetry we only need to show that

$$\begin{aligned} {}_{1}r(\mathfrak {X}) = (q-q^{-1})q^{-1}c_1s(n)T_{w_X}(E_n)\mathfrak {X}\end{aligned}$$

since $T_1T_{w_X}(E_n)$ and $T_nT_{w_X}(E_1)$ commute. We have

$$\begin{aligned} {}_{1}r(T_nT_{w_X}(E_1)^k)&= 0,\\ {}_{1}r(T_1T_{w_X}(E_n)^k)&= q^{-1}(q-q^{-1})\{k\} T_{w_X}(E_n)T_1T_{w_X}(E_n)^{k-1}. \end{aligned}$$

Using this, we obtain

$$\begin{aligned} {}_{1}r(\mathfrak {X})&= \bigg ( \sum _{k \ge 0} \dfrac{ (c_1s(n) )^k}{\{k\}!} {}_{1}r(T_1T_{w_X}(E_n)^k) \bigg ) \bigg ( \sum _{k \ge 0} \dfrac{ (c_ns(1) )^k}{\{k\}!} T_nT_{w_X}(E_1)^k \bigg )\\&= q^{-1}(q-q^{-1})c_1s(n) T_{w_X}(E_n)\mathfrak {X}\end{aligned}$$

as required. $\square$

Remark 3.11

Let $\mathscr {A}=\mathbb {Z}[q,q^{-1}]$ and let ${}_{\mathscr {A}}U^+$ be the $\mathscr {A}$-subalgebra of $U^+$ generated by $E_i^{(n)}=\frac{E_i^n}{[n]!}$ for all $n\in \mathbb {N}_0$, $i\in I$. Set ${}_{\mathscr {A}}\widehat{U^+}=\prod _{\mu \in Q^+}{}_{\mathscr {A}}U^+_\mu$ where ${}_{\mathscr {A}}U^+_\mu = {}_{\mathscr {A}}U^+\cap U^+_\mu$ for all $\mu \in Q^+$. By [5, Theorem 5.3] we have $\mathfrak {X}\in {}_{\mathscr {A}}\widehat{U^+}$ if $c_is(\tau (i))\in \pm q^\mathbb {Z}$ for all $i\in I{{\setminus }} X$. This integrality property is crucial for the theory of canonical bases of quantum symmetric pairs developed in [5].

We observe that the integrality of the quasi K-matrix in rank one can in some cases be read off from the explicit formulas given in this section. Indeed, Lemma 3.7, 3.9 and 3.10 imply that $\mathfrak {X}\in {}_{\mathscr {A}}\widehat{U^+}$ in the rank one cases of type AI, AIII and AIV. The rank one case $AII_3$ is more complicated, and Lemma 3.8 does not give an obvious way to see that $\mathfrak {X}\in {}_{\mathscr {A}}\widehat{U^+}$. Nevertheless, $\mathfrak {X}$ is also integral in this case, as shown in [5, A.5]. Based on the present remark, the integrality of $\mathfrak {X}$ in higher rank is discussed in Remark 3.23.

3.4 Partial quasi K-matrices

All through this section we make the assumption that $\mathbf {s} = {{\mathbf {0}}}= (0, 0, \dotsc ,0) \in \mathcal {S}$. In Sect. 3.5 we discuss the case of general parameters $\mathbf {s} \in \mathcal {S}$. Recall that the Lusztig automorphisms $T_i$ of $U_q(\mathfrak {g})$ for all $i \in I$ give rise to a representation of Br(W) on $U_q(\mathfrak {g})$. Since $\widetilde{W}$ is a subgroup of W, we obtain algebra automorphisms of $U_q(\mathfrak {g})$ defined by

$$\begin{aligned} \widetilde{T}_i := T_{\widetilde{\sigma }_i} \qquad \text{ for } \text{ each } i\in I{{\setminus }} X. \end{aligned}$$

By Theorem 2.4 and Proposition 2.5 the algebra automorphisms $\widetilde{T}_i$ give rise to a representation of $Br(\widetilde{W})$ on $U_q(\mathfrak {g})$.

Recall from Sect. 2.3 that $\widetilde{\Pi }=\{\widetilde{\alpha }_i\,|\,i\in I{{\setminus }} X\}$ and define $Q(2\Sigma )=2\mathbb {Z}\widetilde{\Pi }$ and $Q^+(2\Sigma )=2\mathbb {N}_0 \widetilde{\Pi }$. By (3.25) and the assumption $\mathbf {s}= {{\mathbf {0}}}$ we have

$$\begin{aligned} {}_{i}r(\mathfrak {X}_{\mu }) = -(q-q^{-1})q^{-(\Theta (\alpha _i),\alpha _i)} c_i X_i \mathfrak {X}_{\mu - 2\widetilde{\alpha }_i} \qquad \text{ for } \text{ any } \mu \in Q^+. \end{aligned}$$

Hence we may consider the quasi K-matrix $\mathfrak {X}$ as an element in $\prod _{\mu \in Q^+(2\Sigma )}U_{\mu }^+ \subset \mathscr {U}$. For any $w\in W$ define

$$\begin{aligned} \widetilde{U}^+[w]= \bigoplus \limits _{\mu \in Q^+(2\Sigma )} U^+[w]_{\mu } \end{aligned}$$

and set $\widetilde{U}^+ = \bigoplus _{\mu \in Q^+(2\Sigma )} U_{\mu }^+$. Then $\widetilde{U}^+$ and $\widetilde{U}^+[w]$ are $\mathbb {K}(q)$-subalgebras of $U^+$ and $U^+[w]$, respectively. In particular by Eq. (3.32) we have

Let $\mathbb {K}'$ be a field extension of $\mathbb {K}(q)$ which contains $q^{1/2}$ and elements $\widetilde{c}_i$ such that

$$\begin{aligned} \widetilde{c}_i^{\;2}=c_i c_{\tau (i)}s(i)s(\tau (i)) \qquad \text{ for } \text{ all } i\in I{{\setminus }} X. \end{aligned}$$

(3.44)

We extend $\widetilde{U}^+$ and $\widetilde{U}^+[w]$ for $w\in W$ to $\mathbb {K}'$-algebras $\widetilde{U}^+_{1/2} = \mathbb {K}' \otimes _{\mathbb {K}(q)} \widetilde{U}^+$ and $\widetilde{U}^+_{1/2}[w] = \mathbb {K}' \otimes _{\mathbb {K}(q)} \widetilde{U}^+[w]$. Define an algebra automorphism $\Psi : \widetilde{U}^+_{1/2} \rightarrow \widetilde{U}^+_{1/2}$ by

$$\begin{aligned} \Psi (E_{2\widetilde{\alpha }_i}) = q^{(\widetilde{\alpha }_i, \widetilde{\alpha }_i)} \widetilde{c}_iE_{2\widetilde{\alpha }_i} \qquad \text{ for } \text{ all } E_{2\widetilde{\alpha }_i} \in U^+_{2\widetilde{\alpha }_i}. \end{aligned}$$

(3.45)

For each $i \in I{{\setminus }} X$ define an algebra homomorphism

$$\begin{aligned} \Omega _i = \Psi \circ \widetilde{T}_i \circ \Psi ^{-1}: \widetilde{U}^+_{1/2}[\widetilde{\sigma }_iw_0] \rightarrow \widetilde{U}^{+}_{1/2}. \end{aligned}$$

The role of the conjugation by $\Psi$ in the definition of $\Omega _i$ is discussed in Remark 3.16 below. We consider the restriction of the algebra homomorphism $\Omega _i$ to the subalgebra $\widetilde{U}^+[\widetilde{\sigma }_iw_0]$, and we denote this restriction also by $\Omega _i$. Crucially, by the following proposition, the image of the restriction $\Omega _i$ belongs to $\widetilde{U}^+$ and does not involve any of the adjoined square roots.

Proposition 3.12

For every $i \in I {{\setminus }} X$ the map $\Omega _i: \widetilde{U}^+[\widetilde{\sigma }_iw_0] \rightarrow \widetilde{U}^+$ is a well defined algebra homomorphism.

Proof

It remains to show that the image of $\Omega _i$ is contained in $\widetilde{U}^+$. Observe that $\widetilde{T}_i(\widetilde{U}^+_{\mu }) \subseteq \widetilde{U}^+_{\widetilde{\sigma }_i(\mu )}$ for all $\mu \in Q^+(2\Sigma )$. By Corollary 2.9 we have

$$\begin{aligned} \widetilde{\sigma }_i(\mu ) = \mu - \dfrac{2(\mu , \widetilde{\alpha }_i)}{(\widetilde{\alpha }_i, \widetilde{\alpha }_i)} \widetilde{\alpha }_i \qquad \text{ for } \text{ all } \mu \in Q^+(2\Sigma ). \end{aligned}$$

Hence Eq. (3.45) implies that

$$\begin{aligned} \Omega _i|_{\widetilde{U}^+_{\mu }} = q^{-(\mu , \widetilde{\alpha }_i)} \widetilde{c}_i^{-(\mu ,\widetilde{\alpha }_i)/(\widetilde{\alpha }_i,\widetilde{\alpha }_i)} \widetilde{T}_i|_{\widetilde{U}^+_{\mu }}. \end{aligned}$$

Since $\mu \in Q^+(2\Sigma )$ it follows that the exponent $-(\mu ,\widetilde{\alpha }_i)$ is an integer. Moreover, Corollary 2.9 implies that the exponent $-(\mu , \widetilde{\alpha }_i)/(\widetilde{\alpha }_i,\widetilde{\alpha }_i)$ is an integer.

If $i = \tau (i)$ then Eq. (3.44) and condition (2.2) imply that $\widetilde{c}_i = \pm c_i$. This implies that the image of $\Omega _i$ is contained in $\widetilde{U}^+$ in this case.

Suppose instead that $i\in I{{\setminus }} X$ satisfies $i \ne \tau (i)$. If additionally $(\alpha _i,\Theta (\alpha _i))=0$, then (3.14) implies that $c_i=c_{\tau (i)}$. Moreover in this case $\Theta (\alpha _i)=-\alpha _{\tau (i)}$ and hence $s(i)=s(\tau (i))$ by (2.3). Hence we get $\widetilde{c}_i=\pm c_i s(i)$ in the case $i\ne \tau (i)$, $(\alpha _i,\Theta (\alpha _i))=0$ which implies that the image of $\Omega _i$ is contained in $\widetilde{U}^+$ in this case.

Finally, we consider the case that $i\ne \tau (i)$ and $(\alpha _i,\Theta (\alpha _i))\ne 0$. We are then in Case 3 in [20, p. 17] and hence the restricted root system $\Sigma$ is of type $(BC)_n$ for $n \ge 1$ and $(\widetilde{\alpha }_i, \widetilde{\alpha }_i) = \frac{1}{4}(\alpha _i,\alpha _i)$. Since $\mu \in Q^+(2\Sigma )\subset Q$ we have

$$\begin{aligned} \dfrac{(\mu ,\widetilde{\alpha }_i)}{(\widetilde{\alpha }_i, \widetilde{\alpha }_i)} =4 \dfrac{(\mu ,\alpha _i)}{(\alpha _i, \alpha _i)}\in 2\mathbb {Z}. \end{aligned}$$

Hence the image of $\Omega _i$ is contained in $\widetilde{U}^+$ in all cases as required. $\square$

Consider $\widetilde{w} \in \widetilde{W}$ and let $\widetilde{w} = \widetilde{\sigma }_{i_1}\widetilde{\sigma }_{i_2} \cdots \widetilde{\sigma }_{i_t}$ be a reduced expression. For $k = 1, \dotsc , t$ let

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}}^{[k]} = \Omega _{i_1}\Omega _{i_2} \cdots \Omega _{i_{k-1}}(\mathfrak {X}_{i_k}) = \Psi \circ \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{k-1}} \circ \Psi ^{-1}(\mathfrak {X}_{i_k}). \end{aligned}$$

(3.46)

By Proposition 2.5 we have $U^+[\widetilde{\sigma }_{i_k}] \subset U^+[\widetilde{\sigma }_{i_{k-1}}w_0]$ for $k = 2, \dotsc , t$, and

$$\begin{aligned} \widetilde{T}_{i_l} \cdots \widetilde{T}_{i_{k-1}}(U^+[\widetilde{\sigma }_{i_k}]) \subset U^+[\widetilde{\sigma }_{i_{l-1}}w_0] \qquad \text{ for } l=2, \dotsc , k-1 \end{aligned}$$

and hence the elements $\mathfrak {X}_{\widetilde{w}}^{[k]}$ are well-defined. Moreover, by Proposition 3.12 we have

When clear, we omit the subscript $\widetilde{w}$ and write $\mathfrak {X}^{[k]}$ instead of $\mathfrak {X}_{\widetilde{w}}^{[k]}$.

Definition 3.13

Let $\widetilde{w} \in \widetilde{W}$ and let $\widetilde{w} = \widetilde{\sigma }_{i_1}\widetilde{\sigma }_{i_2} \dots \widetilde{\sigma }_{i_t}$ be a reduced expression. The partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}}$ associated to $\widetilde{w}$ and the given reduced expression is defined by

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}} = \mathfrak {X}^{[k]}\mathfrak {X}^{[k-1]} \cdots \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}. \end{aligned}$$

(3.47)

We expect that the partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}}$ only depends on $\widetilde{w} \in \widetilde{W}$ and not on the chosen reduced expression. As we will see in Theorem 3.17 it suffices to check the independence of the reduced expression in rank two. If the Satake diagram is of rank two then the restricted Weyl group $\widetilde{W}$ is of one of the types $A_1 \times A_1$, $A_2$, $B_2$ or $G_2$. In each case, only the longest word for $\widetilde{W}$ has distinct reduced expressions.

Conjecture 3.14

Assume that $(I,X,\tau )$ is a Satake diagram of rank two. Then the element $\mathfrak {X}_{\widetilde{w}} \in \mathscr {U}$ defined by (3.47) depends only on $\widetilde{w} \in \widetilde{W}$ and not on the chosen reduced expression.

In Appendix A, we prove the following Theorem which confirms Conjecture 3.14 in many cases. The proof is performed by showing that for both reduced expressions of the longest word in $\widetilde{W}$ the resulting elements $\mathfrak {X}_w$ satisfy the relations (3.26).

Theorem 3.15

Assume that $\mathfrak {g} = \mathfrak {sl}_n(\mathbb {C})$ or $X = \emptyset$. Then Conjecture 3.14 holds.

Remark 3.16

The Hopf algebra automorphism $\Psi$ in the definition of $\Omega _i$ turns out to be necessary for the rank two calculations in Appendix A which prove Theorem 3.15. The conjugation by $\Psi$ affects the coefficients in the partial quasi K-matrix associated to a reduced expression of an element $\widetilde{w} \in \widetilde{W}$. In rank two the two partial quasi K-matrices associated to the longest word $\widetilde{w}_0\in \widetilde{W}$ coincide only after this change of coefficients. The effect of the conjugation by $\Psi$ can be seen in particular in Sects. A.3 and A.4 of the appendix which treat type $AIII_n$ for $n\ge 3$.

Once the rank two case is established, we can generalise to higher rank cases.

Theorem 3.17

Suppose that $(I,X,\tau )$ is a Satake diagram such that all subdiagrams $(J,X \cap J, \tau |_J)$ of rank two satisfy Conjecture 3.14. Then the element $\mathfrak {X}_{\widetilde{w}} \in \mathscr {U}$ depends on $\widetilde{w} \in \widetilde{W}$ and not on the chosen reduced expression.

Proof

Let $\widetilde{w}$ and $\widetilde{w}^{\prime }$ be reduced expressions which represent the same element in $\widetilde{W}$. Assume that $\widetilde{w}$ and $\widetilde{w}^{\prime }$ differ by a single braid relation. The following are the possible braid relations:

$$\begin{aligned} \widetilde{\sigma }_p\widetilde{\sigma }_r&= \widetilde{\sigma }_r\widetilde{\sigma }_p, \nonumber \\ \widetilde{\sigma }_p\widetilde{\sigma }_r\widetilde{\sigma }_p&= \widetilde{\sigma }_r\widetilde{\sigma }_p\widetilde{\sigma }_r,\\ (\widetilde{\sigma }_p\widetilde{\sigma }_r)^2&= (\widetilde{\sigma }_r\widetilde{\sigma }_p)^2, \nonumber \\ (\widetilde{\sigma }_p\widetilde{\sigma }_r)^3&= (\widetilde{\sigma }_r\widetilde{\sigma }_p)^3.\nonumber \end{aligned}$$

(3.48)

The argument for each relation is the same, so we only consider the second case. Assume that $\widetilde{w}$ and $\widetilde{w}^{\prime }$ differ by relation (3.48), that is

$$\begin{aligned} \widetilde{w}&= \widetilde{\sigma }_{i_1} \cdots \widetilde{\sigma }_{i_{k-1}} \big (\widetilde{\sigma }_p\widetilde{\sigma }_r\widetilde{\sigma }_p \big ) \widetilde{\sigma }_{i_{k+3}} \cdots \widetilde{\sigma }_{i_t},\\ \widetilde{w}^{\prime }&= \widetilde{\sigma }_{i_1} \cdots \widetilde{\sigma }_{i_{k-1}} \big (\widetilde{\sigma }_r\widetilde{\sigma }_p\widetilde{\sigma }_r \big ) \widetilde{\sigma }_{i_{k+3}} \cdots \widetilde{\sigma }_{i_t} \end{aligned}$$

for some $k = 1, \dotsc , t-2$. For $l = 1, \dotsc ,k-1$, we have

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}}^{[l]}&= \Psi \circ \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{l-1}} \circ \Psi ^{-1}(\mathfrak {X}_{i_l}) = \mathfrak {X}_{\widetilde{w}^{\prime }}^{[l]}. \end{aligned}$$

Since the algebra automorphisms $\widetilde{T}_i$ satisfy braid relations, we have

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}}^{[l]}&= \Psi \circ \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{k-1}} \big ( \widetilde{T}_p\widetilde{T}_r\widetilde{T}_p \big ) \widetilde{T}_{i_{k+3}} \cdots \widetilde{T}_{i_{l-1}} \circ \Psi ^{-1}(\mathfrak {X}_{i_l})\\&=\Psi \circ \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{k-1}} \big ( \widetilde{T}_r\widetilde{T}_p\widetilde{T}_r \big ) \widetilde{T}_{i_{k+3}} \cdots \widetilde{T}_{i_{l-1}} \circ \Psi ^{-1}(\mathfrak {X}_{i_l}) = \mathfrak {X}_{\widetilde{w}^{\prime }}^{[l]} \end{aligned}$$

for $l = k+3, \dotsc ,t$. Finally, consider the rank two subdiagram $(J,X \cap J, \tau |_J)$ obtained by taking $J = J_1 \cup J_2$, where $J_1 = \{r, p, \tau (r), \tau (p)\}$ and $J_2 \subset X$ is the union of connected components of X which are connected to a node of $J_1$. By assumption,

$$\begin{aligned} \mathfrak {X}_{\widetilde{\sigma }_p\widetilde{\sigma }_r\widetilde{\sigma }_p}&= \mathfrak {X}_p \cdot \Psi \widetilde{T}_p\Psi ^{-1}(\mathfrak {X}_r) \cdot \Psi \widetilde{T}_p\widetilde{T}_r\Psi ^{-1}(\mathfrak {X}_p)\\&= \mathfrak {X}_r \cdot \Psi \widetilde{T}_r\Psi ^{-1}(\mathfrak {X}_p) \cdot \Psi \widetilde{T}_r\widetilde{T}_p\Psi ^{-1}(\mathfrak {X}_r) = \mathfrak {X}_{\widetilde{\sigma }_r\widetilde{\sigma }_p\widetilde{\sigma }_r}. \end{aligned}$$

It follows from this that

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}}^{[k]}\mathfrak {X}_{\widetilde{w}}^{[k+1]}\mathfrak {X}_{\widetilde{w}}^{[k+2]}&= \Psi \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{k-1}}\Psi ^{-1} \big (\mathfrak {X}_{\widetilde{\sigma }_p\widetilde{\sigma }_r\widetilde{\sigma }_p} \big )\\&= \Psi \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{k-1}}\Psi ^{-1} \big (\mathfrak {X}_{\widetilde{\sigma }_r\widetilde{\sigma }_p\widetilde{\sigma }_r} \big )\\&= \mathfrak {X}_{\widetilde{w}^{\prime }}^{[k]}\mathfrak {X}_{\widetilde{w}^{\prime }}^{[k+1]} \mathfrak {X}_{\widetilde{w}^{\prime }}^{[k+2]} \end{aligned}$$

Hence we have $\mathfrak {X}_{\widetilde{w}} = \mathfrak {X}_{\widetilde{w}^{\prime }}$ as required.

If $\widetilde{w}$ and $\widetilde{w}^{\prime }$ differ by more than a single relation, then we can find a sequence of reduced expressions

$$\begin{aligned} \widetilde{w} = \widetilde{w}_1, \widetilde{w}_2, \dotsc , \widetilde{w}_n = \widetilde{w}^{\prime } \end{aligned}$$

such that for each $i = 1, \dotsc ,n-1$, the expressions $\widetilde{w}_i$ and $\widetilde{w}_{i+1}$ differ by a single relation. We repeat the above argument at each step and obtain $\mathfrak {X}_{\widetilde{w}} = \mathfrak {X}_{\widetilde{w}^{\prime }}$. $\square$

Recall from (2.9) that there exists a diagram automorphism $\tau _0: I \rightarrow I$ such that the longest element $w_0 \in W$ satisfies

$$\begin{aligned} w_0(\alpha _i) = -\alpha _{\tau _0(i)} \end{aligned}$$

(3.49)

for all $i \in I$.

Proposition 3.18

Let $\widetilde{w}_0 \in \widetilde{W}$ be the longest element with reduced expression $\widetilde{w}_0 = \widetilde{\sigma }_{i_1} \cdots \widetilde{\sigma }_{i_t}$. Then

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0}^{[t]} = \mathfrak {X}_{\tau _0(i_t)}. \end{aligned}$$

(3.50)

Proof

To simplify notation we write $i_t = i$. By construction we have

$$\begin{aligned} \widetilde{w}_0w_X&= w_0, \\ w_X\widetilde{\sigma }_i&= w_{\{i,\tau (i)\} \cup X} \quad \text{ for } \text{ all } i \in I {{\setminus }} X. \end{aligned}$$

As $w_X$ and $w_{ \{i, \tau (i)\} \cup X}$ commute, we get

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0}^{[t]}&= \Psi \circ T_{\widetilde{w}_0} T_{\widetilde{\sigma }_{i}}^{-1} \circ \Psi ^{-1} (\mathfrak {X}_{i})\\&= \Psi \circ T_{w_0}T_{w_X}^{-1} T_{w_X} T_{w_{\{i,\tau (i)\} \cup X}}^{-1} \circ \Psi ^{-1}(\mathfrak {X}_{i})\\&= \Psi \circ T_{w_0}T_{w_{\{i,\tau (i)\} \cup X}}^{-1} \circ \Psi ^{-1}(\mathfrak {X}_{i}). \end{aligned}$$

Recall that

$$\begin{aligned} T_{w_0} = \mathrm {tw}^{-1} \circ \tau _0 \end{aligned}$$

(3.51)

where $\mathrm {tw} : U_q(\mathfrak {g}) \rightarrow U_q(\mathfrak {g})$ is the algebra automorphism defined by

$$\begin{aligned} \mathrm {tw}(E_i) = -K_i^{-1}F_i, \quad \mathrm {tw}(F_i) = -E_iK_i, \quad \mathrm {tw}(K_i) = K_i^{-1} \end{aligned}$$

for $i \in I$, see [3, 7.1]. Analogously we have on $U_q(\mathfrak {g}_{\{i,\tau (i)\} \cup X})$ the relation

$$\begin{aligned} T_{w_{\{i, \tau (i)\}\cup X}} = \mathrm {tw}^{-1} \circ \tau _{0,i} = \tau _{0,i} \circ \mathrm {tw}^{-1} \end{aligned}$$

(3.52)

where $\tau _{0,i}: \{i,\tau (i)\} \cup X \rightarrow \{i, \tau (i)\} \cup X$ is the diagram automorphism satisfying (2.10). We obtain

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0}^{[t]}&= \Psi \circ \mathrm {tw}^{-1} \circ \tau _0 \circ \mathrm {tw} \circ \tau _{0,i} \circ \Psi ^{-1}(\mathfrak {X}_{i})\nonumber \\&= \Psi \circ \tau _0\tau _{0,i} \circ \Psi ^{-1} (\mathfrak {X}_{i}). \end{aligned}$$

(3.53)

Case 1

$\tau (i) = i$.

In this case Lemma 3.3 implies that

$$\begin{aligned} \tau _{0,i}(\mathfrak {X}_i) = \mathfrak {X}_i. \end{aligned}$$

(3.54)

Moreover $s(\tau (i)) = s(i) = 1$ and hence by Lemma 3.6 and by definition of $\Psi$ we have

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0}^{[t]}&= \sum _{n \in \mathbb {N}_0} \Psi \circ \tau _0\tau _{0,i} \circ \Psi ^{-1} (c_i^n\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))})\\&= \sum _{n \in \mathbb {N}_0} q^{-n/2(\alpha _i - \Theta (\alpha _i), \alpha _i)} \Psi \circ \tau _0\tau _{0,i}(\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))})\\&\overset{(3.54)}{=} \sum _{n \in \mathbb {N}_0} q^{-n/2(\alpha _i - \Theta (\alpha _i),\alpha _i)} \Psi \circ \tau _0 (\mathcal {E}_{n(\alpha _i - \Theta (\alpha _i))})\\&= \sum _{n \in \mathbb {N}_0} q^{-n/2(\alpha _i - \Theta (\alpha _i),\alpha _i)} \Psi (\mathcal {E}_{n(\alpha _{\tau _0(i)} - \Theta (\alpha _{\tau _0(i)})}) \end{aligned}$$

where we use the notation from Lemma 3.6 also for $\mathfrak {X}_{\tau _0(i)}$.

As $(\alpha _{\tau _0(i)} - \Theta (\alpha _{\tau _0(i)}), \alpha _{\tau _0(i)})~= (\alpha _i - \Theta (\alpha _i), \alpha _i)$ and $s(\tau _0(i))=1$, formula (3.45) gives us

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0}^{[t]} = \sum _{n \in \mathbb {N}_0} c_{\tau _0(i)}^n \mathcal {E}_{n(\alpha _{\tau _0(i)} - \Theta (\alpha _{\tau _0(i)}))} = \mathfrak {X}_{\tau _0(i)} \end{aligned}$$

(3.55)

which proves the Lemma in this case.

Case 2

$\tau (i) \ne i$.

In this case the rank one Satake subdiagram is either of type $AIII_{11}$ or of type AIV for $n \ge 2$ as in Table 1.

If the rank one Satake subdiagram is of type AIV for $n \ge 2$ then $\tau = \tau _0$ and $\tau _{0,i}$ coincide on $\{i, \tau (i)\} \cup X$ and hence (3.53) implies that

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0}^{[t]} = \mathfrak {X}_i = \mathfrak {X}_{\tau \tau _{0}(i)} = \mathfrak {X}_{\tau _0(i)}. \end{aligned}$$

(3.56)

If the rank one subdiagram is of type $AIII_{11}$ then $\tau _{0,i}(i) = i$. If additionally $\tau _0(i) = i$ then $\tau _0(\tau (i)) = \tau (i)$ and hence (3.53) implies that $\mathfrak {X}_{\widetilde{w}_0}^{[t]} = \mathfrak {X}_i = \mathfrak {X}_{\tau _0(i)}$ in this case.

If on the other hand $\tau _0(i) \ne i$ then $\tau _0 = \tau$ and we invoke the fact that

$$\begin{aligned} s(i) = s(\tau (i)), \quad c_i = c_{\tau (i)} \end{aligned}$$

(3.57)

which hold by (2.3) and (3.14). Relation (3.57) and $\tau = \tau _0$ imply that $\tau _0 \circ \Psi ^{-1}(\mathfrak {X}_i) = \Psi ^{-1} \circ \tau _0(\mathfrak {X}_i)$. Hence Eq. (3.53) implies that $\mathfrak {X}_{\widetilde{w}_0}^{[t]} = \mathfrak {X}_{\tau _0(i)}$ also in this case. $\square$

Lemma 3.19

Let $\widetilde{w}_0 = \widetilde{\sigma }_{i_1} \cdots \widetilde{\sigma }_{i_t}$ be a reduced expression for the longest word in $\widetilde{W}$. Then $\mathfrak {X}_{\widetilde{w}_0}^{[i]} \in \widehat{U^+[\widetilde{\sigma }_{k}\widetilde{w}_0]}$ for $i = 1, \dotsc , t-1$ and $k = \tau _0(i_t)$.

Proof

We have

$$\begin{aligned} \widetilde{\sigma }_k\widetilde{w}_0 = \widetilde{\sigma }_kw_0w_X = w_0\widetilde{\sigma }_{\tau _0(k)}w_X = w_0w_X\widetilde{\sigma }_{\tau _0(k)} = \widetilde{w}_0\widetilde{\sigma }_{i_t} = \widetilde{\sigma }_{i_1} \cdots \widetilde{\sigma }_{i_{t-1}}. \end{aligned}$$

By definition of $U^+[w]$ for each $w \in W$ and Proposition 2.5 we have

$$\begin{aligned} \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{j-1}}( U^+[\widetilde{\sigma }_{i_j}] ) \subseteq U^+[\widetilde{\sigma }_k\widetilde{w}_0] \end{aligned}$$

for $j = 1, \dotsc , t-1$. Now the claim of the lemma follows from Eq. (3.32), Proposition 3.12 and the fact that

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0}^{[j]} = \Psi \circ \widetilde{T}_{i_1} \cdots \widetilde{T}_{i_{j-1}} \circ \Psi ^{-1} ( \mathfrak {X}_{i_j}) \end{aligned}$$

for $j = 1, \dotsc , t-1$. $\square$

With the above preparations we are ready to prove the main result of the paper.

Theorem 3.20

Suppose that $(I,X,\tau )$ is a Satake diagram such that all subdiagrams $(J,X \cap J, \tau |_J)$ of rank two satisfy Conjecture 3.14. Then $\mathfrak {X}_{\widetilde{w}_0}$ coincides with the quasi K-matrix $\mathfrak {X}$.

Proof

It suffices to show that

$$\begin{aligned} {}_{i}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})q^{-(\Theta (\alpha _i), \alpha _i)}c_is(\tau (i))T_{w_X}(E_{\tau (i)})\mathfrak {X}_{\widetilde{w}_0} \end{aligned}$$

(3.58)

for all $i \in I {{\setminus }} X$. By Theorem 3.17 the element $\mathfrak {X}_{\widetilde{w}_0}$ is independent of the chosen reduced expression for $\widetilde{w}_0$. Fix $i\in I{{\setminus }} X$ and choose a reduced expression $\widetilde{w}_0 = \widetilde{\sigma }_{i_1} \cdots \widetilde{\sigma }_{i_t}$ such that $i_t=\tau _0(i)$. Proposition 3.18 implies that

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0} =\mathfrak {X}_{i} \mathfrak {X}^{[t-1]} \cdots \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}. \end{aligned}$$

By the previous lemma we have $\mathfrak {X}^{[j]} \in \widehat{U^{+}[\widetilde{\sigma }_i\widetilde{w}_0]}$ for $j = 1, \dotsc , t-1$. By [11, 8.26, (4)] this implies that ${}_{i}r(\mathfrak {X}^{[j]}) = 0$ for $j = 1, \dotsc , t-1$. By Eq. (3.26) and the skew derivative property (3.9) we obtain

$$\begin{aligned} {}_{i}r(\mathfrak {X}_{\widetilde{w}_0})&= {}_{i}r(\mathfrak {X}_i)\mathfrak {X}^{[t-1]} \cdots \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1}) q^{-(\Theta (\alpha _i),\alpha _i)}c_is(\tau (i))T_{w_X}(E_{\tau (i)})\mathfrak {X}_{i} \mathfrak {X}^{[t-1]} \cdots \mathfrak {X}^{[2]}\mathfrak {X}^{[1]} \end{aligned}$$

which proves (3.58). $\square$

Combining Theorems 3.15 and 3.20 we obtain the following result.

Corollary 3.21

Let $\mathfrak {g}$ be of type $A_n$ or $X = \emptyset$. Then the quasi K-matrix $\mathfrak {X}$ is given by $\mathfrak {X}= \mathfrak {X}_{\widetilde{w}_0}$ for any reduced expression of the longest word $\widetilde{w}_0 \in \widetilde{W}$.

Conjecture 3.22

The statement of Corollary 3.21 holds for any Satake diagram of finite type.

Remark 3.23

We continue the discussion of the integrality of the quasi K-matrix $\mathfrak {X}$ from Remark 3.11 under the assumption that $c_is(\tau (i))\in \pm q^\mathbb {Z}$ for all $i\in I{{\setminus }} X$. In this case $\mathfrak {X}^{[k]}_{\widetilde{w}} \in {}_{\mathscr {A}}\widehat{U^+}$ for $k=1, \dots , t$ if $\widetilde{w} \in \widetilde{W}$ has a reduced expression $\widetilde{w}=\widetilde{\sigma }_{i_1} \widetilde{\sigma }_{i_2} \dots \widetilde{\sigma }_{i_t}$. Indeed, the discussion in the proof of Proposition 3.12 shows that $\mathfrak {X}^{[k]}_{\widetilde{w}}$ differs from $\widetilde{T}_{i_1} \dots \widetilde{T}_{i_1}(\mathfrak {X}_{i_k})$ by a factor in $\pm q^\mathbb {Z}$. Hence we obtain $\mathfrak {X}_{\widetilde{w}}\in {}_{\mathscr {A}}\widehat{U^+}$ for all $\widetilde{w} \in \widetilde{W}$. By Corollary 3.21, choosing $\widetilde{w}=\widetilde{w}_0$, we obtain $\mathfrak {X}\in {}_{\mathscr {A}}\widehat{U^+}$ whenever $\mathfrak {g}$ is of type $A_n$ or $X=\emptyset$. In these cases we have hence reproduced [5, Theorem 5.3] for ${{\mathbf {s}}}={{\mathbf {0}}}$ without the use of canonical bases. The case of general Satake diagrams hinges on Conjecture 3.22 and the integrality in rank one from [5, Appendix A].

3.5 Quasi K-matrices for general parameters

We now give a description of the quasi K-matrix $\mathfrak {X}$ for general parameters $\mathbf {s}\in \mathcal {S}$. Recall that we denote the generators $B_i$ by $B_i^{{{\mathbf {c}}},{{\mathbf {s}}}}$ if we need to specify the dependence on the parameters. By [18, Theorem 7.1], [12, Theorem 7.1] the algebra $B_{\mathbf {c},\mathbf {s}}$ can be given in terms of the generators $B_i$ for $i\in I{{\setminus }} X$ and $\mathcal {M}_X U^0_\Theta$ by relations which are independent of ${{\mathbf {s}}}$. Hence there exists an algebra isomorphism $\varphi _{\mathbf {s}}:B_{{{\mathbf {c}}},{{\mathbf {0}}}}\rightarrow B_{\mathbf {c},\mathbf {s}}$ given by

$$\begin{aligned} \varphi _{\mathbf {s}}(B_i^{{{\mathbf {c}}},{{\mathbf {0}}}})=B_i^{{{\mathbf {c}}},{{\mathbf {s}}}}, \quad \varphi _{\mathbf {s}}(b)=b \qquad \text{ for } \text{ all } i\in I {{\setminus }} X, b\in \mathcal {M}_X U^0_\Theta . \end{aligned}$$

This algebra isomorphism allows us to define a one dimensional representation $\chi _{{\mathbf {s}}}:B_{{{\mathbf {c}}},{{\mathbf {0}}}}\rightarrow \mathbb {K}(q)$ by $\chi _{{\mathbf {s}}}=\varepsilon \circ \varphi _{{\mathbf {s}}}$. By definition we have

$$\begin{aligned} \chi _{{{\mathbf {s}}}}(B_i^{{{\mathbf {c}}},{{\mathbf {0}}}})=s_i \quad \text{ for } \text{ all } i\in I {{\setminus }} X, \qquad \chi _{{{\mathbf {s}}}}|_{\mathcal {M}_X U^0_\Theta }=\varepsilon |_{\mathcal {M}_X U^0_\Theta }. \end{aligned}$$

By [12, (5.5)] we have

$$\begin{aligned} \Delta (B_i) - B_i\otimes K_i^{-1} \in \mathcal {M}_X U^0_\Theta \otimes U_q(\mathfrak {g})\end{aligned}$$

(3.59)

which implies that

$$\begin{aligned} \varphi _{{{\mathbf {s}}}}=(\chi _{{\mathbf {s}}}\otimes \mathrm {id})\circ \Delta \end{aligned}$$

(3.60)

on $B_{{{\mathbf {c}}},{{\mathbf {0}}}}$. For later use we observe the following compatibility with the bar involution.

Lemma 3.24

For all $b\in B_{{{\mathbf {c}}},{{\mathbf {0}}}}$ we have

(3.61)

Proof

As and are $\mathbb {K}$-algebra homomorphisms, it suffices to check equation (3.61) on the generators $B_i^{{{\mathbf {c}}},{{\mathbf {0}}}}$ for $i\in I{{\setminus }} X$ and on $\mathcal {M}_X U^0_\Theta$. If $b\in \mathcal {M}_X U^0_\Theta$ then both sides of (3.61) coincide with $\overline{b}^U$. If $b=B_i^{{{\mathbf {c}}}, {{\mathbf {0}}}}$ for some $i\notin \{j\in I_{ns}\,|\, a_{jk}\in -2\mathbb {N}_0 \text{ for } \text{ all } k \in I_{ns}{\setminus }\{j\}\}$ then $s_i=0$ by the definition of $\mathcal {S}$ in (3.15) and hence $B_i^{{{\mathbf {c}}},{{\mathbf {0}}}}=B_i^{{{\mathbf {c}}},{{\mathbf {s}}}}$. Using the membership property (3.59) we get

which proves (3.61) in this case. Finally, if $i\in \{j\in I_{ns}\,|\, a_{jk}\in -2\mathbb {N}_0 \text{ for } \text{ all } k \in I_{ns}{{{\setminus }}}\{j\}\}$, then the definition (3.13) of $I_{ns}$ implies that

$$\begin{aligned} B_i^{{{\mathbf {c}}},{{\mathbf {s}}}}=F_i-c_iE_i K_i^{-1} + s_i K_i^{-1}. \end{aligned}$$

Hence, using $s_i=\overline{s_i}^U$ from (3.19), we get

which completes the proof of the lemma. $\square$

As in [3, 3.2] we consider the algebra

$$\begin{aligned} \mathscr {U}^{(2)}_0=\mathrm {End}(\mathcal {F}or\circ \otimes :\mathcal {O}_{int}\times \mathcal {O}_{int} \rightarrow \mathcal {V}ect) \end{aligned}$$

and observe that $\prod _{\mu \in Q^+} U^-_\mu \otimes U^+_\mu$ is a subalgebra of $\mathscr {U}^{(2)}_0$. Let $R\in \prod _{\mu \in Q^+} U^-_\mu \otimes U^+_\mu$ be the quasi R-matrix for $U_q(\mathfrak {g})$, see [23, Theorem 4.1.2]. Following [4, 3.1] we define an element

$$\begin{aligned} R^\theta = \Delta (\mathfrak {X})\cdot R \cdot (\mathfrak {X}^{-1} \otimes 1) \in \mathscr {U}_{0}^{(2)}, \end{aligned}$$

(3.62)

see also [13, 3.4]. In [4] the element $R^\theta$ is called the quasi R-matrix for $B_{\mathbf {c},\mathbf {s}}$. By [4, Proposition 3.2] it satisfies the following intertwiner property

(3.63)

in $\mathscr {U}^{(2)}_0$. Moreover, by [4, Proposition 3.5], [13, Proposition 3.10] we can write $R^\theta$ as an infinite sum

$$\begin{aligned} R^\theta =\sum _{\mu \in Q^+}R^\theta _\mu \qquad \text{ with } R^\theta _\mu \in B_{\mathbf {c},\mathbf {s}}\otimes U^+_\mu . \end{aligned}$$

(3.64)

Similarly to the notation $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$ introduced at the end of Sect. 3.2, we write $R^\theta _{{{\mathbf {c}}},{{\mathbf {s}}}}$ if we need to specify the dependence on the parameters. Observe that once we have an explicit formula for $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {0}}}}$, Eq. (3.62) provides us with an explicit formula for $R^\theta _{{{\mathbf {c}}},{{\mathbf {0}}}}$. This in turn provides a formula for the quasi K-matrix $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$ for general parameters ${{\mathbf {s}}}\in \mathcal {S}$. Indeed, by Eq. (3.64) we can apply the character $\chi _{{\mathbf {s}}}$ to the first tensor factor of $R^\theta _{{{\mathbf {c}}},{{\mathbf {0}}}}$ to obtain an element $\mathfrak {X}'=(\chi _{{\mathbf {s}}}\otimes \mathrm {id})(R^{\theta }_{{{\mathbf {c}}},{{\mathbf {0}}}})$ which can be written as

$$\begin{aligned} \mathfrak {X}'=\sum _{\mu \in Q^+}\mathfrak {X}'_\mu \qquad \text{ with } \mathfrak {X}'_\mu \in U_{\mu }^+. \end{aligned}$$

Moreover, Eq. (3.64) implies that $\mathfrak {X}'_0=1$. By the following proposition the element $\mathfrak {X}'\in \mathscr {U}$ is the quasi K-matrix for $B_{\mathbf {c},\mathbf {s}}$.

Proposition 3.25

For any ${{\mathbf {c}}}\in \mathcal {C}$, ${{\mathbf {s}}}\in \mathcal {S}$ we have $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}=(\chi _{{{\mathbf {s}}}} \otimes \mathrm {id})(R^{\theta }_{{{\mathbf {c}}},{{\mathbf {0}}}})$.

Proof

We keep the notation $\mathfrak {X}'=(\chi _{{\mathbf {s}}}\otimes \mathrm {id})(R^{\theta }_{{{\mathbf {c}}},{{\mathbf {0}}}})$ from above. By Eq. (3.63) we have

Applying $\chi _{{\mathbf {s}}}\otimes \mathrm {id}$ to both sides of this relation, we obtain in view of Eq. (3.60) the relation

By Lemma 3.24 the above relation implies that

$$\begin{aligned} \varphi _{{\mathbf {s}}}(\overline{b}^{B_{{{\mathbf {c}}},{{\mathbf {0}}}}}) \mathfrak {X}'= \mathfrak {X}'\overline{\varphi _{{\mathbf {s}}}(b)}^U \qquad \text{ for } \text{ all } b\in B_{{{\mathbf {c}}},{{\mathbf {0}}}}. \end{aligned}$$

This gives in particular $B_i^{{{\mathbf {c}}},{{\mathbf {s}}}} \mathfrak {X}' = \mathfrak {X}' \overline{B_i^{{{\mathbf {c}}},{{\mathbf {s}}}}}^U$ for all $i\in I$ and $b \mathfrak {X}'=\mathfrak {X}' b$ for all $b\in \mathcal {M}_XU^\Theta _0$. This means that $\mathfrak {X}'$ satisfies the defining relation (3.22) of $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$ and hence, in view of the normalisation $\mathfrak {X}'_0=1$ observed above, we get $\mathfrak {X}'=\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$.   $\square$

Remark 3.26

The existence of the quasi K-matrix $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$ was established in [3, Theorem 6.10] by fairly involved calculations. It was noted in [3, Remark 6.9] that these calculations simplify significantly if one restricts to the case ${{\mathbf {s}}}={{\mathbf {0}}}$. Proposition 3.25 now shows that in the presence of (3.64) the existence of $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {0}}}}$ implies the existence of $\mathfrak {X}_{{{\mathbf {c}}},{{\mathbf {s}}}}$ for any ${{\mathbf {s}}}\in \mathcal {S}$ satisfying (3.19). Relation (3.64) was established in [13].

Appendix A: Rank two calculations

In rank two, there are two distinct reduced expressions for the longest word $\widetilde{w}_0 \in \widetilde{W}$. All irreducible rank two Satake diagrams for simple $\mathfrak {g}$ are shown in Table 2. Using the explicit formulas from Sect. 3.3, we confirm that the partial quasi K-matrices for the two reduced expressions of $\widetilde{w}_0$ coincide with the quasi K-matrix if the rank two Satake diagram is of type $A_n$ or $B_2$. We also state this result in type $G_2$, but we refer to [6, 6.6] for the proof, which involves significantly longer calculations. The calculations in this appendix and [6] prove Theorem 3.15.

Table 2 Irreducible Satake diagrams of rank two for simple $\mathfrak {g}$

1.1 Type $AI_2$

Consider the Satake diagram of type $AI_2$.

Since $\Theta = -\text {id}$ the restricted Weyl group $\widetilde{W}$ coincides with the Weyl group W. The longest word of the Weyl group has two reduced expressions given by

$$\begin{aligned} w_0&= \sigma _1\sigma _2\sigma _1,\quad w_0^{\prime } = \sigma _2\sigma _1\sigma _2. \end{aligned}$$

Proposition A.1

In this case, the partial quasi K-matrices $\mathfrak {X}_{w_0}$ and $\mathfrak {X}_{w_0^{\prime }}$ coincide with the quasi K-matrix $\mathfrak {X}$. Hence $\mathfrak {X}_{w_0} = \mathfrak {X}_{w_0^{\prime }}$.

Before we prove this, we need to know how the Lusztig skew derivations ${}_1r$ and ${}_2r$ act on certain elements and their powers, and also some commutation relations. These are given in the following two lemmas, whose proofs are obtained by straightforward computation.

Lemma A.2

For any $n \in \mathbb {N}$, the relations

$$\begin{aligned} E_2^n E_1&= q^n E_1 E_2^n - q\{ n \} E_2^{n-1} T_{1}(E_2),\\ E_1^n E_2&= q^n E_2 E_1^n - q\{ n \} E_1^{n-1} T_{2}(E_1),\\ T_{1}(E_2)^n E_1&= q^{-n} E_1 T_{1}(E_2)^n,\\ T_{2}(E_1)^n E_2&= q^{-n} E_2 T_{2}(E_1)^n \end{aligned}$$

hold in $U_q(\mathfrak {sl}_3)$.

Lemma A.3

For any $n \in \mathbb {N}$, the relations

$$\begin{aligned}&{}_{ 1 }r( E_2^n ) = \, {}_{ 1 }r( T_{2}(E_1)^n ) = \, {}_{ 2 }r( E_1^n ) = \, {}_{2}r( T_{1}(E_2)^n ) = 0,\\&{}_{ 1 }r( E_1^n ) = \{ n \} E_1^{n-1},\\&{}_{ 2 }r( E_2^n ) = \{ n \} E_2^{n-1},\\&{}_{ 1 }r( T_{1}(E_2)^n ) = (1 - q^{-2}) \{ n \} E_2 T_{1}(E_2)^{n-1},\\&{}_{ 2 }r( T_{2}(E_1)^n ) = (1 - q^{-2}) \{ n \} E_1 T_{2}(E_1)^{n-1} \end{aligned}$$

hold in $U_q(\mathfrak {sl}_3)$.

Proof of Proposition A.1

Consider first the element $\mathfrak {X}_{w_0}$. Using (3.47) and Lemma 3.7, we write

$$\begin{aligned} \mathfrak {X}_{w_0} = \mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} \end{aligned}$$

(A.1)

where

$$\begin{aligned}&\mathfrak {X}^{[3]} \overset{(3.50)}{=} \mathfrak {X}_2 = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}!!}(q^2c_2)^nE_2^{2n},\\&\mathfrak {X}^{[2]} = \Psi \circ T_1 \circ \Psi ^{-1}(\mathfrak {X}_2) = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}!!}(q^2c_1)^n(q^2c_2)^nT_1(E_2)^{2n},\\&\mathfrak {X}^{[1]} = \mathfrak {X}_1 = \sum _{n \ge 0}\dfrac{(q-q^{-1})^n}{\{2n\}!!}(q^2c_1)^nE_1^{2n}. \end{aligned}$$

For $i = 1,2,3$, let $\mathfrak {X}_{[i]} = K_1\mathfrak {X}^{[i]}K_1^{-1}$. The difference between $\mathfrak {X}_{[i]}$ and $\mathfrak {X}^{[i]}$ is the occurrence of a q-power in each summand of the infinite series. By Eq. (3.26), to show that $\mathfrak {X}_{w_0}$ coincides with the quasi K-matrix $\mathfrak {X}$ we show that

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{w_0})&= (q-q^{-1})(q^2c_1)E_1\mathfrak {X}_{w_0},\\ {}_{2}r(\mathfrak {X}_{w_0})&= (q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{w_0}. \end{aligned}$$

By Lemmas A.3 and 3.7, we see that

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{w_0})&= {}_{2}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})(q^2c_2)E_2\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{w_0}. \end{aligned}$$

By the property (3.9) of the skew derivative ${}_1r$, we have

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{w_0}) = \mathfrak {X}_{[3]}{}_{1}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} + \mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{1}r(\mathfrak {X}^{[1]}) \end{aligned}$$

(A.2)

Using Lemma A.3, we have

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[2]})&= \sum _{n \ge } \dfrac{(q-q^{-1})^n}{\{2n\}!!} (q^2c_1)^n(q^2c_2)^n {}_{1}r(T_1(E_2)^{2n})\\&= q^{-1}(q-q^{-1})E_2 \sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{\{2n-2\}!!} (q^2c_1)^n(q^2c_2)^n T_1(E_2)^{2n-1}\\&= (q-q^{-1})^2 q^{-1} (q^2c_1)(q^2c_2) E_2T_1(E_2) \mathfrak {X}^{[2]},\\ {}_{1}r(\mathfrak {X}^{[1]})&= (q-q^{-1})(q^2c_1) E_1 \mathfrak {X}^{[1]}. \end{aligned}$$

The second summand of Eq. (A.2) is of the form

$$\begin{aligned} (q-q^{-1})(q^2c_1) \mathfrak {X}_{[3]}\mathfrak {X}_{[2]}E_1\mathfrak {X}^{[1]}. \end{aligned}$$

We use Lemma A.2 to bring the $E_1$ in the above summand to the front. We have

$$\begin{aligned} \mathfrak {X}_{[2]}E_1&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}!!} (q^2c_1)^n(q^2c_2)^n q^{2n} T_1(E_2)^{2n}E_1\\&= E_1 \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}!!} (q^2c_1)^n(q^2c_2)^n T_1(E_2)^{2n}\\&= E_1\mathfrak {X}^{[2]},\\ \mathfrak {X}_{[3]}E_1&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}!!} (q^2c_2)^n q^{-2n} E_2^{2n}E_1\\&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}!!} (q^2c_2)^n q^{-2n} (q^{2n}E_1E_2^{2n} - q\{2n\}E_2^{2n-1}T_1(E_2))\\&= E_1\mathfrak {X}^{[3]} - q\sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{\{2n-2\}!!} (q^2c_2)^nq^{-2n}E_2^{2n-1}T_1(E_2)\\&= E_1\mathfrak {X}^{[3]} - (q-q^{-1})q^{-1}(q^2c_2)\mathfrak {X}_{[3]}E_2T_1(E_2). \end{aligned}$$

Hence,

$$\begin{aligned} (q-q^{-1})(q^2c_1) \mathfrak {X}_{[3]}\mathfrak {X}_{[2]}E_1\mathfrak {X}^{[1]}&= (q-q^{-1})(q^2c_1) \mathfrak {X}_{[3]}E_1\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})(q^2c_1)E_1\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad {}- (q-q^{-1})^2 q^{-1}(q^2c_1)(q^2c_2)\mathfrak {X}_{[3]}E_2T_1(E_2) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})(q^2c_1) E_1\mathfrak {X}_{w_0} - \mathfrak {X}_{[3]}{}_{1}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]}. \end{aligned}$$

It follows from (A.2) that ${}_{1}r(\mathfrak {X}_{w_0}) = (q-q^{-1})(q^2c_1) E_1\mathfrak {X}_{w_0}$, so $\mathfrak {X}_{w_0}$ coincides with the quasi K-matrix $\mathfrak {X}$. Instead of repeating the same calculation for $\mathfrak {X}_{w_0^{\prime }}$, we use the underlying symmetry in type $AI_2$, which implies that $\mathfrak {X}_{w_0^{\prime }}$ also coincides with the quasi K-matrix $\mathfrak {X}$. $\square$

1.2 Type $AII_5$

Consider the Satake diagram of type $AII_{5}$.

In this case the involutive automorphism $\Theta : \mathfrak {h}^{*} \rightarrow \mathfrak {h}^{*}$ is given by

$$\begin{aligned} \Theta = -\sigma _1\sigma _3\sigma _5. \end{aligned}$$

There are two $\tau$-orbits of white nodes given by the sets $\{2\}$ and $\{4\}$. The restricted root system is of type $\textit{A}_2$ since the restricted roots

$$\begin{aligned} \widetilde{\alpha }_2 = \dfrac{\alpha _1 + 2\alpha _2 + \alpha _3}{2}, \qquad {} \widetilde{\alpha }_4 = \dfrac{\alpha _3 + 2\alpha _4 + \alpha _5}{2} \end{aligned}$$

have the same length. The subgroup $\widetilde{W} \subset W^{\Theta }$ is generated by the elements

$$\begin{aligned} \widetilde{\sigma }_2 = \sigma _2\sigma _1\sigma _3\sigma _2, \qquad {} \widetilde{\sigma }_4 = \sigma _4\sigma _3\sigma _5\sigma _4. \end{aligned}$$

The longest word of the restricted Weyl group has two reduced expressions given by

$$\begin{aligned} \widetilde{w}_0 = \widetilde{\sigma }_2\widetilde{\sigma }_4\widetilde{\sigma }_2, \qquad {} \widetilde{w}_0^{\prime } = \widetilde{\sigma }_4\widetilde{\sigma }_2\widetilde{\sigma }_4. \end{aligned}$$

By Lemma 3.8 we have

$$\begin{aligned} \mathfrak {X}_2&= \sum _{n \ge 0} \dfrac{(qc_2)^n}{\{n\}!} [E_2, T_{13}(E_2)]_{q^{-2}}^n,\\ \mathfrak {X}_4&= \sum _{n \ge 0} \dfrac{(qc_4)^n}{\{n\}!} [E_4, T_{35}(E_4)]_{q^{-2}}^n. \end{aligned}$$

Proposition A.4

The partial quasi K-matrices $\mathfrak {X}_{\widetilde{w}_0}$ and $\mathfrak {X}_{\widetilde{w}_0^{\prime }}$ coincide with the quasi K-matrix $\mathfrak {X}$.

We have the following relations needed for the proof of Proposition A.4. These are proved by induction.

Lemma A.5

For any $n \in \mathbb {N}$ the relations

$$\begin{aligned}&[E_4, T_{35}(E_4)]_{q^{-2}}^n T_{13}(E_2) = q^nT_{13}(E_2)[E_4, T_{35}(E_4)]_{q^{-2}}^n \nonumber \\&\quad - q\{n\}[E_4, T_{35}(E_4)]^{n-1}_{q^{-2}}[T_3(E_4), T_{1235}(E_4)]_{q^{-2}}, \end{aligned}$$

(A.3)

$$\begin{aligned}&[T_{23}(E_4), T_{1235}(E_4)]_{q^{-2}}^n T_{13}(E_2) = q^{-n}T_{13}(E_2)[T_{23}(E_4), T_{1235}(E_4)]_{q^{-2}}^n \end{aligned}$$

(A.4)

hold in $U_q(\mathfrak {sl}_6)$.

Lemma A.6

For any $n \in \mathbb {N}$ the relation

$$\begin{aligned} \begin{aligned}&{}_{2}r([T_{23}(E_4), T_{1235}(E_4)]_{q^{-2}}^n)\\&\quad = q^{-1}(q-q^{-1})\{n\} [T_3(E_4), T_{1235}(E_4)]_{q^{-2}} [T_{23}(E_4), T_{1235}(E_4)]_{q^{-2}}^{n-1} \end{aligned} \end{aligned}$$

(A.5)

holds in $U_q(\mathfrak {sl}_6)$.

Proof of Proposition A.4

We only confirm that $\mathfrak {X}_{\widetilde{w}_0}$ coincides with the quasi K-matrix $\mathfrak {X}$. By the underlying symmetry in type $AII_5$, the calculation for $\mathfrak {X}_{\widetilde{w}_0^{\prime }}$ is the same up to a change of indices. By definition, we have

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0} = \mathfrak {X}^{[3]} \mathfrak {X}^{[2]} \mathfrak {X}^{[1]} \end{aligned}$$

where

$$\begin{aligned}&\mathfrak {X}^{[3]} \overset{(3.50)}{=} \mathfrak {X}_4, \\&\mathfrak {X}^{[2]} = \Psi \circ T_{2132} \circ \Psi ^{-1}(\mathfrak {X}_4) = \sum _{n \ge 0} \dfrac{(q^2c_2c_4)^n}{\{n\}!} [T_{23}(E_4), T_{1235}(E_4)]_{q^{-2}}^n,\\&\mathfrak {X}^{[1]} = \mathfrak {X}_2. \end{aligned}$$

Using Lemma 3.8 and [11, 8.26, (4)] we see that

$$\begin{aligned} {}_{4}r(\mathfrak {X}_{\widetilde{w}_0})&= {}_{4}r(\mathfrak {X}_4) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})c_4T_{35}(E_4)\mathfrak {X}_{4}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})c_4T_{35}(E_4)\mathfrak {X}_{\widetilde{w}_0}. \end{aligned}$$

We want to show that

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})c_2T_{13}(E_2)\mathfrak {X}_{\widetilde{w}_0}. \end{aligned}$$

For $i = 1, 2, 3$, let $\mathfrak {X}_{[i]} = K_2\mathfrak {X}^{[i]}K_2^{-1}$. By property (3.9) of the skew derivative ${}_2r$ and [11, 8.26, (4)] we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0}) = \mathfrak {X}_{[3]} {}_{2}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} + \mathfrak {X}_{[3]}\mathfrak {X}_{[2]} {}_{2}r(\mathfrak {X}^{[1]}). \end{aligned}$$

(A.6)

By Lemma 3.8 we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[1]}) = (q-q^{-1})c_2T_{13}(E_2)\mathfrak {X}^{[1]}. \end{aligned}$$

By Lemma A.6 we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[2]})&= \sum _{n \ge 1} \dfrac{(q^2c_2c_4)^n}{\{n\}!} {}_{2}r([T_{23}(E_4), T_{1235}(E_4)]_{q^{-2}}^n) \\&\overset{(\mathrm{A.5})}{=} q(q-q^{-1})c_2c_4 [T_3(E_4), T_{1235}(E_4)]_{q^{-2}} \mathfrak {X}^{[2]}. \end{aligned}$$

The second summand of Eq. (A.6) is of the form $(q-q^{-1})c_2\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}T_{13}(E_2)\mathfrak {X}^{[1]}$. Using Lemma A.5, we bring the $T_{13}(E_2)$ term in this expression to the front. We have

$$\begin{aligned} \mathfrak {X}_{[2]}T_{13}(E_2)&\overset{(\mathrm{A.4})}{=} T_{13}(E_2)\mathfrak {X}^{[2]}, \end{aligned}$$

(A.7)

$$\begin{aligned} \mathfrak {X}_{[3]}T_{13}(E_2)&\overset{(\mathrm{A.3})}{=} T_{13}(E_2)\mathfrak {X}^{[3]} - qc_4\mathfrak {X}_{[3]}[T_3(E_4), T_{1235}(E_4)]_{q^{-2}}. \end{aligned}$$

(A.8)

Substituting (A.7) and (A.8) into $(q-q^{-1})c_2\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}T_{13}(E_2)\mathfrak {X}^{[1]}$, we obtain

$$\begin{aligned} (q-q^{-1})c_2\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}T_{13}(E_2)\mathfrak {X}^{[1]}&\overset{(\mathrm{A.7})}{=} (q-q^{-1})c_2\mathfrak {X}_{[3]}T_{13}(E_2)\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\overset{(\mathrm{A.8})}{=} (q-q^{-1})c_2T_{13}(E_2)\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&- (q-q^{-1})qc_2c_4\mathfrak {X}^{[3]}[T_3(E_4), T_{1235}(E_4)]_{q^{-2}} \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})c_2T_{13}(E_2)\mathfrak {X}_{\widetilde{w}_0} - \mathfrak {X}_{[3]}{}_{2}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]}. \end{aligned}$$

It follows from (A.6) that ${}_{2}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})c_2T_{13}(E_2)\mathfrak {X}_{\widetilde{w}_0}$ as required. $\square$

1.3 Type $AIII_3$

We consider the diagram of type $AIII_3$ with non-trivial diagram automorphism $\tau$ and no black dots.

Here, we see that there are 2 nodes in the restricted Dynkin diagram, corresponding to the restricted roots

$$\begin{aligned} \widetilde{\alpha }_1 = \dfrac{\alpha _1 +\alpha _3}{2}, \qquad {} \widetilde{\alpha }_2 = \alpha _2. \end{aligned}$$

A quick check confirms that $1/2(\alpha _1 + \alpha _3)$ is the short root, and hence the restricted root system is of type $B_2$. The subgroup $\widetilde{W}$ is generated by the elements

$$\begin{aligned} \widetilde{\sigma }_1 = \sigma _1\sigma _3, \qquad {} \widetilde{\sigma }_2 = \sigma _2. \end{aligned}$$

The longest word of the restricted Weyl group has two reduced expressions given by

$$\begin{aligned} \widetilde{w}_0 = \widetilde{\sigma }_1\widetilde{\sigma }_2\widetilde{\sigma }_1\widetilde{\sigma }_2, \qquad {} \widetilde{w}_0^{\prime } = \widetilde{\sigma }_2\widetilde{\sigma }_1\widetilde{\sigma }_2\widetilde{\sigma }_1. \end{aligned}$$

The definition (3.14) and condition (3.18) imply that $c_1=c_3=\overline{c_1}$. By Lemmas 3.7 and 3.9 we have

$$\begin{aligned} \mathfrak {X}_1&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{n \}! } c_1^n(E_1E_3)^n, \end{aligned}$$

(A.9)

$$\begin{aligned} \mathfrak {X}_2&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{2n \}!! } (q^2c_2)^n E_2^{2n}. \end{aligned}$$

(A.10)

Proposition A.7

The partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}_0}$ coincides with the quasi K-matrix $\mathfrak {X}$.

The following relations are needed for the proof of Proposition A.7. They are checked by induction.

Lemma A.8

For any $n \in \mathbb {N}$, the relations

$$\begin{aligned} T_{13}(E_2)^{n} E_3&= q^{-n}E_3T_{13}(E_2)^n, \end{aligned}$$

(A.11)

$$\begin{aligned} \big ( T_1(E_2)T_3(E_2) \big )^nE_3&= E_3 \big ( T_1(E_2)T_3(E_2) \big )^n \nonumber \\&\qquad - q\{ n \} \big ( T_1(E_2)T_3(E_2) \big )^{n-1} T_3(E_2)T_{13}(E_2), \end{aligned}$$

(A.12)

$$\begin{aligned} E_2^nE_3&= q^nE_3E_2^n - q\{ n \} E_2^{n-1}T_3(E_2) \end{aligned}$$

(A.13)

hold in $U_q(\mathfrak {sl}_4)$.

Lemma A.9

For any $n \in \mathbb {N}$, the relations

$$\begin{aligned} {}_{1}r(T_{13}(E_2)^n )&= q^{-1}(q-q^{-1}) \{n \} T_3(E_2)T_{13}(E_2)^{n-1}, \end{aligned}$$

(A.14)

$$\begin{aligned} {}_{1}r( \big ( T_1(E_2)T_3(E_2) \big )^n)&= q^{-1}(q-q^{-1}) \{ n \} E_2T_3(E_2) \big ( T_1(E_2)T_3(E_2) \big )^{n-1} \end{aligned}$$

(A.15)

hold in $U_q(\mathfrak {sl}_4)$.

Proof of Proposition A.7

Take $\widetilde{w}_0 = \widetilde{\sigma }_1 \widetilde{\sigma }_2 \widetilde{\sigma }_1 \widetilde{\sigma }_1$. Then we have

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0} = \mathfrak {X}^{[4]}\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} \end{aligned}$$

where

$$\begin{aligned}&\mathfrak {X}^{[4]} \overset{(3.50)}{=} \mathfrak {X}_{\tau _0(2)} = \mathfrak {X}_2, \\&\mathfrak {X}^{[3]} = \Psi \circ \widetilde{T}_1\widetilde{T}_2 \circ \Psi ^{-1}(\mathfrak {X}_1) = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{n \}! } (q^2c_1c_2)^n (T_3(E_2)T_1(E_2))^n,\\&\mathfrak {X}^{[2]} = \Psi \circ \widetilde{T}_1 \circ \Psi ^{-1}(\mathfrak {X}_2) = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{2n \}!!} (q^4c_1^2c_2)^n T_{13}(E_2)^{2n},\\&\mathfrak {X}^{[1]} = \mathfrak {X}_1. \end{aligned}$$

By Lemma 3.7, property (3.9) of the skew derivative ${}_2r$ and [11, 8.26, (4)], we see that ${}_{2}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})q^2c_2E_2\mathfrak {X}_{\widetilde{w}_0}$. Due to the underlying symmetry in this case, we only need to show that

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})c_1E_3\mathfrak {X}_{\widetilde{w}_0}. \end{aligned}$$

For each $i = 1, 2, 3, 4$, let $\mathfrak {X}_{[i]} = K_1\mathfrak {X}^{[i]}K_1^{-1}$. Then by the property (3.9) of the skew derivation ${}_1r$, we have

$$\begin{aligned} {}_{1}r(\mathfrak {X}_w) = \mathfrak {X}_{[4]} {}_{1}r(\mathfrak {X}^{[3]}) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]} {}_{1}r(\mathfrak {X}^{[2]}) \mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]} {}_{1}r(\mathfrak {X}^{[1]}). \end{aligned}$$

Using Lemmas 3.9 and A.9, it follows that

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[3]})&\overset{(\mathrm{A.15})}{=} q^{-1}(q-q^{-1})^2(q^2c_1c_2)E_2T_3(E_2)\mathfrak {X}^{[3]}, \end{aligned}$$

(A.16)

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[2]})&\overset{(\mathrm{A.14})}{=} q^{-1}(q-q^{-1})^2(q^4c_1^2c_2)T_3(E_2)T_{13}(E_2)\mathfrak {X}^{[2]}, \end{aligned}$$

(A.17)

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[1]})&\overset{(3.42)}{=} (q-q^{-1})E_3\mathfrak {X}^{[1]}. \end{aligned}$$

(A.18)

Using Lemma A.8, we look at the term $\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]} {}_{1}r(\mathfrak {X}^{[1]})$ in more detail. We have

$$\begin{aligned} \mathfrak {X}_{[2]}E_3&\overset{(\mathrm{A.11})}{=} E_3\mathfrak {X}^{[2]}, \end{aligned}$$

(A.19)

$$\begin{aligned} \mathfrak {X}_{[3]}E_3&\overset{(\mathrm{A.12})}{=} E_3\mathfrak {X}^{[3]} - q(q-q^{-1})(q^2c_1c_2)\mathfrak {X}_{[3]}T_3(E_2)T_{13}(E_2), \end{aligned}$$

(A.20)

$$\begin{aligned} \mathfrak {X}_{[4]}E_3&\overset{(\mathrm{A.13})}{=} E_3\mathfrak {X}^{[4]} - q^{-1}(q-q^{-1})(q^2c_2)\mathfrak {X}_{[4]}E_2T_3(E_2). \end{aligned}$$

(A.21)

It follows that

$$\begin{aligned}&\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]} {}_{1}r(\mathfrak {X}^{[1]}) \\&\quad \overset{(\mathrm{A.19})}{=} (q-q^{-1})c_1\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}E_3\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad \overset{(\mathrm{A.20})}{=} (q-q^{-1})c_1\mathfrak {X}_{[4]} \big (E_3\mathfrak {X}^{[3]} - q(q-q^{-1})(q^2c_1c_2)\mathfrak {X}_{[3]}T_3(E_2)T_{13}(E_2) \big )\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad \overset{(\mathrm{A.21})}{=} (q-q^{-1})c_1 \big ( E_3\mathfrak {X}^{[4]} - q^{-1}(q-q^{-1})(q^2c_2)\mathfrak {X}_{[4]}E_2T_3(E_2) \big )\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad \qquad - \mathfrak {X}_{[4]}\mathfrak {X}_{[3]} {}_{1}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} \\&\quad = (q-q^{-1})c_1E_3\mathfrak {X}_{\widetilde{w}_0} - \mathfrak {X}_{[4]}{}_{1}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} - \mathfrak {X}_{[4]}\mathfrak {X}_{[3]} {}_{1}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} \end{aligned}$$

by Eqs. (A.16), (A.17) and (A.18). Hence, it follows that ${}_{1}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})c_1E_3\mathfrak {X}_{\widetilde{w}_0}$, as required. $\square$

Proposition A.10

The partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}_0^\prime }$ coincides with the quasi K-matrix $\mathfrak {X}$.

The following relations are needed for the proof of Proposition A.10. They are checked by induction.

Lemma A.11

For any $n \in \mathbb {N}$, the relations

$$\begin{aligned}&\big (T_{2}(E_3)T_{2}(E_1) \big )^n E_2 = q^{-2n}E_2 \big (T_{2}(E_3)T_{2}(E_1) \big )^n, \end{aligned}$$

(A.22)

$$\begin{aligned}&T_{213}(E_2)^nE_2 = E_2T_{213}^n(E_2) - (q-q^{-1}) \{n \} T_{213}(E_2)^{n-1}T_{2}(E_3)T_{2}(E_1), \end{aligned}$$

(A.23)

$$\begin{aligned}&\big (E_1E_3\big )^n E_2 = q^{2n}E_2 \big (E_1E_3 \big )^n - q \{ n \}\big (E_1E_3\big )^{n-1}(E_3T_{2}(E_1)+ E_1 T_2(E_3))\nonumber \\&\qquad {}- q^2 \{n \}^2 \big (E_1E_3 \big )^{n-1} T_{213}(E_2) \end{aligned}$$

(A.24)

hold in $U_q(\mathfrak {sl}_4)$.

Lemma A.12

For any $n \in \mathbb {N}$, the relations

$$\begin{aligned} {}_{2}r(T_{213}(E_2)^n)&= q^{-2}(q-q^{-1})^2\{n\}E_1E_3T_{213}(E_2)^{n-1}, \end{aligned}$$

(A.25)

$$\begin{aligned} {}_{2}r(T_{2}(E_3)^n T_{2}(E_1)^n )&= q^{-1}(q-q^{-1})\{n\} E_3T_2(E_3)^{n-1}T_2(E_1)^n\nonumber \\&\quad + q^{-1}(q-q^{-1})\{n\}E_1T_2(E_3)^nT_2(E_1)^{n-1} \nonumber \\&\quad + (q-q^{-1}) \{n \}^2 T_{213}(E_2) \big ( T_2(E_1)T_2(E_3) \big )^{n-1} \end{aligned}$$

(A.26)

hold in $U_q(\mathfrak {sl}_4)$.

Proof of Proposition A.10

For $\widetilde{w}_0^{\prime } = \widetilde{\sigma }_2\widetilde{\sigma }_1\widetilde{\sigma }_2\widetilde{\sigma }_1$ we have

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0^{\prime }} = \mathfrak {X}^{[4]}\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}, \end{aligned}$$

where

$$\begin{aligned}&\mathfrak {X}^{[4]} \overset{(3.50)}{=} \mathfrak {X}_{\tau _0(1)} = \mathfrak {X}_1,\\&\mathfrak {X}^{[3]} = \Psi \circ \widetilde{T}_2\widetilde{T}_1 \circ \Psi ^{-1}(\mathfrak {X}_2) = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{ 2n \}!!}(q^4c_1^2c_2)^n T_{213}(E_2)^{2n},\\&\mathfrak {X}^{[2]} =\Psi \circ \widetilde{T}_2 \circ \Psi ^{-1}(\mathfrak {X}_1) = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{n \}!}(q^2c_1c_2)^n T_2(E_1)^nT_2(E_3)^n,\\&\mathfrak {X}^{[1]} = \mathfrak {X}_2. \end{aligned}$$

By Lemma 3.9, property (3.9) of the skew derivative ${}_1r$ and [11, 8.26, (4)] we have ${}_{1}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }})= (q-q^{-1})c_1E_3\mathfrak {X}_{\widetilde{w}_0^{\prime }}$. We want to show that

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }}) = (q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{\widetilde{w}_0^{\prime }}. \end{aligned}$$

For $i = 1, 2, 3, 4$, we let $\mathfrak {X}_{[i]} = K_2\mathfrak {X}^{[i]}K_2^{-1}$. Note that we have $\mathfrak {X}_{[3]} = \mathfrak {X}^{[3]}$. By the property (3.9) of the skew derivative ${}_2r$, we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }}) = \mathfrak {X}_{[4]} {}_{2}r(\mathfrak {X}^{[3]}) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]} {}_{2}r(\mathfrak {X}^{[2]}) \mathfrak {X}^{[1]} + \mathfrak {X}_{[4]} \mathfrak {X}_{[3]}\mathfrak {X}_{[2]} {}_{2}r(\mathfrak {X}^{[1]}). \end{aligned}$$

Using Lemma A.12, we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[3]})&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{ 2n \}!!}(q^4c_1^2c_2)^n {}_{2}r(T_{213}(E_2)^{2n}) \nonumber \\&\overset{(\mathrm{A.25})}{=} q^{-2}(q-q^{-1})^2 \sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{ \{ 2n \}!!}(q^4c_1^2c_2)^n \{ 2n \} E_1E_3T_{213}(E_2)^{2n-1} \nonumber \\&= q^{-2}(q-q^{-1})^2 E_1E_3 \sum _{n \ge 0} \dfrac{(q-q^{-1})^{n+1}}{ \{2n \}!!}(q^4c_1^2c_2)^{n+1} T_{213}(E_2)^{2n+1} \nonumber \\&=(q-q^{-1})^3(q^2c_1^2c_2) E_1E_3T_{213}(E_2) \mathfrak {X}^{[3]}. \end{aligned}$$

(A.27)

Similarly, we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[2]})&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{ n \}! } (q^2c_1c_2)^n {}_{2}r( T_2(E_1)^nT_2(E_3)^n) \\&\overset{(\mathrm{A.26})}{=} q^{-1}(q-q^{-1})^2(q^2c_1c_2) \Big ( E_3T_2(E_1) + E_1T_2(E_3) \Big ) \mathfrak {X}^{[2]} \\&\qquad {} + (q-q^{-1})T_{213}(E_2) \sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{ \{ n \}!}(q^2c_1c_2)^n \{n \}^2 T_2(E_1)^{n-1}T_2(E_3)^{n-1} \\&= (q-q^{-1})^2(qc_1c_2) \Big ( E_3T_2(E_1) + E_1T_2(E_3) \Big ) \mathfrak {X}^{[2]} \nonumber \\&\qquad {} + (q-q^{-1})^2(q^2c_1c_2)T_{213}(E_2) \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{ n \}! } (q^2c_1c_2)^n \{n+1 \} T_2(E_1)^nT_2(E_3)^n. \end{aligned}$$

We want to write the last summand in terms of $\mathfrak {X}^{[2]}$. To do this, we use the fact that $\{n+1\} = 1 + q^2\{n\}$ for $n \ge 1$. This is a useful fact that will be used again in future calculations. Using this, we have

$$\begin{aligned}&\sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{ \{ n \}! } (q^2c_1c_2)^n \{n+1 \} T_2(E_1)^nT_2(E_3)^n\\&\qquad {}= 1 + \sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{ \{ n \}! } (q^2c_1c_2)^n ( 1 + q^2\{n\} ) T_2(E_1)^nT_2(E_3)^n \\&\qquad {}= \mathfrak {X}^{[2]} + q^2\sum _{n \ge 1} \dfrac{(q-q^{-1})^n}{ \{ n-1 \}! } (q^2c_1c_2)^n T_2(E_1)^nT_2(E_3)^n \\&\qquad {}= \mathfrak {X}^{[2]} + q^2\sum _{n \ge 0} \dfrac{(q-q^{-1})^{n+1}}{ \{ n \}! } (q^2c_1c_2)^{n+1} T_2(E_1)^{n+1}T_2(E_3)^{n+1}\\&\qquad {}= \mathfrak {X}^{[2]} + (q-q^{-1})(q^4c_1c_2)T_2(E_1)T_2(E_3)\mathfrak {X}^{[2]}. \end{aligned}$$

Inserting this equation into the expression for ${}_{2}r(\mathfrak {X}^{[2]})$, we obtain

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[2]})&=(q-q^{-1})^2(qc_1c_2) \Big ( E_3T_2(E_1) + E_1T_2(E_3) \Big ) \mathfrak {X}^{[2]} \nonumber \\&\quad + (q-q^{-1})^2(q^2c_1c_2)T_{213}(E_2)\mathfrak {X}^{[2]} \nonumber \\&\quad + (q-q^{-1})^3(q^6c_1^2c_2^2)T_{213}(E_2)T_2(E_1)T_2(E_3)\mathfrak {X}^{[2]}. \end{aligned}$$

(A.28)

Finally, we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[1]}) \overset{(3.42)}{=} (q-q^{-1})(q^2c_2)E_2\mathfrak {X}^{[1]}. \end{aligned}$$

(A.29)

Now, we use Lemma A.11 to rewrite the term $\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]})$. By calculations similar to those leading to (A.27) and (A.28), we obtain

$$\begin{aligned} \mathfrak {X}_{[2]}E_2&= E_2\mathfrak {X}^{[2]}, \end{aligned}$$

(A.30)

$$\begin{aligned} \mathfrak {X}_{[3]}E_2&= E_2\mathfrak {X}^{[3]} - (q-q^{-1})^2(q^4c_1^2c_2)\mathfrak {X}_{[3]}T_{213}(E_2)T_2(E_1)T_2(E_3), \end{aligned}$$

(A.31)

$$\begin{aligned} \mathfrak {X}_{[4]}E_2&= E_2\mathfrak {X}^{[4]} - q^{-1}(q-q^{-1})c_1\mathfrak {X}_{[4]} \big (E_1T_2(E_3) + E_3T_2(E_1) \big ) \nonumber \\&\qquad - (q-q^{-1})c_1 \big ( \mathfrak {X}_{[4]} + (q-q^{-1})c_1\mathfrak {X}_{[4]}E_1E_3 \big )T_{213}(E_2). \end{aligned}$$

(A.32)

We use these to obtain the following.

$$\begin{aligned}&\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]}) \\&\quad \overset{(\mathrm{A.29})}{=} (q-q^{-1})(q^2c_2)\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}E_2\mathfrak {X}^{[1]}\\&\quad \overset{(\mathrm{A.30})}{=} (q-q^{-1})(q^2c_2)\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}E_2\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad \overset{(\mathrm{A.31})}{=} (q-q^{-1})(q^2c_2)\mathfrak {X}_{[4]}E_2\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad \qquad - (q-q^{-1})^3(q^6c_1^2c_2^2)\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}T_{213}(E_2)T_2(E_1)T_2(E_3) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad \overset{(\mathrm{A.32})}{=} (q-q^{-1})(q^2c_2)\Big (E_2\mathfrak {X}^{[4]} - q^{-1}(q-q^{-1})c_1\mathfrak {X}_{[4]} \big (E_1T_2(E_3) + E_3T_2(E_1) \big )\\&\quad \qquad - (q-q^{-1})c_1 \big ( \mathfrak {X}_{[4]} + (q-q^{-1})c_1\mathfrak {X}_{[4]}E_1E_3 \big ) T_{213}(E_2) \Big ) \mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad \qquad -(q-q^{-1})^3(q^6c_1^2c_2^2)\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}T_{213}(E_2)T_2(E_1)T_2(E_3) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}. \end{aligned}$$

We gather terms now and get

$$\begin{aligned}&\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]})\\&\quad = (q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{\widetilde{w}_0'} \\&\qquad -(q-q^{-1})^2(qc_1c_2)\mathfrak {X}_{[4]} \big (E_1T_2(E_3) + E_3T_2(E_1) \big ) \mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\qquad - (q-q^{-1})^2(q^2c_1c_2)\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}T_{213}(E_2) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\qquad + (q-q^{-1})^3(q^6c_1^2c_2^2)\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}T_{213} (E_2)T_2(E_1)T_2(E_3)\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\qquad -(q-q^{-1})^3(q^2c_1^2c_2)\mathfrak {X}_{[4]}E_1E_3T_{213}(E_2) \mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad = (q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{\widetilde{w}_0^{\prime }} - \mathfrak {X}_{[4]}\mathfrak {X}_{[3]} {}_{2}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} - \mathfrak {X}_{[4]} {}_{2}r(\mathfrak {X}^{[3]}) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]} \end{aligned}$$

where we use the fact that $E_1T_2(E_3)$ and $E_3T_2(E_1)$ both commute with $T_{213}(E_2)$, and hence with $\mathfrak {X}^{[3]}$. It follows that

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }})&= \mathfrak {X}_{[4]} {}_{2}r(\mathfrak {X}^{[3]}) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]} {}_{2}r(\mathfrak {X}^{[2]}) \mathfrak {X}^{[1]} + \mathfrak {X}_{[4]} \mathfrak {X}_{[3]}\mathfrak {X}_{[2]} {}_{2}r(\mathfrak {X}^{[1]})\\&= (q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{\widetilde{w}_0^{\prime }} \end{aligned}$$

as required. $\square$

1.4 Type $AIII_n$ for $n \ge 4$

Consider the Satake diagram of type $AIII_n$ for $n \ge 4$.

In this case the restricted root system is of type $B_2$ with

$$\begin{aligned} \widetilde{\alpha }_1 = \dfrac{\alpha _1 + \alpha _n}{2}, \qquad \widetilde{\alpha }_2 = \dfrac{\alpha _2 + \alpha _3 + \cdots + \alpha _{n-1}}{2}. \end{aligned}$$

The subgroup $\widetilde{W} \subset W^{\Theta }$ is generated by the elements

$$\begin{aligned} \widetilde{\sigma }_1 = \sigma _1\sigma _n, \qquad \widetilde{\sigma }_2 = w_Xw_{\{2,n-1\} \cup X} = \sigma _2\sigma _3 \dots \sigma _{n-1} \dots \sigma _3\sigma _2. \end{aligned}$$

The longest word of the restricted Weyl group has two reduced expressions given by

$$\begin{aligned} \widetilde{w}_0 = \widetilde{\sigma }_1\widetilde{\sigma }_2\widetilde{\sigma }_1\widetilde{\sigma }_2, \qquad \widetilde{w}_0^{\prime } = \widetilde{\sigma }_2\widetilde{\sigma }_1\widetilde{\sigma }_2\widetilde{\sigma }_1. \end{aligned}$$

The definition (3.14) and condition (3.18) imply that $c_1=c_n=\overline{c_1}$. By Lemmas 3.9 and 3.10 we have

$$\begin{aligned} \mathfrak {X}_1&= \sum _{k \ge 0} \dfrac{ (q-q^{-1})^k }{\{k\}!}c_1^k (E_1E_n)^k,\\ \mathfrak {X}_2&= \bigg ( \sum _{k \ge 0}\dfrac{(c_{2}s(n-1))^k}{\{k\}!} T_2T_{w_X}(E_{n-1})^k \bigg ) \bigg ( \sum _{k \ge 0}\dfrac{(c_{n-1}s(2))^k}{\{k\}!} T_{n-1}T_{w_X}(E_2)^k \bigg ). \end{aligned}$$

Proposition A.13

The partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}_0}$ coincides with the quasi K-matrix $\mathfrak {X}$.

We have the following relations needed in the proof of Proposition A.13, proved by induction.

Lemma A.14

For any $k \in \mathbb {N}$ the relations

$$\begin{aligned}&T_{1n(n-1)}T_{w_X}(E_2)^k E_n = q^{-k} E_n T_{1n(n-1)}T_{w_X}(E_2)^k, \nonumber \\&T_{12n}T_{w_X}(E_{n-1})^kE_n = q^{-k} E_n T_{12n}T_{w_X}(E_{n-1})^k, \nonumber \\&T_{n \cdots 3}(E_2)^k E_n = q^{-k} E_n T_{n \cdots 3}(E_2)^k, \nonumber \\&T_{1 \cdots (n-2)}(E_{n-1})^k E_n = q^k E_n T_{1 \cdots (n-2)}(E_{n-1})^k \nonumber \\&\qquad - q \{k \} T_{1 \cdots (n-2)}(E_{n-1})^{k-1} T_{12n}T_{w_X}(E_{n-1}), \nonumber \\&T_{n-1}T_{w_X}(E_2)^k E_n = q^k E_n T_{n-1}T_{w_X}(E_2)^k \nonumber \\&\qquad - q \{k \} T_{n-1}T_{w_X}(E_2)^{k-1} T_{n \cdots 3}(E_2), \nonumber \\&T_2T_{w_X}(E_{n-1})^k E_n = q^k E_n T_2T_{w_X}(E_{n-1})^k \nonumber \\&\qquad - q \{k \} T_2T_{w_X}(E_{n-1})^{k-1} T_{2n}T_{w_X}(E_{n-1}), \nonumber \\&T_{2n}T_{w_X}(E_{n-1}) T_{n-1}T_{w_X}(E_2)^k = q^{-k}T_{n-1}T_{w_X}(E_2)^k T_{2n}T_{w_X}(E_{n-1})\nonumber \\&\qquad + q^{1-k} (q-q^{-1}) \{k \}T_{n-1}T_{w_X}(E_2)^{k-1}T_2T_{w_X}(E_{n-1})T_{n \cdots 3}(E_2), \end{aligned}$$

(A.33)

$$\begin{aligned}&T_{12n}T_{w_X}(E_{n-1})^{k} T_{n \cdots 3}(E_2) \nonumber \\&\quad = \big ( q^{-k} + (q-q^{-1})q^{1-k}\{ k \} \big ) T_{n \cdots 3}(E_2) T_{12n}T_{w_X}(E_{n-1})^{k} \end{aligned}$$

(A.34)

hold in $U_q(\mathfrak {sl}_{n+1})$.

Lemma A.15

For any $k \in \mathbb {N}$ the relations

$$\begin{aligned}&{}_{1}r(T_{1 \cdots (n-2)}(E_{n-1})^k) \nonumber \\&\quad = q^{-1}(q-q^{-1}) \{k \} T_{2 \cdots (n-2)}(E_{n-1})T_{1 \cdots (n-2)}(E_{n-1})^{k-1}, \end{aligned}$$

(A.35)

$$\begin{aligned}&{}_{1}r(T_{12n}T_{w_X}(E_{n-1})^k) \nonumber \\&\quad = q^{-1}(q-q^{-1}) \{k \} T_{2n}T_{w_X}(E_{n-1})T_{12n}T_{w_X}(E_{n-1})^{k-1}, \end{aligned}$$

(A.36)

$$\begin{aligned}&{}_{1}r(T_{1n(n-1)}T_{w_X}(E_2)^k) \nonumber \\&\quad = q^{-1}(q-q^{-1}) \{k \} T_{n(n-1)}T_{w_X}(E_2)T_{1n(n-1)}T_{w_X}(E_2)^{k-1} \end{aligned}$$

(A.37)

hold in $U_q(\mathfrak {sl}_{n+1})$.

One can show that the above Lemmas still hold if we consider the case where we have no black dots. In this situation, the calculations differ slightly but the results still hold.

Proof of Proposition A.13

We write

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0} = \mathfrak {X}^{[4]}\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}, \end{aligned}$$

(A.38)

where, recalling the notation $\widetilde{c}_2^{\;2} = c_2c_{n-1}s(n-1)s(2)$, we have

$$\begin{aligned}&\mathfrak {X}^{[4]} \overset{(3.50)}{=} \mathfrak {X}_2,\\&\mathfrak {X}^{[3]} = \sum _{k \ge 0}\dfrac{(q-q^{-1})^k}{\{k\}!}(qc_1\widetilde{c}_2^{\;2})^kT_{n \cdots 3}(E_2)^kT_{1 \cdots (n-2)}(E_{n-1})^k,\\&\mathfrak {X}^{[2]} = \bigg (\sum _{k \ge 0} \dfrac{(qc_1c_2s(n-1))^{k}}{\{k\}!}T_{12n}T_{w_X}(E_{n-1})^{k} \bigg )\\&\qquad \qquad \bigg ( \sum _{k \ge 0} \dfrac{(qc_1c_{n-1}s(2))^{k}}{\{k\}!} T_{1n(n-1)}T_{w_X}(E_2)^{k} \bigg ),\\&\mathfrak {X}^{[1]} =\mathfrak {X}_1. \end{aligned}$$

When necessary, since $\mathfrak {X}^{[4]}$ and $\mathfrak {X}^{[2]}$ are a product of two infinite sums, we write $\mathfrak {X}^{[4]} = \mathfrak {X}^{[4;1]}\mathfrak {X}^{[4;2]}$ and $\mathfrak {X}^{[2]} = \mathfrak {X}^{[2;1]}\mathfrak {X}^{[2;2]}$.

For each $i = 1, 2, 3, 4$, let $\mathfrak {X}_{[i]} = K_1\mathfrak {X}^{[i]}K_1^{-1}$. We write $\mathfrak {X}_{[4]} = \mathfrak {X}_{[4;1]}\mathfrak {X}_{[4;2]}$ and $\mathfrak {X}_{[2]}= \mathfrak {X}_{[2;1]}\mathfrak {X}_{[2;2]}$. Now, by the rank one case for $\mathfrak {X}_2$ given in Lemma 3.10 and [11, 8.26, (4)], we obtain

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})c_2s(n-1)T_{w_X}(E_{n-1})\mathfrak {X}_{\widetilde{w}_0}. \end{aligned}$$

(A.39)

The underlying symmetry implies that we only need to show that

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})c_1 E_n \mathfrak {X}_{\widetilde{w}_0}. \end{aligned}$$

By property (3.9) of the skew derivative ${}_1r$ and [11, 8.26, (4)] we have

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{\widetilde{w}_0}) = \mathfrak {X}_{[4]} {}_{1}r(\mathfrak {X}^{[3]}) \mathfrak {X}^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}{}_{1}r(\mathfrak {X})^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{1}r(\mathfrak {X}^{[1]}).\nonumber \\ \end{aligned}$$

(A.40)

Using Lemma A.15 and Lemma 3.9 we find that

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[3]})&\overset{(\mathrm{A.35})}{=} (q-q^{-1})^2 c_1 \widetilde{c}_2^{\;2}T_{n \cdots 3}(E_2)T_{2 \cdots (n-2)}(E_{n-1})\mathfrak {X}^{[3]}, \end{aligned}$$

(A.41)

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[1]})&\overset{(3.42)}{=} (q-q^{-1})c_1E_n\mathfrak {X}^{[1]}. \end{aligned}$$

(A.42)

To write an expression for ${}_{1}r(\mathfrak {X}^{[2]})$, we use the splitting of $\mathfrak {X}^{[2]}$ into a product of two infinite sums. By Eqs. (A.36) and (A.37) of Lemma A.15 we have

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[2]})&= {}_{1}r(\mathfrak {X}^{[2;1]})\mathfrak {X}^{[2;2]} + \mathfrak {X}_{[2;1]}{}_{1}r(\mathfrak {X}^{[2;2]}) \nonumber \\&= (q-q^{-1})c_1c_2s(n-1)T_{2n}T_{w_X}(E_{n-1})\mathfrak {X}^{[2]} \nonumber \\&\qquad {}+ (q-q^{-1})c_1c_{n-1}s(2)\mathfrak {X}_{[2;1]} T_{n(n-1)}T_{w_X}(E_2)\mathfrak {X}^{[2;2]}. \end{aligned}$$

(A.43)

We would like the last summand of this expression to be in terms of $\mathfrak {X}^{[2]}$. To this end, we use Eq. (A.34) in Lemma A.14 to obtain

$$\begin{aligned}&\mathfrak {X}_{[2;1]}T_{n(n-1)}T_{w_X}(E_2)\\&\quad = \bigg ( 1 + \sum _{k \ge 1} \dfrac{(qc_1c_2s(n-1))^{k}}{\{k\}!}q^{k}T_{12n}T_{w_X}(E_{n-1})^{k} \bigg ) T_{n(n-1)}T_{w_X}(E_2)\\&\overset{(\mathrm{A.34})}{=} T_{n(n-1)}T_{w_X}(E_2)\mathfrak {X}^{[2;1]}\\&\quad \qquad + q^2(q-q^{-1})c_1c_2s(n-1)T_{n(n-1)}T_{w_X}(E_2)T_{12n} T_{w_X}(E_{n-1})\mathfrak {X}^{[2;1]}. \end{aligned}$$

Substituting this into (A.43) it follows that

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[2]})&= (q-q^{-1})c_1c_2s(n-1)T_{2n}T_{w_X}(E_{n-1})\mathfrak {X}^{[2]}\nonumber \\&\quad + (q-q^{-1})c_1c_{n-1}s(2)T_{n(n-1)}T_{w_X}(E_2)\mathfrak {X}^{[2]} \nonumber \\&\quad + q^2(q-q^{-1})^2 c_1^2\widetilde{c}_2^{\;2}T_{n(n-1)}T_{w_X}(E_2)T_{12n}T_{w_X}(E_{n-1})\mathfrak {X}^{[2]}. \end{aligned}$$

(A.44)

When we calculate ${}_{1}r(\mathfrak {X}_{\widetilde{w}_0})$, we obtain a component of the form $\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{1}r(\mathfrak {X}^{[1]})$. From Eq. (A.42), we see that we obtain an $E_n$ term that we want to pass to the front of this expression. We use Lemma A.14 to do this. We have

$$\begin{aligned} \mathfrak {X}_{[2]}E_n&= E_n\mathfrak {X}^{[2]}, \end{aligned}$$

(A.45)

$$\begin{aligned} \mathfrak {X}_{[3]}E_n&= E_n\mathfrak {X}_{[3]} - q^2(q-q^{-1})c_1 \widetilde{c}_2^{\;2} T_{n \cdots 3}(E_2)\mathfrak {X}^{[3]}T_{12n}T_{w_X}(E_{n-1}), \end{aligned}$$

(A.46)

$$\begin{aligned} \mathfrak {X}_{[4;2]}E_n&= E_n\mathfrak {X}^{[4;2]} - c_1c_{n-1}s(2)\mathfrak {X}_{[4;2]} T_{n \cdots 3}(E_2), \end{aligned}$$

(A.47)

$$\begin{aligned} \mathfrak {X}_{[4;1]}E_n&= E_n\mathfrak {X}^{[4;1]} - c_1c_2s(n-1)\mathfrak {X}_{[4;1]}T_{2n} T_{w_X}(E_{n-1}). \end{aligned}$$

(A.48)

Now, using Eq. (A.33), we obtain the following expression for $\mathfrak {X}_{[4]}E_n$.

$$\begin{aligned} \mathfrak {X}_{[4]}E_n&\overset{(\mathrm{A.47})}{=} \mathfrak {X}_{[4;1]} \big ( E_n\mathfrak {X}^{[4;2]} - c_1 c_{n-1}s(2)\mathfrak {X}_{[4;2]}T_{n \cdots 3}(E_2) \big ) \nonumber \\&\overset{(\mathrm{A.48})}{=} \big ( E_n\mathfrak {X}^{[4;1]} - c_1c_2s(n-1)\mathfrak {X}_{[4;1]}T_{2n}T_{w_X}(E_{n-1})\big )\mathfrak {X}^{[4;2]} \nonumber \\&\qquad - c_1c_{n-1}s(2)\mathfrak {X}_{[4]}T_{n \cdots 3}(E_2)\nonumber \\&\overset{(\mathrm{A.33})}{=} E_n\mathfrak {X}^{[4]} - c_1c_{n-1}s(2)\mathfrak {X}_{[4]}T_{n \cdots 3}(E_2)\nonumber \\&\qquad - c_1c_2s(n-1)\mathfrak {X}_{[4;1]}\mathfrak {X}_{[4;2]}T_{2n}T_{w_X}(E_{n-1})\nonumber \\&\qquad + (q- q^{-1})c_1^2c_2c_{n-1}s(2)s(n-1)\mathfrak {X}_{[4;1]}\mathfrak {X}_{[4;2]}T_2T_{w_X}(E_{n-1})T_{n \cdots 3}(E_2) \nonumber \\&= E_n\mathfrak {X}^{[4]} - c_1c_{n-1}s(2)\mathfrak {X}_{[4]}T_{n \cdots 3}(E_2) - c_1c_2s(n-1)\mathfrak {X}_{[4]}T_{2n}T_{w_X}(E_{n-1}) \nonumber \\&\qquad - (q-q^{-1})c_1^2\widetilde{c}_2^{\;2}\mathfrak {X}_{[4]}T_2T_{w_X}(E_{n-1})T_{n \cdots 3}(E_2). \end{aligned}$$

(A.49)

Using Eqs. (A.45), (A.46) and (A.49), and comparing with Eqs. (A.41), (A.42) and (A.44), one finds that

$$\begin{aligned} \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]} {}_{1}r(\mathfrak {X}^{[1]})&= (q-q^{-1})\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}E_n\mathfrak {X}^{[1]}\\&= (q-q^{-1})E_n\mathfrak {X}_{\widetilde{w}_0} - \mathfrak {X}_{[4]} {}_{1}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&\quad - \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}{}_{1}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]}, \end{aligned}$$

and hence it follows that

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{\widetilde{w}_0}) = (q-q^{-1})E_n\mathfrak {X}_{\widetilde{w}_0}, \end{aligned}$$

as required. This completes the proof. $\square$

Proposition A.16

The partial quasi K-matrix $\mathfrak {X}_{\widetilde{w}_0^{\prime }}$ coincides with the quasi K-matrix $\mathfrak {X}$.

We have the following relations needed in the proof of Proposition A.16, proved by induction.

Lemma A.17

For any $k \in \mathbb {N}$ the relations

$$\begin{aligned}&T_{2 \cdots (n-1)}(E_n)^kT_{w_X}(E_{n-1}) = T_{w_X}(E_{n-1})T_{2 \cdots (n-1)}(E_n)^k, \\&T_{(n-1) \cdots 2}(E_1)^kT_{w_X}(E_{n-1}) = q^{-k}T_{w_X}(E_{n-1})T_{(n-1) \cdots 2}(E_1)^k, \\&T_{2 \cdots (n-1)}T_{12n}T_{w_X}(E_{n-1})^kT_{w_X}(E_{n-1}) = T_{w_X}(E_{n-1})T_{2 \cdots (n-1)}T_{12n}T_{w_X}(E_{n-1})^k, \\&T_{(n-1) \cdots 2}T_{1n(n-1)}T_{w_X}(E_2)^kT_{w_X}(E_{n-1}) = T_{w_X}(E_{n-1})T_{(n-1) \cdots 2}T_{1n(n-1)}T_{w_X}(E_2)^k \\&\qquad - (q-q^{-1})\{k\}T_{(n-1) \cdots 2}T_{1n(n-1)}T_{w_X}(E_2)^{k-1}T_{3 \cdots (n-1)}(E_n)T_{(n-1) \cdots 2}(E_1), \\&E_1^kT_{w_X}(E_{n-1}) = T_{w_X}(E_{n-1})E_1^k, \\&E_n^kT_{w_X}(E_{n-1}) = q^kT_{w_X}(E_{n-1})E_n^k - q\{k\}E_n^{k-1}T_{w_X}T_{n-1}(E_n) \end{aligned}$$

hold in $U_q(\mathfrak {sl}_{n+1})$.

Lemma A.18

For any $k \in \mathbb {N}$ the relations

$$\begin{aligned}&{}_{2}r(T_{2 \cdots (n-1)}(E_n)^k) = q^{-1}(q-q^{-1})\{k\}T_{3 \cdots (n-1)}(E_n)T_{2 \cdots (n-1)}(E_n)^{k-1}, \\&{}_{2}r(T_{2 \cdots (n-1)}T_{12n}T_{w_X}(E_{n-1})^k) \\&\qquad = q^{-2}(q-q^{-1})^2\{k\}E_1T_{3 \cdots (n-1)}(E_n)T_{2 \cdots (n-1)}T_{12n}T_{w_X}(E_{n-1})^{k-1} \end{aligned}$$

hold in $U_q(\mathfrak {sl}_{n+1})$.

Proof of Proposition A.16

We write

$$\begin{aligned} \mathfrak {X}_{\widetilde{w}_0^{\prime }} = \mathfrak {X}^{[4]}\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}, \end{aligned}$$

where we have

$$\begin{aligned}&\mathfrak {X}^{[4]} \overset{(3.50)}{=} \mathfrak {X}_1, \\&\mathfrak {X}^{[3]} = \bigg ( \sum _{k_1 \ge 0}\dfrac{ (qc_1c_2s(n-1))^{k_1} }{ \{ k_1 \}! } T_{2 \cdots (n-1)}T_{12n}T_{w_X}(E_{n-1})^{k_1} \bigg )\\&\qquad \qquad \bigg ( \sum _{k_2 \ge 0}\dfrac{ (qc_1c_{n-1}s(2))^{k_2} }{ \{ k_2 \}! } T_{(n-1) \cdots 2}T_{1n(n-1)}T_{w_X}(E_{2})^{k_2} \bigg ),\\&\mathfrak {X}^{[2]} = \sum _{k \ge 0} \dfrac{(q-q^{-1})^k}{\{k\}!}(qc_1\widetilde{c}_2^{\;2})^k T_{(n-1) \cdots 2}(E_1)^k T_{2 \cdots (n-1)}(E_n)^k,\\&\mathfrak {X}^{[1]} = \mathfrak {X}_2. \end{aligned}$$

For $i = 1, 2, 3, 4$ let $\mathfrak {X}_{[i]} = K_2\mathfrak {X}^{[i]}K_2^{-1}$. By Lemma 3.9 and [11, 8.26, (4)] it follows that

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }})&= {}_{1}r(\mathfrak {X}^{[4]}) \mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})E_n\mathfrak {X}_{\widetilde{w}_0^{\prime }}. \end{aligned}$$

Hence, we only need to check that

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }}) = q^{-1}(q-q^{-1})c_2s(n-1)T_{w_X}(E_{n-1})\mathfrak {X}_{\widetilde{w}_0^{\prime }}. \end{aligned}$$

By [11, 8.26, (4)] and property (3.9) of the skew derivative ${}_2r$ we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }}) = \mathfrak {X}_{[4]}{}_{2}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}{}_{2}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]}). \end{aligned}$$

Using Lemmas 3.10 and A.18, we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[3]})&= q^{-1}(q-q^{-1})^2c_1c_2s(n-1)E_1T_{3 \cdots (n-1)}(E_n)\mathfrak {X}^{[3]},\\ {}_{2}r(\mathfrak {X}^{[2]})&= (q-q^{-1})^2c_1\widetilde{c}_2^{\;2}T_{3 \cdots (n-1)}(E_n)T_{(n-1) \cdots 2}(E_1)\mathfrak {X}^{[2]},\\ {}_{2}r(\mathfrak {X}^{[1]})&= q^{-1}(q-q^{-1})c_2s(n-1)T_{w_X}(E_{n-1})\mathfrak {X}^{[1]}. \end{aligned}$$

By Lemma A.17, we have

$$\begin{aligned} \mathfrak {X}_{[2]}T_{w_X}(E_{n-1})&= T_{w_X}(E_{n-1})\mathfrak {X}^{[2]},\\ \mathfrak {X}_{[3]}T_{w_X}(E_{n-1})&= T_{w_X}(E_{n-1})\mathfrak {X}^{[3]}\\&\quad {}- q(q-q^{-1})c_1c_{n-1}s(2)\mathfrak {X}_{[3]}T_{3 \cdots (n-1)}(E_n)T_{(n-1) \cdots 2}(E_1),\\ \mathfrak {X}_{[4]}T_{w_X}(E_{n-1})&= T_{w_X}(E_{n-1})\mathfrak {X}^{[4]} - (q-q^{-1})c_1\mathfrak {X}_{[4]}E_1T_{3 \cdots (n-1)}(E_n). \end{aligned}$$

It follows that

$$\begin{aligned} \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]})&= q^{-1}(q-q^{-1})c_2s(n-1)T_{w_X}(E_{n-1})\mathfrak {X}_{w^{\prime }} \\&\quad {}- \mathfrak {X}_{[4]}{}_{2}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} - \mathfrak {X}_{[4]} \mathfrak {X}_{[3]}{}_{2}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} \end{aligned}$$

and therefore we obtain

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{\widetilde{w}_0^{\prime }}) = q^{-1}(q-q^{-1})c_2s(n-1)T_{w_X}(E_{n-1})\mathfrak {X}_{\widetilde{w}_0^{\prime }} \end{aligned}$$

as required. $\square$

1.5 Type $CI_2$

Consider the Satake diagram of type $CI_2$.

Since $\Theta = -\text {id}$ the subgroup $\widetilde{W}$ coincides with W. The longest word of the Weyl group has two reduced expressions given by

$$\begin{aligned} w_0 = \sigma _1\sigma _2\sigma _1\sigma _2,\qquad w_0^{\prime } = \sigma _2\sigma _1\sigma _2\sigma _1. \end{aligned}$$

By Lemma 3.7 we have

$$\begin{aligned} \mathfrak {X}_1&= \sum _{n \ge 0} \dfrac{(q^{2} - q^{-2})^n}{\{2n\}_1!!} (q^4c_1)^nE_1^{2n},\\ \mathfrak {X}_2&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}_2!!} (q^2c_2)^nE_2^{2n}. \end{aligned}$$

Proposition A.19

The partial quasi K-matrix $\mathfrak {X}_{w_0}$ coincides with the quasi K-matrix $\mathfrak {X}$.

The following relations are necessary for the proof of Proposition A.19, proved by induction.

Lemma A.20

For any $n \in \mathbb {N}$ the relations

$$\begin{aligned}&E_2^nE_1 = q^{2n}E_1E_2^{n} - q^2\{n\}_2E_2^{n-1}T_1(E_2) - q^3\{n\}_2\{n-1\}_2E_2^{n-1}T_{12}(E_1), \end{aligned}$$

(A.50)

$$\begin{aligned}&T_1(E_2)^nE_1 = q^{-2n}E_1T_1(E_2)^n, \end{aligned}$$

(A.51)

$$\begin{aligned}&T_{12}(E_1)^nE_1 = E_1T_{12}(E_1)^n - \frac{(q^2-1)}{[2]_2}\{n\}_1T_{12}(E_1)^{n-1}T_1(E_2)^2 \end{aligned}$$

(A.52)

hold in $U_q(\mathfrak {so}_5)$.

Lemma A.21

For any $n \in \mathbb {N}$ the relations

$$\begin{aligned}&{}_{1}r(T_1(E_2)^{n+1}) = q^{-2}(q^{2} - q^{-2}) \{n+1\}_2 E_2T_1(E_2)^{n}\nonumber \\&\quad + q^{-1}(q^{2} - q^{-2}) \{n+1\}_2\{n\}_2T_{12}(E_1)T_1(E_2)^{n-1}, \end{aligned}$$

(A.53)

$$\begin{aligned}&{}_{1}r(T_{12}(E_1)^n) = q^{-3}(q-q^{-1})^2\{n\}_1E_2^2T_{12}(E_1)^{n-1} \end{aligned}$$

(A.54)

hold in $U_q(\mathfrak {so}_5)$.

Proof of Proposition A.19

We have

$$\begin{aligned} \mathfrak {X}_{w_0} = \mathfrak {X}^{[4]}\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} \end{aligned}$$

where

$$\begin{aligned} \mathfrak {X}^{[4]}&\overset{(3.50)}{=} \mathfrak {X}_2, \\ \mathfrak {X}^{[3]}&= \Psi \circ T_{12} \circ \Psi ^{-1}(\mathfrak {X}_1) =\sum _{n \ge 0} \dfrac{(q^{2} - q^{-2})^n}{\{2n\}_1!!} (q^4c_1)^n(q^2c_2)^{2n}T_{12}(E_1)^{2n},\\ \mathfrak {X}^{[2]}&= \Psi \circ T_1 \circ \Psi ^{-1}(\mathfrak {X}_2) = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}_2!!} (q^4c_1)^n(q^2c_2)^n T_1(E_2)^{2n},\\ \mathfrak {X}^{[1]}&= \mathfrak {X}_1. \end{aligned}$$

By Lemma 3.7 and [11, 8.26, (4)] we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{w_0})&= {}_{2}r(\mathfrak {X}_2)\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})(q^2c_2) E_2 \mathfrak {X}_2\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}\\&= (q-q^{-1})(q^2c_2) E_2 \mathfrak {X}_{w_0}. \end{aligned}$$

We want to show that

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{w_0}) = (q^{2} - q^{-2})(q^4c_1)E_1\mathfrak {X}_{w_0}. \end{aligned}$$

For each $i = 1,2,3,4$ let $\mathfrak {X}_{[i]} = K_1\mathfrak {X}^{[i]}K_1^{-1}$. Note that $\mathfrak {X}_{[3]} = \mathfrak {X}^{[3]}$. By property (3.9) of the skew derivative ${}_1r$ we see that

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{w_0}) = \mathfrak {X}_{[4]}{}_{1}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}{}_{1}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{1}r(\mathfrak {X}^{[1]}).\nonumber \\ \end{aligned}$$

(A.55)

Lemma A.21 gives

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[3]})&= q^{-3}(q-q^{-1})^2 (q^{2} - q^{-2}) (q^4c_1)(q^2c_2)^2 E_2^2T_{12}(E_1)\mathfrak {X}^{[3]}, \end{aligned}$$

(A.56)

$$\begin{aligned} {}_{1}r(\mathfrak {X}^{[1]})&= (q^{2} - q^{-2})(q^4c_1)E_1\mathfrak {X}^{[1]}. \end{aligned}$$

(A.57)

By Eq. (A.53) we have

$$\begin{aligned} \begin{aligned} {}_{1}r(\mathfrak {X}^{[2]})&= q^{-2}(q-q^{-1})(q^{2} - q^{-2})(q^4c_1)(q^2c_2)E_2T_1(E_2)\mathfrak {X}^{[2]}\\&\quad {}+ q^{-1}(q-q^{-1})(q^{2} - q^{-2})(q^6c_1c_2)T_{12}(E_1) \widehat{\mathfrak {X}}^{[2]} \end{aligned} \end{aligned}$$

(A.58)

where

$$\begin{aligned} \widehat{\mathfrak {X}}^{[2]} = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}_2!!}(q^6c_1c_2)^n \{2n+1\}_2T_1(E_2)^{2n}. \end{aligned}$$

For any $n \ge 1$ we have

$$\begin{aligned} \{2n+1\}_2 = 1 + q^2\{2n\}_2. \end{aligned}$$

(A.59)

It hence follows that

$$\begin{aligned} \begin{aligned}&\sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}_2!!}(q^4c_1)^n(q^2c_2)^n \{2n+1\}_2T_1(E_2)^{2n}\\&\quad = \mathfrak {X}^{[2]} + q^2(q-q^{-1})(q^4c_1)(q^2c_2)T_1(E_2)^2\mathfrak {X}^{[2]}. \end{aligned} \end{aligned}$$

Substituting this into Eq. (A.58) we see that

$$\begin{aligned} \begin{aligned} {}_{1}r(\mathfrak {X}^{[2]})&= q^{-2}(q-q^{-1})(q^{2} - q^{-2})(q^4c_1)(q^2c_2)E_2T_1(E_2)\mathfrak {X}^{[2]}\\&\quad + q^{-1}(q-q^{-1})(q^{2} - q^{-2})(q^4c_1)(q^2c_2)T_{12}(E_1)\mathfrak {X}^{[2]}\\&\quad + q (q-q^{-1})^2 (q^{2} - q^{-2}) (q^4c_1)^2(q^2c_2)^2T_{12}(E_1)T_1(E_2)^2\mathfrak {X}^{[2]} \end{aligned} \end{aligned}$$

(A.60)

When we calculate ${}_{1}r(\mathfrak {X}_{w_0})$, we obtain a component of the form $\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{1}r(\mathfrak {X}^{[1]})$. From Eq. (A.57), this contains an $E_1$ term that we pass to the front using Lemma A.20. We have

$$\begin{aligned} \mathfrak {X}_{[2]}E_1&\overset{(\mathrm{A.51})}{=} E_1\mathfrak {X}^{[2]}, \end{aligned}$$

(A.61)

$$\begin{aligned} \mathfrak {X}_{[3]}E_1&\overset{(\mathrm{A.52})}{=} E_1\mathfrak {X}^{[3]} - q(q-q^{-1})^2(q^4c_1)(q^2c_2)^2\mathfrak {X}_{[3]}T_{12}(E_1)T_1(E_2)^2, \end{aligned}$$

(A.62)

$$\begin{aligned} \mathfrak {X}_{[4]}E_1&\overset{(\mathrm{A.50})}{=} E_1\mathfrak {X}^{[4]} - q^{-2}(q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{[4]}T_1(E_2)\nonumber \\&\qquad - q^{-1}(q-q^{-1})(q^2c_2) \mathfrak {X}_{[4]}T_{12}(E_1) \nonumber \\&\qquad - q^{-3}(q-q^{-1})^2(q^2c_2)^2\mathfrak {X}_{[4]}E_2^2T_{12}(E_1) \end{aligned}$$

(A.63)

where we also use (A.59) to obtain (A.63). Note that $\mathfrak {X}^{[3]}$ commutes with $E_2T_1(E_2)$. Using (A.61), (A.62) and (A.63), and comparing with (A.56) and (A.60) we obtain

$$\begin{aligned}&\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{1}r(\mathfrak {X}^{[1]})\\&\quad =(q^{2} - q^{-2})(q^4c_1)E_1\mathfrak {X}_{w_0} - \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}{}_{1}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} - \mathfrak {X}_{[4]}{}_{1}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]}. \end{aligned}$$

Hence by (A.55) we have

$$\begin{aligned} {}_{1}r(\mathfrak {X}_{w_0}) = (q^{2} - q^{-2})(q^4c_1)E_1\mathfrak {X}_{w_0} \end{aligned}$$

as required. $\square$

We now consider the reduced expression $w_0^{\prime } = \sigma _2\sigma _1\sigma _2\sigma _1$.

Proposition A.22

The partial quasi K-matrix $\mathfrak {X}_{w_0^{\prime }}$ coincides with the quasi K-matrix $\mathfrak {X}$.

The following relations are needed for the proof and are obtained by induction.

Lemma A.23

For any $n \in \mathbb {N}$ the relations

$$\begin{aligned} E_1^nE_2&= q^{2n}E_2E_1^n - q^2\{n\}_1 E_1^{n-1}T_{21}(E_2), \end{aligned}$$

(A.64)

$$\begin{aligned} T_2(E_1)^nE_2&= q^{-2n}E_2T_2(E_1)^n, \end{aligned}$$

(A.65)

$$\begin{aligned} T_{21}(E_2)^nE_2&= E_2T_{21}(E_2)^n - [2]_2\{n\}_2T_{21}(E_2)^{n-1}T_2(E_1) \end{aligned}$$

(A.66)

hold in $U_q(\mathfrak {so}_5)$.

Lemma A.24

For any $n \in \mathbb {N}$ the relations

$$\begin{aligned} {}_{2}r(T_2(E_1)^n)&= (q-q^{-1})\{n\}_1T_{21}(E_2)T_2(E_1)^{n-1}, \end{aligned}$$

(A.67)

$$\begin{aligned} {}_{2}r(T_{21}(E_2)^n)&= q^{-2}(q^{2} - q^{-2})\{n\}_2E_1T_{21}(E_2)^{n-1} \end{aligned}$$

(A.68)

hold in $U_q(\mathfrak {so}_5)$.

Proof of Proposition A.22

We have

$$\begin{aligned} \mathfrak {X}_{w_0^{\prime }} = \mathfrak {X}^{[4]}\mathfrak {X}^{[3]}\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} \end{aligned}$$

where

$$\begin{aligned}&\mathfrak {X}^{[4]} \overset{(3.50)}{=} \mathfrak {X}_1,\\&\mathfrak {X}^{[3]} = \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}_2!!} (q^4c_1)^n(q^2c_2)^nT_{21}(E_2)^{2n},\\&\mathfrak {X}^{[2]} = \sum _{n \ge 0} \dfrac{(q^{2} - q^{-2})^n}{\{2n\}_1!!} (q^4c_1)^n(q^2c_2)^{2n}T_2(E_1)^{2n},\\&\mathfrak {X}^{[1]} = \mathfrak {X}_2. \end{aligned}$$

By Lemma 3.7 and [11, 8.26, (4)] we have ${}_{1}r(\mathfrak {X}_{w_0^{\prime }})=(q^{2} - q^{-2})(q^4c_1)E_1\mathfrak {X}_{w_0^{\prime }}$. We want to show that

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{w_0^{\prime }}) = (q-q^{-1})(q^2c_2)E_2 \mathfrak {X}_{w_0^{\prime }}. \end{aligned}$$

For each $i = 1,2,3,4$ let $\mathfrak {X}_{[i]} = K_2\mathfrak {X}^{[i]}K_2^{-1}$. By property (3.9) of the skew derivative ${}_2r$ we have

$$\begin{aligned} {}_{2}r(\mathfrak {X}_{w_0^{\prime }}) = \mathfrak {X}_{[4]}{}_{2}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}{}_{2}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]} + \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]}).\nonumber \\ \end{aligned}$$

(A.69)

Using Lemma A.24 we obtain

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[3]})&\overset{(\mathrm{A.68})}{=} q^{-2}(q-q^{-1})(q^{2} - q^{-2})(q^4c_1)(q^2c_2)E_1T_{21}(E_2)\mathfrak {X}^{[3]}, \end{aligned}$$

(A.70)

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[2]})&\overset{(\mathrm{A.67})}{=} (q-q^{-1})(q^{2} - q^{-2})(q^4c_1)(q^2c_2)^2T_{21}(E_2)T_2(E_1)\mathfrak {X}^{[2]}, \end{aligned}$$

(A.71)

$$\begin{aligned} {}_{2}r(\mathfrak {X}^{[1]})&\overset{(3.39)}{=} (q-q^{-1})(q^2c_2) E_2\mathfrak {X}^{[1]}. \end{aligned}$$

(A.72)

By Lemma A.23 we have

$$\begin{aligned} \mathfrak {X}_{[2]}E_2&\overset{(\mathrm{A.65})}{=} E_2\mathfrak {X}^{[2]}, \end{aligned}$$

(A.73)

$$\begin{aligned} \mathfrak {X}_{[3]}E_2&\overset{(\mathrm{A.66})}{=} E_2\mathfrak {X}^{[3]} - (q^{2} - q^{-2})(q^4c_1)(q^2c_2)\mathfrak {X}_{[3]}T_{21}(E_2)T_2(E_1), \end{aligned}$$

(A.74)

$$\begin{aligned} \mathfrak {X}_{[4]}E_2&\overset{(\mathrm{A.64})}{=} E_2\mathfrak {X}^{[4]} - q^{-2}(q^{2} - q^{-2})(q^4c_1)\mathfrak {X}_{[4]}E_1T_{21}(E_2). \end{aligned}$$

(A.75)

Using Eqs. (A.73), (A.74) and (A.75), and comparing with Eqs. (A.70) and (A.71) we can rewrite the term $\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]})$ as

$$\begin{aligned}&\mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}{}_{2}r(\mathfrak {X}^{[1]}) \\&\overset{(\mathrm{A.72})}{=} (q-q^{-1})(q^2c_2) \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}\mathfrak {X}_{[2]}E_2\mathfrak {X}^{[1]}\\&\quad = (q-q^{-1})(q^2c_2) E_2\mathfrak {X}_{w_0^{\prime }} - \mathfrak {X}_{[4]}{}_{2}r(\mathfrak {X}^{[3]})\mathfrak {X}^{[2]}\mathfrak {X}^{[1]} - \mathfrak {X}_{[4]}\mathfrak {X}_{[3]}{}_{2}r(\mathfrak {X}^{[2]})\mathfrak {X}^{[1]}. \end{aligned}$$

It hence follows from (A.69) that ${}_{2}r(\mathfrak {X}_{w_0^{\prime }}) = (q-q^{-1})(q^2c_2)E_2\mathfrak {X}_{w_0^{\prime }}$ as required. $\square$

1.6 Type $G_2$

Consider the Satake diagram of type $G_2$.

Here the subgroup $\widetilde{W}$ coincides with W and the longest element of the Weyl group has two reduced expressions given by

$$\begin{aligned} w_0 = \sigma _1\sigma _2\sigma _1\sigma _2\sigma _1\sigma _2, \quad w_0' = \sigma _2\sigma _1\sigma _2\sigma _1\sigma _2\sigma _1. \end{aligned}$$

By Lemma 3.7 we have

$$\begin{aligned} \mathfrak {X}_1&= \sum _{n \ge 0} \dfrac{(q-q^{-1})^n}{\{2n\}_1!!} (q^2c_1)^n E_1^{2n},\\ \mathfrak {X}_2&= \sum _{n \ge 0} \dfrac{(q^3-q^{-3})^n}{\{2n\}_2!!} (q^6c_2)^n E_2^{2n}. \end{aligned}$$

The following proposition is proved in [6, 6.6].

Proposition A.25

The partial quasi K-matrices $\mathfrak {X}_{w_0}$ and $\mathfrak {X}_{w_0'}$ coincide with the quasi K-matrix $\mathfrak {X}$.