# Linkage principle for small quantum groups

Cristian Vay
Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional
de Córdoba. CIEM–CONICET. Medina Allende s/n, Ciudad Universitaria 5000
Córdoba, Argentina<EMAIL_ADDRESS>
###### Abstract.

We consider small quantum groups with root systems of Cartan, super and
modular type, among others. These are constructed as Drinfeld doubles of
finite-dimensional Nichols algebras of diagonal type. We prove a linkage
principle for them by adapting techniques from the work of Andersen, Jantzen
and Soergel in the context of small quantum groups at roots of unity.
Consequently we characterize the blocks of the category of modules. We also
find a notion of (a)typicality similar to the one in the representation theory
of Lie superalgebras. The typical simple modules turn out to be the simple and
projective Verma modules. Moreover, we deduce a character formula for
$1$-atypical simple modules.

_2020 Mathematics Subject Classification._ 16T05, 17B37, 22E47, 17B10, 17B35,
20G05.
_Key words._ Quantum group, Nichols algebra, Weyl groupoid, Root system,
Representation theory

This work is partially supported by CONICET PIP 11220200102916CO, Foncyt PICT
2020-SERIEA-02847 and Secyt (UNC)

##  1\. Introduction

The small quantum groups at roots of unity introduced by Lusztig are beautiful
examples of finite-dimensional Hopf algebras with relevant applications in
several subjects. They are instrumental in Lusztig’s program concerning the
representation theory of algebraic groups in positive characteristic [33, 34,
35, 36]. In this framework, he conjectured a character formula for their
simple modules, and that this formula implies an analogous one for restricted
Lie algebras in positive characteristic. The validity of the “quantum formula”
was confirmed by the works of Kazhdan–Lusztig [30, 31] and Kashiwara–Tanisaki
[29]. Andersen, Jantzen and Soergel [1] then proved that the “quantum formula”
implies the “modular formula” for large enough characteristics (although, we
have to mention that 20 years later Williamson [40] showed that this “enough”
could be very big). More recently, Fiebig [19] gave a new proof for the
“quantum formula” building up on certain categories defined in [1] and, joint
to Lanini, they have carried out a further development on these categories,
cf. [20].

Small quantum group at a root of unity$u_{q}(\mathfrak{g})$Restricted
enveloping algebra$U^{[p]}(\mathfrak{g}_{k})$ Figure 1. Algebras involved in
Lusztig’s conjectures. Given a finite root system $\Delta$, we write
$\mathfrak{g}$ and $\mathfrak{g}_{k}$ for the associated Lie algebras over
$\mathbb{C}$ and over an algebraically closed field $k$ of characteristic $p$
odd ($\neq 3$ if $\mathfrak{g}$ has a component of type $G_{2}$). We write $q$
for a $p$-th root of unity in $\mathbb{C}$.

Here, we adapt tools and techniques from [1] to investigate the representation
theory of a more general class of small quantum groups: the Drinfeld doubles
of bosonizations of finite-dimensional Nichols algebras of diagonal type over
abelian groups, see Figure 2. The interest in Nichols algebras have been
continually increasing since Andruskiewitsch and Schneider assigned them a
central role in the classification of pointed Hopf algebras [10]. Notably, the
classification and the structure of the Nichols algebras of diagonal type are
governed by Lie-type objects introduced by Heckenberger [21], see also [25, 9,
24]. In particular, they have associated (generalized) root systems and,
besides those of Cartan type, we find among the examples root systems of
finite-dimensional contragredient Lie superalgebras in characteristic $0$, $3$
and $5$, and root systems of finite-dimensional contragredient Lie algebras in
positive characteristic, as it was observed by Andruskiewitsch, Angiono and
Yamane [8, 3]. Consequently, we get small quantum groups with these more
general root systems.

$\mathfrak{q}\in\mathbb{C}^{\theta\times\theta}$$\mathfrak{B}_{\mathfrak{q}}$ptNichols
algebra$\mathfrak{B}_{\mathfrak{q}}\\#\mathbb{C}\Gamma$ptBosonization$\mathcal{D}(\mathfrak{B}_{\mathfrak{q}}\\#\mathbb{C}\Gamma)$ptDrinfeld
double$u_{\mathfrak{q}}$$\mathcal{D}(\mathfrak{B}_{\mathfrak{q}}\\#\mathbb{C}\Gamma)$ptSmall
quantum group Figure 2. We construct small quantum groups from matrices with
finite-dimensional Nichols algebras of diagonal type; $\Gamma$ is a group
quotient of $\mathbb{Z}^{\theta}$. For instance, the positive part of
$u_{q}(\mathfrak{g})$ is the Nichols algebra of
$\mathfrak{q}=(q^{d_{i}c_{ij}})_{i,j}$ where $C=(c_{ij})_{i,j}$ is the Cartan
matrix of $\mathfrak{g}$ and $(d_{i}c_{ij})_{i,j}$ is symmetric. Thus, the
corresponding bosonization is the Borel subalgebra and $u_{q}(\mathfrak{g})$
is a quotient of $u_{\mathfrak{q}}$ by a central group subalgebra.

###  1.1. Main results

Let $u_{\mathfrak{q}}$ be as in Figure 2. Then it admits a triangular
decomposition
$u_{\mathfrak{q}}=u_{\mathfrak{q}}^{-}u_{\mathfrak{q}}^{0}u_{\mathfrak{q}}^{+}$
which gives rise to Verma type modules $M(\pi)$ for every algebra map
$\pi:u_{\mathfrak{q}}^{0}\longrightarrow\mathbb{C}$. Moreover, every $M(\pi)$
has a unique simple quotient $L(\pi)$ and any simple $u_{\mathfrak{q}}$-module
can be obtained in this way. For instance, all these were computed for
$\mathfrak{q}$ of type $\mathfrak{ufo}(7)$ in [5]. The linkage principle gives
us information about the composition factors of the Verma modules as we
explain after introducing some notation.

Let $\\{\alpha_{1},...,\alpha_{\theta}\\}$ be the canonical $\mathbb{Z}$-basis
of $\mathbb{Z}^{\mathbb{I}}$ with $\mathbb{I}=\\{1,...,\theta\\}$. The matrix
$\mathfrak{q}$ defines a bicharacter
$\mathbb{Z}^{\mathbb{I}}\times\mathbb{Z}^{\mathbb{I}}\longrightarrow\mathbb{C}^{\times}$
which we denote also $\mathfrak{q}$. Given $\beta\in\mathbb{Z}^{\mathbb{I}}$,
we set $q_{\beta}=\mathfrak{q}(\beta,\beta)$ and
$b^{\mathfrak{q}}(\beta)=\operatorname{{\mathrm{ord}}}q_{\beta}$. We denote
$\rho^{\mathfrak{q}}:\mathbb{Z}^{\mathbb{I}}\longrightarrow\mathbb{C}^{\times}$
the group homomorphism such that
$\rho^{\mathfrak{q}}(\alpha_{i})=q_{\alpha_{i}}$ for all $i\in\mathbb{I}$. We
notice $u_{\mathfrak{q}}^{0}$ is an abelian group algebra generated by
$K_{i},L_{i}$, $i\in\mathbb{I}$ (in Figure 2, $\Gamma$ is generated by the
$K_{i}$’s) and unlike $u_{q}(\mathfrak{g})$, they yield two copies of the
(finite) torus. If
$\alpha=n_{1}\alpha_{1}+\cdots+n_{\theta}\alpha_{\theta}\in\mathbb{Z}^{\mathbb{I}}$,
we denote $K_{\alpha}=K_{1}^{n_{1}}\cdots K_{\theta}^{n_{\theta}}$ and
$L_{\alpha}=L_{1}^{n_{1}}\cdots L_{\theta}^{n_{\theta}}$. The algebra map
$\pi\widetilde{\mu}:u_{\mathfrak{q}}^{0}\longrightarrow\mathbb{C}$ is defined
by
$\pi\widetilde{\mu}(K_{\alpha}L_{\beta})=\frac{\mathfrak{q}(\alpha,\mu)}{\mathfrak{q}(\mu,\beta)}\pi(K_{\alpha}L_{\beta})$,
$\alpha,\beta\in\mathbb{Z}^{\mathbb{I}}$. Let
$\Delta^{\mathfrak{q}}_{+}\subset\mathbb{Z}_{\geq 0}^{\mathbb{I}}$ be the set
of positive roots of the Nichols algebra $\mathfrak{B}_{\mathfrak{q}}$. If
$\beta\in\Delta^{\mathfrak{q}}_{+}$, we define

$\displaystyle\beta\downarrow\mu=\mu-n^{\pi}_{\beta}(\mu)\beta$

where $n^{\pi}_{\beta}(\mu)$ is the unique
$n\in\\{1,...,b^{\mathfrak{q}}(\beta)-1\\}$ such that
$q_{\beta}^{n}-\rho^{\mathfrak{q}}(\beta)\,\pi\widetilde{\mu}(K_{\beta}L_{\beta}^{-1})=0$,
if it exists, and otherwise $n^{\pi}_{\beta}(\mu)=0$.

######  Theorem 1.1 (Strong linkage principle).

Let $L$ be a composition factor of $M(\pi\widetilde{\mu})$,
$\mu\in\mathbb{Z}^{\mathbb{I}}$. Then $L\simeq L(\pi\widetilde{\mu})$ or
$L\simeq L(\pi\widetilde{\lambda})$ with
$\lambda=\beta_{r}\downarrow\cdots\beta_{1}\downarrow\mu$ for some
$\beta_{1},...,\beta_{r}\in\Delta_{+}^{\mathfrak{q}}$.

A first consequence of this principle is that it defines an equivalence
relation which completely characterizes the blocks of the category. It is also
the starting point to imagine character formulas for the simple modules. In
this direction, we deduce that $M(\pi\widetilde{\mu})=L(\pi\widetilde{\mu})$
is simple if and only if

$\displaystyle\prod_{\begin{subarray}{c}\beta\in\Delta_{+}^{\mathfrak{q}}\\\
1\leq
t<b^{\mathfrak{q}}(\beta)\end{subarray}}\left(q_{\beta}^{t}-\rho^{\mathfrak{q}}(\beta)\,\pi\widetilde{\mu}(K_{\beta}L_{\beta}^{-1})\right)\neq
0;$

this was also proved in [26, Proposition 5.16]. Moreover, we give a character
formula for $L(\pi\widetilde{\mu})$ if the above expression is zero for a
unique $\beta\in\Delta_{+}^{\mathfrak{q}}$ and for $t=n_{\beta}^{\pi}(\mu)$.
Explicitly,

$\displaystyle{\mathrm{ch}}L(\pi\widetilde{\mu})=\quad\frac{1-e^{-n_{\beta}^{\pi}(\mu)\beta}}{1-e^{-\beta}}\prod_{\gamma\in\Delta_{+}^{\mathfrak{q}}\setminus\\{\beta\\}}\frac{1-e^{-b^{\mathfrak{q}}(\gamma)\gamma}}{1-e^{-\gamma}}.$

By analogy with the theory of Lie superalgebras [27, 39], we say that the
number of zeros of the former product measures the degree of atypicality. Like
in [41] where Yamane gives a Weyl–Kac character formula for typical simple
modules over quantum groups of Nichols algebras of diagonal type with finite
root systems (when the Nichols algebra is finite-dimensional, his typical
modules coincides with ours).

When I discussed a preliminary version of Theorem 1.1 with Nicolás
Andruskiewitsch and Simon Riche, they respectively asked me about the Weyl
groupoid and the affine Weyl group, how these objects come into play. In fact,
the linkage principle is usually encoded in the dot action of the affine Weyl
group, and the Weyl groupoid is its replacement in the theory of Nichols
algebras [21]. We partially answer their questions in the next corollary. We
discover also a phenomenon similar to what occurs in the setting of Lie
superalgebras. There, the action of the affine Weyl group generated by the
even reflections as well as the translations by odd roots take part in the
linkage principle, see e. g. [17].

Let $\Delta_{+,\operatorname{car}}^{\mathfrak{q}}$ be the set of positive
Cartan roots of $\mathfrak{q}$ and $s_{\beta}$ the associated reflection. We
set
$\varrho^{\mathfrak{q}}=\frac{1}{2}\sum_{\beta\in\Delta^{\mathfrak{q}}_{+}}(b^{\mathfrak{q}}(\beta)-1)\beta$.
For $\mu\in\mathbb{Z}^{\mathbb{I}}$,
$\beta\in\Delta_{+,\operatorname{car}}^{\mathfrak{q}}$ and $m\in\mathbb{Z}$,
we define

$\displaystyle
s_{\beta,m}\bullet\mu=s_{\beta}(\mu+mb^{\mathfrak{q}}(\beta)\beta-\varrho^{\mathfrak{q}})+\varrho^{\mathfrak{q}}.$

We denote $\mathcal{W}_{\operatorname{link}}^{\mathfrak{q}}$ the group
generated by all the affine reflections $s_{\beta,m}$. We recall that
$\mathfrak{q}$ is of Cartan type if
$\Delta_{+}^{\mathfrak{q}}=\Delta_{+,\operatorname{car}}^{\mathfrak{q}}$, and
$\mathfrak{q}$ is of super type if its root system is isomorphic to the root
system of a finite-dimensional contragredient Lie superalgebra in
characteristic 0. If $\mathfrak{q}$ is of super type, then
$\Delta_{+,\operatorname{odd}}^{\mathfrak{q}}:=\Delta^{\mathfrak{q}}_{+}\setminus\Delta_{+,\operatorname{car}}^{\mathfrak{q}}$
is not empty and $\operatorname{{\mathrm{ord}}}q_{\beta}=2$ for every root
$\beta\in\Delta_{+,\operatorname{odd}}^{\mathfrak{q}}$; in this case
$\beta\downarrow\mu=\mu$ or $\mu-\beta$.

######  Corollary 1.2 (Linkage principle).

Assume $\pi$ is the trivial algebra map. Let $L(\pi\widetilde{\lambda})$ be a
composition factor of $M(\pi\widetilde{\mu})$. Then

  1. (1)

$\lambda\in\mathcal{W}_{\operatorname{link}}^{\mathfrak{q}}\bullet\mu$ if
$\mathfrak{q}$ is of Cartan type.

  2. (2)

$\lambda\in\mathcal{W}^{\mathfrak{q}}_{\operatorname{link}}\bullet(\mu+\mathbb{Z}\Delta_{+,\operatorname{odd}}^{\mathfrak{q}})$
if $\mathfrak{q}$ is of super type.

This corollary holds more generally for matrices of standard type, those with
constant bundle of root system. However, there are matrices which are not, as
the matrices of modular type. The assumption on $\pi$ is not particularly
restrictive as we deal with all the simple modules of highest-weight
$\widetilde{\mu}:u_{\mathfrak{q}}^{0}\longrightarrow\mathbb{C}$,
$K_{\alpha}L_{\beta}\mapsto\frac{\mathfrak{q}(\alpha,\mu)}{\mathfrak{q}(\mu,\beta)}$,
for all $\mu\in\mathbb{Z}^{\mathbb{I}}$. In the case of $u_{q}(\mathfrak{q})$,
these form the category of modules of type $1$ in the sense of Lusztig, cf.
[1, §2.4].

###  1.2. Sketch of the proof

As we mentioned at the beginning, we imitate the ideas of [1, §1-§7]. There,
the authors consider $\mathbb{Z}^{\mathbb{I}}$-graded algebras admitting a
triangular decomposition $U=U^{-}U^{0}U^{+}$ with $U^{0}$ commutative and
satisfying additional conditions which are fulfilled by $u_{q}(\mathfrak{g})$
and $U^{[p]}(\mathfrak{g}_{k})$. Then, given a Noetherian commutative algebra
$\mathbf{A}$ and an algebra map $\pi:U^{0}\longrightarrow\mathbf{A}$, they
define certain categories $\mathcal{C}_{\mathbf{A}}$ of
$\mathbb{Z}^{\mathbb{I}}$-graded $(U,\mathbf{A})$-bimodules. We observe here
that we can consider these categories also for $u_{\mathfrak{q}}$. Roughly
speaking, in the case $\mathbf{A}=\mathbb{C}$, this identifies with the
abelian subcategory generated by the simple modules $L(\pi\widetilde{\mu})$
for $\mu\in\mathbb{Z}^{\mathbb{I}}$.

A powerful tool used in loc. cit. to study the categories
$\mathcal{C}_{\mathbf{A}}$ are the so-called Lusztig automorphisms $T_{w}$ of
$u_{q}(\mathfrak{g})$, where $w$ runs in the Weyl group of $\mathfrak{g}$. We
find a difference between $u_{q}(\mathfrak{g})$ and $u_{\mathfrak{q}}$ at this
point. Indeed, we have Lusztig isomorphisms but connecting possibly different
algebras, i.e. $T_{w}:u_{w^{-*}\mathfrak{q}}\longrightarrow u_{\mathfrak{q}}$
and the matrices $w^{-*}\mathfrak{q}$ and $\mathfrak{q}$ are not necessarily
equal. These isomorphisms were defined in [23] for each
$w\in{}^{\mathfrak{q}}\mathcal{W}$, the Weyl groupoid of $\mathfrak{q}$ [21].
Nevertheless, we can carefully use them as in [1].

First, we produce different triangular decompositions on $u_{\mathfrak{q}}$
and then each triangular decomposition gives rise to new Verma modules.
Namely, for $w\in{}^{\mathfrak{q}}\mathcal{W}$ and
$\mu\in\mathbb{Z}^{\mathbb{I}}$, we denote $Z_{\mathbb{C}}^{w}(\mu)$ the Verma
module induced by $\pi\widetilde{\mu}$ using the triangular decomposition
$u_{\mathfrak{q}}=T_{w}(u_{w^{-*}\mathfrak{q}}^{-})\,u_{\mathfrak{q}}^{0}\,T_{w}(u_{w^{-*}\mathfrak{q}}^{+})$.
Let $L_{\mathbb{C}}^{w}(\mu)$ denote the unique simple quotient of
$Z_{\mathbb{C}}^{w}(\mu)$. For instance,
$Z_{\mathbb{C}}(\mu):=Z_{\mathbb{C}}^{\mathrm{id}}(\mu)=M(\pi\widetilde{\mu})$
and
$L_{\mathbb{C}}(\mu):=L_{\mathbb{C}}^{\mathrm{id}}(\mu)=L(\pi\widetilde{\mu})$.

Now, we set $\mu\langle
w\rangle=\mu+w(\varrho^{w^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}}\in\mathbb{Z}^{\mathbb{I}}$;
we notice that $\varrho^{w^{-*}\mathfrak{q}}$ and $\varrho^{\mathfrak{q}}$
could be different. We show that the Verma modules
$Z^{w}_{\mathbb{C}}(\mu\langle w\rangle)$ and $Z^{x}_{\mathbb{C}}(\mu\langle
x\rangle)$ have identical characters and the Hom-space between them is one-
dimensional, for all $w,x\in{}^{\mathfrak{q}}\mathcal{W}$. Moreover, we
construct inductively a generator for each space and compute its kernel. We
also prove that the image of $\Phi:Z_{\mathbb{C}}(\mu)\longrightarrow
Z^{w_{0}}_{\mathbb{C}}(\mu\langle w_{0}\rangle)$ is isomorphic to
$L_{\mathbb{C}}(\mu)$, cf. Figure 3.

$Z_{\mathbb{C}}(\mu)$$Z^{w}_{\mathbb{C}}(\mu\langle
w\rangle)$$Z^{w\sigma_{i_{s}}}_{\mathbb{C}}(\mu\langle
w\sigma_{i_{s}}\rangle)$$Z^{w_{0}}_{\mathbb{C}}(\mu\langle
w_{0}\rangle)$$Z^{w_{s}}_{\mathbb{C}}(\beta\downarrow\mu\langle
w\rangle)$$Z_{\mathbb{C}}(\beta\downarrow\mu)$$\Phi$$\circlearrowleft$$\varphi$$\psi$
Figure 3. Here $w_{0}=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{n}}$ is a
reduced expression of the longest element in ${}^{\mathfrak{q}}\mathcal{W}$,
$w=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{s-1}}$ and
$\beta=w\alpha_{i_{s}}\in\Delta_{+}^{\mathfrak{q}}$. The horizontal maps are
generators of the corresponding Hom-spaces and
${\mathrm{Im}}\psi=\operatorname{{\mathrm{Ker}}}\varphi$.

We are ready to outline the last step of the proof of Theorem 1.1. Let
$L_{\mathbb{C}}(\lambda)$ be a composition factor of $Z_{\mathbb{C}}(\mu)$ not
isomorphic to $L_{\mathbb{C}}(\mu)$. Then $L_{\mathbb{C}}(\lambda)$ is a
composition factor of $\operatorname{{\mathrm{Ker}}}\Phi$ and hence also of
$\operatorname{{\mathrm{Ker}}}\varphi={\mathrm{Im}}\psi$, cf. Figure 3 (for
some $s$). Since the Verma modules in the same row of Figure 3 have identical
characters, we conclude that $L_{\mathbb{C}}(\lambda)$ is a composition factor
of $Z_{\mathbb{C}}(\beta\downarrow\mu)$ as well. We observe that
$-\beta^{\mathfrak{q}}_{top}=\sum_{\beta\in\Delta^{\mathfrak{q}}_{+}}(b^{\mathfrak{q}}(\beta)-1)\beta\leq\lambda\leq\beta\downarrow\mu<\mu$
because $\beta^{\mathfrak{q}}_{top}$ is the maximum
$\mathbb{Z}^{\mathbb{I}}$-degree of $\mathfrak{B}_{\mathfrak{q}}$. Therefore,
by repeating this procedure, we will find
$\beta_{1},...,\beta_{r}\in\Delta_{+}^{\mathfrak{q}}$ such that
$\lambda=\beta_{r}\downarrow\cdots\beta_{1}\downarrow\mu$ as desired.

###  1.3. Relations with other algebras in the literature

We would like to remark that the representations of the present small quantum
groups can be helpful for other algebras. For instance, this was pointed out
by Andruskiewitsch, Angiono and Yakimov [7, §1.3.3] for the large quantum
groups studied in [11, 13, 7] which are analogous to the quantized enveloping
algebras of De Concini–Kac–Procesi. The small quantum groups are particular
quotients of them. As in [7], we highlight that our results apply to small
quantum groups at even roots of unity.

Another example is given by the braided Drinfeld doubles of Nichols algebras
recently constructed by Laugwitz and Sanmarco [32]. These are quotients of
$u_{\mathfrak{q}}$ with only one copy of the finite torus. Since the
corresponding projections preserve the triangular decompositions [32,
Proposition 3.16], a linkage principle for them can be deduced from Theorem
1.1.

Finally, it is worth noting that Pan and Shu [37] have obtained results
comparable to ours, as well as their proofs, for modular Lie superalgebras.
This could suggest a relationship between small quantum groups of super type
and the corresponding modular Lie superalgebras as it happens between
$u_{q}(\mathfrak{g})$ and $U^{[p]}(\mathfrak{g}_{k})$.

###  1.4. Organization

The exposition is mostly self-contained. In Section 2 we set up general
conventions. In Section 3 we collect the main concepts regarding Nichols
algebras (PBW basis, Weyl groupoid, root system) and their properties. We
illustrate them in super type $A(1|1)$. In Section 4 we recall the
construction of the Drinfeld doubles and their Lusztig isomorphisms. In
Section 5 we introduce the categories defined in [1] and summarize their
general features. Next we investigate these categories over the Drinfeld
doubles of Section 4 using their Lusztig isomorphisms: in Section 6, we
construct the different Verma modules mentioned previously in this
introduction, and we study the morphisms between them in Section 7. Finally,
we prove our main results in Section 8.

###  1.5. Acknowledgments

I am very grateful to Nicolas Andruskiewitsch, for motivating me to study the
representations of Hopf algebras since my PhD, to Ivan Angiono, for patiently
answering all my questions about Nichols algebras of diagonal type, and to
Simon Riche, for teaching me so much about representation theory. I also thank
them for the fruitful discussions.

##  2\. Conventions

Throughout our work $\Bbbk$ denotes an algebraically closed field of
characteristic zero. For $q\in\Bbbk^{\times}$ and $n\in\mathbb{N}$, we recall
the quantum numbers

$\displaystyle(n)_{q}=\sum_{j=0}^{n-1}q^{j}\quad\mbox{and the
identity}\quad(n)_{q}=q^{n-1}(n)_{q^{-1}}.$

Let $\theta\in\mathbb{N}$. We set
$\mathbb{I}=\mathbb{I}_{\theta}=\\{1,2,\dots,\theta\\}$. We denote
$\Pi=\\{\alpha_{1},\dots,\alpha_{\theta}\\}$ the canonical $\mathbb{Z}$-basis
of $\mathbb{Z}^{\mathbb{I}}$. We will write
$0=0\alpha_{1}+\cdots+0\alpha_{\theta}$. We will use $\Pi$ to identify the
matrices of size $\theta\times\theta$ and the bicharacters on
$\mathbb{Z}^{\mathbb{I}}$ with values in $\Bbbk^{\times}$. Explicitly, given
$\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}}\in(\Bbbk^{\times})^{\mathbb{I}\times\mathbb{I}}$,
by abuse of notation, we will denote
$\mathfrak{q}:\mathbb{Z}^{\mathbb{I}}\times\mathbb{Z}^{\mathbb{I}}\longrightarrow\Bbbk^{\times}$
the bicharacter defined by

$\displaystyle\mathfrak{q}(\alpha_{i},\alpha_{j})=q_{ij}\quad\forall\,i,j\in\mathbb{I}.$

Let $\beta\in\mathbb{Z}^{\mathbb{I}}$. We will write
$q_{\beta}=\mathfrak{q}(\beta,\beta)$. In particular, $q_{\alpha_{i}}=q_{ii}$
for all $i\in\mathbb{I}$. The bound function [26, $(2.12)$] is

(2.1) $\displaystyle
b^{\mathfrak{q}}(\beta)=\begin{cases}\min\\{m\in\mathbb{N}\mid(m)_{q_{\beta}}=0\\}&\mbox{if
$(m)_{q_{\beta}}=0$ for some $m\in\mathbb{N}$},\\\
\infty&\mbox{otherwise}.\end{cases}$

Of course, $b^{\mathfrak{q}}(\beta)$ is finite if and only if $q_{\beta}$ is a
primitive root of $1$ of order $b^{\mathfrak{q}}(\beta)$.

We will consider the dual action of
$\operatorname{Aut}_{\mathbb{Z}}(\mathbb{Z}^{\mathbb{I}})$ on bicharacters.
That is, if $w\in\operatorname{Aut}_{\mathbb{Z}}(\mathbb{Z}^{\mathbb{I}})$,
then the bicharacter
$w^{*}{\mathfrak{q}}:\mathbb{Z}^{\mathbb{I}}\times\mathbb{Z}^{\mathbb{I}}\longrightarrow\Bbbk^{\times}$
is defined by

$\displaystyle
w^{*}{\mathfrak{q}}(\alpha,\beta)={\mathfrak{q}}(w^{-1}(\alpha),w^{-1}(\beta))\quad\forall\alpha,\beta\in\mathbb{Z}^{\mathbb{I}}.$

The partial order $\leq^{w}$ in $\mathbb{Z}^{\mathbb{I}}$ is defined by
$\lambda\leq^{w}\mu$ if and only if $w^{-1}(\mu-\lambda)\in\mathbb{Z}_{\geq
0}^{\mathbb{I}}$. When $w={\mathrm{id}}$, we simply write $\leq$.

Let $M=\oplus_{\nu\in\mathbb{Z}^{\mathbb{I}}}M_{\nu}$ be a
$\mathbb{Z}^{\mathbb{I}}$-graded module over a ring $\mathbf{A}$. We call
weight space to the homogeneous component $M_{\nu}$. The support $\sup(M)$ is
the set of all $\nu\in\mathbb{Z}^{\mathbb{I}}$ such that $M_{\nu}\neq 0$. If
$w\in\operatorname{Aut}_{\mathbb{Z}}(\mathbb{Z}^{\mathbb{I}})$, the
$w$-twisted $M[w]$ is the $\mathbb{Z}^{\mathbb{I}}$-graded $\mathbf{A}$-module
whose weight spaces are $M[w]_{\nu}=M_{w^{-1}\nu}$ for all
$\nu\in\mathbb{Z}^{\mathbb{I}}$. This an endofunctors in the category of
$\mathbb{Z}^{\mathbb{I}}$-graded $\mathbf{A}$-modules. If the weight spaces
are $\mathbf{A}$-free, the formal character of $M$ is

$\displaystyle{\mathrm{ch}}M=\sum_{\mu\in\mathbb{Z}^{\mathbb{I}}}{\mathrm{rank}}_{\mathbf{A}}(M_{\mu})\,e^{\mu}$

which belongs to the $\mathbb{Z}$-algebra generated by the symbols $e^{\mu}$
with multiplication $e^{\mu}\cdot e^{\nu}=e^{\mu+\nu}$. We denote
$\overline{(-)}$ and $w(-)$ the automorphisms of it given by
$\overline{e^{\mu}}=e^{-\mu}$ and $w(e^{\mu})=e^{w\mu}$ for all
$\mu\in\mathbb{Z}^{\mathbb{I}}$ and
$w\in\operatorname{Aut}_{\mathbb{Z}}(\mathbb{Z}^{\mathbb{I}})$. Therefore

(2.2) $\displaystyle{\mathrm{ch}}(M[w])=w({\mathrm{ch}}\,M).$

##  3\. Finite-dimensional Nichols algebras of diagonal type

The finite-dimensional Nichols algebras of diagonal type were classified by
Heckenberger in [22]111Indeed, he classifies a more large family but we only
consider here finite-dimensional Nichols algebras.. They are parameterized by
matrices of scalars. The classification and the structure of them are governed
by certain Lie type objects, such as, PBW basis, the generalized Cartan
matrix, the Weyl groupoid and the generalized root system, among others. We
recall here their main features needed for our work. We refer to [3] for an
overview of the theory.

We fix $\theta\in\mathbb{N}$ and set $\mathbb{I}=\mathbb{I}_{\theta}$.
Throughout this section
$\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}}\in(\Bbbk^{\times})^{\mathbb{I}\times\mathbb{I}}$
denotes a matrix with finite-dimensional Nichols algebra
$\mathfrak{B}_{\mathfrak{q}}$. This is constructed as a quotient of the free
$\Bbbk$-algebra generated by $E_{1},...,E_{\theta}$, that is

$\displaystyle\mathfrak{B}_{\mathfrak{q}}=\Bbbk\langle
E_{1},...,E_{\theta}\rangle/{\mathcal{J}}_{\mathfrak{q}}$

for certain ideal ${\mathcal{J}}_{\mathfrak{q}}$. It is a
$\mathbb{Z}^{\mathbb{I}}$-graded algebra with

$\displaystyle\deg E_{i}=\alpha_{i}\quad\forall i\in\mathbb{I}$

and $\mathbb{Z}$-graded with $\deg E_{i}=1$ for all $i\in\mathbb{I}$.
Moreover, $\mathfrak{B}_{\mathfrak{q}}$ is a braided Hopf algebra and a
minimal set of generators of ${\mathcal{J}}_{\mathfrak{q}}$ is known [11, 12]
but we do not need more details.

######  Example 3.1.

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over
$\mathbb{C}$ with Cartan matrix $C=(c_{ij})_{i,j\in\mathbb{I}}$ and
$D=(d_{i})_{i\in\mathbb{I}}$ such that $DC$ is symmetric. Let $q$ be a
primitive root of unity and
$\mathfrak{q}=(q^{d_{i}c_{ij}})_{i,j\in\mathbb{I}}$. Then
$\mathfrak{B}_{\mathfrak{q}}$ is finite-dimensional. Moreover, if
$\operatorname{{\mathrm{ord}}}q$ is an odd prime, not $3$ if $\mathfrak{g}$
has a component of type $G_{2}$, then $\mathfrak{B}_{\mathfrak{q}}$ is the
positive part of the small quantum group $u_{q}(\mathfrak{g})$, one of the
algebras analyzed by Andersen–Jantzen–Soergel [1, Section 1.3. Case 2].

######  Example 3.2.

Let $q\in\Bbbk$ be a primitive root of order $N>2$ and

$\displaystyle\mathfrak{q}=\begin{pmatrix}-1&-q\\\ -1&-1\end{pmatrix}.$

Set $E_{12}=\operatorname{ad}_{c}E_{1}(E_{2})=(E_{1}E_{2}-q_{12}E_{2}E_{1})$.
Then

$\displaystyle\mathfrak{B}_{\mathfrak{q}}=\langle E_{1},E_{2}\mid
E_{1}^{2}=E_{2}^{2}=E_{12}^{N}=0\rangle.$

This is a Nichols algebra of super type $A(1|1)$. This term refers to the root
system which will be introduced in §3.3; see Example 3.5. Through this section
we will use this example to illustrate the different concept. See also [3,
§5.1.11].

###  3.1. PBW basis

In [28], Karchencko proves the existence of homogeneous elements
$E_{\beta_{1}},...,E_{\beta_{n}}\in\mathfrak{B}_{\mathfrak{q}}$ with $\deg
E_{\beta_{\nu}}=\beta_{\nu}\in\mathbb{Z}_{\geq 0}^{\mathbb{I}}$ and
$b^{\mathfrak{q}}(\beta_{\nu})<\infty$, recall (2.1), such that

$\displaystyle\left\\{E_{\beta_{1}}^{m_{1}}\cdots E_{\beta_{n}}^{m_{n}}\mid
0\leq m_{\nu}<b^{\mathfrak{q}}(\beta_{\nu}),\,1\leq\nu\leq n\right\\}$

is a linear basis of $\mathfrak{B}_{\mathfrak{q}}$. We will see in the next
section a way of constructing these elements $E_{\beta_{\nu}}$. The set of
positive roots of $\mathfrak{q}$ is

(3.1) $\displaystyle\Delta_{+}^{\mathfrak{q}}=\\{\beta_{1},...,\beta_{n}\\}.$

It turns out that the elements of $\Delta_{+}^{\mathfrak{q}}$ are pairwise
different [18, Proposition 2.12]. The set of (positive) simple roots is
$\Pi^{\mathfrak{q}}=\\{\alpha_{1},...,\alpha_{\theta}\\}$. We set

(3.2)
$\displaystyle\beta_{top}^{\mathfrak{q}}=\sum_{\beta\in\Delta^{\mathfrak{q}}_{+}}(b^{\mathfrak{q}}(\beta)-1)\beta.$

This is the weight of the homogeneous component of maximum
$\mathbb{Z}^{\mathbb{I}}$-degree of $\mathfrak{B}_{\mathfrak{q}}$.

######  Example 3.3.

The positive roots of $\mathfrak{q}$ in Example 3.2 are

$\displaystyle\Delta_{+}^{\mathfrak{q}}=\\{\alpha_{1},\alpha_{1}+\alpha_{2},\alpha_{2}\\}$

with

$\displaystyle b^{\mathfrak{q}}(\alpha_{1})=2,\quad
b^{\mathfrak{q}}(\alpha_{1}+\alpha_{2})=N,\quad
b^{\mathfrak{q}}(\alpha_{2})=2.$

The associated PBW basis is

$\displaystyle\\{E_{1}^{m_{1}}E_{12}^{m_{2}}E_{2}^{m_{3}}\mid 0\leq
m_{1},m_{3}<2,\,0\leq m_{2}<N\\}.$

###  3.2. Weyl groupoid

For distinct $i,j\in\mathbb{I}$, by [38] there exists $m\in\mathbb{N}_{0}$
such that

$\displaystyle(m+1)_{q_{ii}}(q_{ii}^{m}q_{ij}q_{ji}-1)=0.$

Thus, it is possible to define the generalized Cartan matrix
$C^{\mathfrak{q}}=(c_{ij}^{\mathfrak{q}})_{i,j\in\mathbb{I}}$ of
$\mathfrak{q}$ as

(3.3) $\displaystyle c_{ij}^{\mathfrak{q}}=\begin{cases}2,&\mbox{if $i=j$};\\\
-\min\\{m\in\mathbb{N}_{0}\mid(m+1)_{q_{ii}}(q_{ii}^{m}q_{ij}q_{ji}-1)=0\\},&\mbox{otherwise.}\end{cases}$

This matrix leads to the reflection
$\sigma_{i}^{\mathfrak{q}}\in\operatorname{Aut}_{\mathbb{Z}}(\mathbb{Z}^{\mathbb{I}})$,
$i\in\mathbb{I}$, given by

(3.4)
$\displaystyle\sigma_{i}^{\mathfrak{q}}(\alpha_{j})=\alpha_{j}-c_{ij}^{\mathfrak{q}}\alpha_{i}\quad\forall\,j\in\mathbb{I}.$

It turns out that the Nichols algebra of
$(\sigma_{i}^{\mathfrak{q}})^{*}\mathfrak{q}$ is also finite-dimensional for
all $i\in\mathbb{I}$. However,
$\mathfrak{B}_{(\sigma_{i}^{\mathfrak{q}})^{*}\mathfrak{q}}$ is not
necessarily isomorphic to $\mathfrak{B}_{\mathfrak{q}}$.

Let ${\mathcal{H}}$ be the family of matrices with finite-dimensional Nichols
algebras and $r_{i}:{\mathcal{H}}\longrightarrow{\mathcal{H}}$ the bijection
given by

(3.5) $\displaystyle
r_{i}(\mathfrak{q})=(\sigma_{i}^{{\mathfrak{q}}})^{*}{\mathfrak{q}}.$

for all $\mathfrak{q}\in{\mathcal{H}}$ and $i\in\mathbb{I}$. For instance,
$r_{p}(\mathfrak{q})(\alpha_{i},\alpha_{j})=q_{ij}q_{ip}^{-c_{pj}^{\mathfrak{q}}}q_{pj}^{-c_{pi}^{\mathfrak{q}}}q_{pp}^{c_{pi}^{\mathfrak{q}}c_{pj}^{\mathfrak{q}}}$
for all $p,i,j\in\mathbb{I}$ and hence

(3.6) $\displaystyle c_{ij}^{r_{i}(\mathfrak{q})}=c_{ij}^{\mathfrak{q}}.$

It is immediate that
$\sigma_{i}^{r_{i}(\mathfrak{q})}=\sigma_{i}^{\mathfrak{q}}$ and
$r_{i}^{2}(\mathfrak{q})=\mathfrak{q}$. We notice that

$\displaystyle
r_{i_{k}}(\cdots(r_{i_{1}}(\mathfrak{q})))=\left(\sigma_{i_{k}}^{r_{i_{k-1}}\cdots
r_{i_{1}}(\mathfrak{q})}\cdots\sigma_{i_{2}}^{r_{1}(\mathfrak{q})}\sigma_{i_{1}}^{\mathfrak{q}}\right)^{*}\mathfrak{q}.$

We denote $\mathcal{G}=\langle r_{i}\mid i\in\mathbb{I}\rangle$ which is a
subgroup of the group of bijections on ${\mathcal{H}}$.

Let $\mathcal{X}$ be a $\mathcal{G}$-orbit. The Weyl groupoid $\mathcal{W}$ of
$\mathcal{X}$ is the category whose objects are the matrices belong to
$\mathcal{X}$, and their morphisms are generated by the arrows
$\sigma_{i}^{{\mathfrak{q}}}:{\mathfrak{q}}\rightarrow r_{i}({\mathfrak{q}})$
for all $\mathfrak{q}\in\mathcal{X}$ and $i\in\mathbb{I}$. We denote
$1^{\mathfrak{q}}$ the identity of $\mathfrak{q}$ and set
$\mathcal{W}_{\mathfrak{q}}=\operatorname{Hom}_{\mathcal{W}}(\mathfrak{q},\mathfrak{q})$.
Thus, a morphism in $\mathcal{W}$ is of the form

$\displaystyle\sigma_{i_{k}}^{r_{i_{k-1}}\cdots
r_{i_{1}}(\mathfrak{q})}\cdots\sigma_{i_{2}}^{r_{1}(\mathfrak{q})}\sigma_{i_{1}}^{\mathfrak{q}}$
$\displaystyle:\mathfrak{q}\longrightarrow
r_{i_{k}}(\cdots(r_{i_{1}}(\mathfrak{q}))).$

To shorten notation, we observe that a morphism in $\mathcal{W}$ is determined
either by specifying the source,
$\sigma_{i_{k}}\cdots\sigma_{i_{1}}^{\mathfrak{q}}:\mathfrak{q}\longrightarrow
r_{i_{k}}\cdots r_{i_{1}}(\mathfrak{q})$, or by specifying the target,
$1^{\mathfrak{q}}\sigma_{i_{k}}\cdots\sigma_{i_{1}}:r_{i_{1}}\cdots
r_{i_{k}}(\mathfrak{q})\longrightarrow\mathfrak{q}$. For
$w=\sigma_{i_{k}}\cdots\sigma_{i_{1}}^{\mathfrak{q}}$, we will write

$\displaystyle
w^{*}\mathfrak{q}=r_{i_{k}}(\cdots(r_{i_{1}}(\mathfrak{q})))\quad\mbox{and}\quad
w^{-*}\mathfrak{q}=(w^{-1})^{*}\mathfrak{q}.$

We set ${}^{\mathfrak{q}}\mathcal{W}$, resp. $\mathcal{W}^{\mathfrak{q}}$, the
family of morphisms in $\mathcal{W}$ whose target, resp. source, is
$\mathfrak{q}$.

Clearly,
$(\sigma_{p}^{\mathfrak{q}})^{*}\mathfrak{q}(\sigma_{p}^{\mathfrak{q}}(\alpha),\sigma_{p}^{\mathfrak{q}}(\alpha))=\mathfrak{q}(\alpha,\alpha)$.
Hence, for all $w\in\mathcal{W}^{\mathfrak{q}}$ and
$\alpha\in\mathbb{Z}^{\mathbb{I}}$, it holds

(3.7) $\displaystyle b^{w^{*}\mathfrak{q}}(w\alpha)=b^{\mathfrak{q}}(\alpha).$

By [25, Theorem 1], the defining relations of the Weyl groupoid are of Coxeter
type:

$\displaystyle(\sigma_{i}\sigma_{i})1^{\mathfrak{q}}=1^{\mathfrak{q}}\quad\mbox{and}\quad(\sigma_{i}\sigma_{j})^{m_{ij}^{\mathfrak{q}}}1^{\mathfrak{q}}=1^{\mathfrak{q}}\quad\forall\mathfrak{q}\in\mathcal{X},\,\forall
i,j\in\mathbb{I},\,i\neq j;$

for certain exponents $m_{ij}^{\mathfrak{q}}$. Similar to Coxeter groups,
there exists a length function $\ell:\mathcal{W}\longrightarrow\mathbb{N}_{0}$
given by

$\displaystyle\ell(w)=\min\bigl{\\{}k\in\mathbb{N}_{0}\mid\exists
i_{1},\dots,i_{k}\in\mathbb{I},\mathfrak{q}\in\mathcal{X}:w=\sigma_{i_{k}}\cdots\sigma_{i_{1}}^{\mathfrak{q}}\bigr{\\}}$

for $w\in\mathcal{W}$. Also, for each $\mathfrak{q}\in\mathcal{X}$, there
exists a unique morphism $w_{0}\in{}^{\mathfrak{q}}\mathcal{W}$ of maximal
length. Let $w_{0}=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{n}}$ be a
reduced expression of the longest element in ${}^{\mathfrak{q}}\mathcal{W}$.
The set of positive roots can be constructed as follows

(3.8)
$\displaystyle\Delta_{+}^{\mathfrak{q}}=\biggl{\\{}\beta_{\nu}=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{\nu-1}}(\alpha_{i_{\nu}})\mid
1\leq\nu\leq n\biggr{\\}},$

see for instance [18, Proposition 2.12].

######  Example 3.4.

Let $\mathfrak{q}$ be as in Example 3.2. We set

$\displaystyle\mathfrak{p}=\left(\begin{array}[]{cc}-1&q^{-1}\\\
1&q\end{array}\right)\quad\mbox{and}\quad\mathfrak{r}=\left(\begin{array}[]{cc}q&q^{-1}\\\
1&-1\end{array}\right).$

Then
$\mathcal{X}=\\{\mathfrak{q},\mathfrak{p},\mathfrak{r},\mathfrak{q}^{t},\mathfrak{p}^{t},\mathfrak{r}^{t}\\}$
is a $\mathcal{G}$-orbit. Indeed, the generalized Cartan matrices of them are
all equal to

$\displaystyle\left(\begin{array}[]{rr}2&-1\\\ -1&2\end{array}\right)$

and the Weyl groupoid of $\mathcal{X}$ is

$\mathfrak{q}$$\mathfrak{p}$$\mathfrak{r}$$\mathfrak{q}^{t}$$\mathfrak{p}^{t}$$\mathfrak{r}^{t}$$\sigma_{2}^{\mathfrak{q}}$$\sigma_{2}^{\mathfrak{q}}$$\sigma_{1}^{\mathfrak{q}}$$\sigma_{1}^{\mathfrak{p}}$$\sigma_{2}^{\mathfrak{q}^{t}}$$\sigma_{2}^{\mathfrak{q}^{t}}$$\sigma_{1}^{\mathfrak{q}^{t}}$$\sigma_{1}^{\mathfrak{p}^{t}}$$\sigma_{2}^{\mathfrak{p}}$$\sigma_{2}^{\mathfrak{p}^{t}}$$\sigma_{1}^{\mathfrak{p}}$$\sigma_{1}^{\mathfrak{p}^{t}}$

The corresponding Nichols algebras are not necessarily isomorphic. For
instance,

$\displaystyle\mathfrak{B}_{\mathfrak{p}}$ $\displaystyle=\langle
E_{1},E_{2}\mid E_{1}^{2}=E_{2}^{N}=E_{221}^{2}=0\rangle\quad\mbox{with}\quad
E_{221}=(\operatorname{ad}_{c}E_{2})^{2}(E_{1}),$
$\displaystyle\mathfrak{B}_{\mathfrak{r}}$ $\displaystyle=\langle
E_{1},E_{2}\mid E_{1}^{N}=E_{2}^{2}=E_{112}^{2}=0\rangle\quad\mbox{with}\quad
E_{112}=(\operatorname{ad}_{c}E_{1})^{2}(E_{2}).$

###  3.3. Root systems

The bundle
${\mathcal{R}}=\\{\Delta^{\mathfrak{q}}\\}_{\mathfrak{q}\in\mathcal{X}}$ with

$\displaystyle\Delta^{\mathfrak{q}}=\Delta^{\mathfrak{q}}_{+}\cup-\Delta^{\mathfrak{q}}_{+}$

is the so-called (generalized) root system of $\mathcal{X}$ (or of
$\mathfrak{q}\in\mathcal{X}$)

We highlight that, unlike classical root systems, $\Delta^{\mathfrak{q}}$ and
$\Delta^{\mathfrak{p}}$ are not necessarily equals for distinct
$\mathfrak{q},\mathfrak{p}\in\mathcal{X}$. When it is needed, we will write
$\beta^{\mathfrak{q}}$ to emphasize that $\beta\in\mathbb{Z}^{\mathbb{I}}$
belongs to $\Delta^{\mathfrak{q}}$. However, they share other characteristics
with classical root systems which will be useful, cf. [18, 26]. In particular,
for all $i\in\mathbb{I}$, it holds that

(3.9)
$\displaystyle\sigma_{i}^{\mathfrak{q}}(\Delta_{+}^{\mathfrak{q}}\setminus\\{\alpha_{i}\\})=\Delta_{+}^{r_{i}(\mathfrak{q})}\setminus\\{\alpha_{i}\\},\quad\sigma_{i}^{\mathfrak{q}}(\alpha_{i})=-\alpha_{i}\quad\mbox{and}\quad
w(\Delta^{w^{-*}\mathfrak{q}})=\Delta^{\mathfrak{q}}.$

Also,
$\ell(w)=\left|w(\Delta_{+}^{\mathfrak{p}})\cap-\Delta_{+}^{\mathfrak{q}}\right|$
for any $w:\mathfrak{p}\rightarrow\mathfrak{q}$ [25, Lemma 8 $(iii)$]. This
implies that

(3.10) $\displaystyle
w_{0}(\beta_{top}^{w_{0}^{-*}\mathfrak{q}})=-\beta_{top}^{\mathfrak{q}}.$

######  Example 3.5.

The generalized root system of $\mathfrak{q}$ from Example 3.2 is constant:
$\Delta^{\mathfrak{z}}=\\{\pm\alpha_{1},\pm(\alpha_{1}+\alpha_{2}),\pm\alpha_{2}\\}$
for all $\mathfrak{z}\in\mathcal{X}$.

We have introduced here generalized root systems for Nichols algebras.
However, this is a combinatorial object appearing in other context. According
to [3, Theorem 2.34] contragredient Lie superalgebras have associated
generalized root systems. For instance, that associated to
$\mathfrak{sl}(1|1)$ is equal to ${\mathcal{R}}$.

###  3.4. A shift operation on weights

Recall the element $\beta_{top}^{\mathfrak{q}}$ in (3.2). We set

(3.11)
$\displaystyle\varrho^{\mathfrak{q}}=\frac{1}{2}\beta^{\mathfrak{q}}_{top}.$

This element will play the role of the semi-sum of the positive roots in Lie
theory. The following definition is analogous to [1, 4.7$(4)$].

######  Definition 3.6.

For $\mu\in\mathbb{Z}^{\mathbb{I}}$ and
$w:w^{-*}\mathfrak{q}\rightarrow\mathfrak{q}$ in $\mathcal{W}$, we set

$\displaystyle\mu\langle
w\rangle=\mu+w(\varrho^{w^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}}\in\mathbb{Z}^{\mathbb{I}}.$

We observe that

(3.12) $\displaystyle
w(\varrho^{w^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}}=-\sum_{\beta\in\Delta_{+}^{\mathfrak{q}}\,:\,w^{-1}\beta\in\Delta_{-}^{w^{-*}\mathfrak{q}}}(b^{\mathfrak{q}}(\beta)-1)\beta$

since $w(\Delta^{w^{-*}\mathfrak{q}})=\Delta^{\mathfrak{q}}$ and
$b^{w^{-*}\mathfrak{q}}(w^{-1}\beta)=b^{\mathfrak{q}}(\beta)$. For instance,
$\mu\langle w_{0}\rangle=\mu-\beta^{\mathfrak{q}}_{top}$. This operation
satisfies that

(3.13) $\displaystyle\mu\langle w\sigma_{i}\rangle=\mu\langle
w\rangle-(b^{\mathfrak{q}}(w\alpha_{i})-1)w\alpha_{i}$

where $w\sigma_{i}:\sigma_{i}^{*}w^{-*}\mathfrak{q}\rightarrow\mathfrak{q}$
and $i\in\mathbb{I}$. Indeed,

$\displaystyle
w\sigma_{i}\left(\varrho^{\sigma_{i}^{*}w^{-*}\mathfrak{q}}\right)-\varrho^{\mathfrak{q}}$
$\displaystyle=w\left(\sigma_{i}(\varrho^{\sigma_{i}^{*}w^{-*}\mathfrak{q}})-\varrho^{w^{-*}\mathfrak{q}}+\varrho^{w^{-*}\mathfrak{q}}\right)-\varrho^{\mathfrak{q}}$
$\displaystyle=w\left(-(b^{w^{-*}\mathfrak{q}}(\alpha_{i})-1)\alpha_{i}+\varrho^{w^{-*}\mathfrak{q}}\right)-\varrho^{\mathfrak{q}}$
$\displaystyle=w\left(\varrho^{w^{-*}\mathfrak{q}}\right)-\varrho^{\mathfrak{q}}-\left(b^{w^{-*}\mathfrak{q}}(\alpha_{i})-1\right)w\alpha_{i}.$

######  Example 3.7.

Even if the generalized root system is constant, the corresponding elements
$\varrho^{\mathfrak{q}}$ could be different. For instance,

$\displaystyle\varrho^{\mathfrak{q}}=\frac{1}{2}\left(\alpha_{1}+(N-1)(\alpha_{1}+\alpha_{2})+\alpha_{2}\right)$
$\displaystyle\varrho^{\mathfrak{p}}=\frac{1}{2}\left(\alpha_{1}+(\alpha_{1}+\alpha_{2})+(N-1)\alpha_{2}\right)$
$\displaystyle\varrho^{\mathfrak{r}}=\frac{1}{2}\left((N-1)\alpha_{1}+(\alpha_{1}+\alpha_{2})+\alpha_{2}\right)$

where $\mathfrak{q}$, $\mathfrak{p}$ and $\mathfrak{r}$ are as in Example 3.4.

##  4\. Small quantum groups

In this section, we introduce the Hopf algebras in which we are interested in.
We will follow [23]. We fix $\theta\in\mathbb{N}$ and a matrix
$\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}}\in(\Bbbk^{\times})^{\mathbb{I}\times\mathbb{I}}$
with finite-dimensional Nichols algebra $\mathfrak{B}_{\mathfrak{q}}$, where
$\mathbb{I}=\mathbb{I}_{\theta}$. Let $\mathcal{X}$ be the $\mathcal{G}$-orbit
of $\mathfrak{q}$ and $\mathcal{W}$ its Weyl groupoid.

We denote $U_{\mathfrak{q}}$ the Hopf algebra generated by the symbols
$K_{i}$, $K_{i}^{-1}$, $L_{i}$, $L_{i}^{-1}$, $E_{i}$ and $F_{i}$, with
$i\in\mathbb{I}$, subject to the relations:

$\displaystyle K_{i}E_{j}=q_{ij}E_{j}K_{i},\quad$ $\displaystyle
L_{i}E_{j}=q_{ji}^{-1}E_{j}L_{i}$ $\displaystyle
K_{i}F_{j}=q_{ij}^{-1}F_{j}K_{i},\quad$ $\displaystyle
L_{i}F_{j}=q_{ji}F_{j}L_{i}$ (4.1) $\displaystyle E_{i}F_{j}-F_{j}E_{i}$
$\displaystyle=\delta_{i,j}(K_{i}-L_{i})$ $\displaystyle XY$
$\displaystyle=YX,$ $\displaystyle K_{i}K_{i}^{-1}$
$\displaystyle=L_{i}L_{i}^{-1}=1$

for all $i,j\in\mathbb{I}$ and $X,Y\in\\{K_{i}^{\pm 1},L_{i}^{\pm 1}\mid
i\in\mathbb{I}\\}$. Also, the generators $E_{i}$ (resp. $F_{i}$) are subject
to the relations given by ${\mathcal{J}}_{\mathfrak{q}}$ (resp.
$\tau({\mathcal{J}}_{\mathfrak{q}})$, cf. (4.13) below). However, these last
relations will not be needed and neither the counit, nor he comultiplication
and nor the antipode of $U_{\mathfrak{q}}$.

We have that
$U_{\mathfrak{q}}=\oplus_{\mu\in\mathbb{Z}^{\mathbb{I}}}\,(U_{\mathfrak{q}})_{\mu}$
is a $\mathbb{Z}^{\mathbb{I}}$-graded Hopf algebra with

$\displaystyle\deg E_{i}=-\deg F_{i}=\alpha_{i}\quad\mbox{and}\quad\deg
K_{i}^{\pm 1}=\deg L_{i}^{\pm 1}=0\quad\forall\,i\in\mathbb{I}.$

For
$\alpha=n_{1}\alpha_{1}+\cdots+n_{\theta}\alpha_{\theta}\in\mathbb{Z}^{\mathbb{I}}$,
we set

$\displaystyle K_{\alpha}=K_{1}^{n_{1}}\cdots
K_{\theta}^{n_{\theta}}\quad\mbox{and}\quad L_{\alpha}=L_{1}^{n_{1}}\cdots
L_{\theta}^{n_{\theta}}.$

In particular, $K_{\alpha_{i}}=K_{i}$ for $i\in\mathbb{I}$.

Given $\mu\in\mathbb{Z}^{\mathbb{I}}$, we define
$\mathfrak{q}_{\mu}\in\operatorname{Alg}(U_{\mathfrak{q}}^{0},\Bbbk)$ and
$\widetilde{\mu}\in\operatorname{Aut}_{\Bbbk-alg}(U_{\mathfrak{q}}^{0})$ by

(4.2)
$\displaystyle\mathfrak{q}_{\mu}(K_{\alpha}L_{\beta})=\frac{\mathfrak{q}(\alpha,\mu)}{\mathfrak{q}(\mu,\beta)}\quad\mbox{and}\quad\widetilde{\mu}(K_{\alpha}L_{\beta})=\mathfrak{q}_{\mu}(K_{\alpha}L_{\beta})\,K_{\alpha}L_{\beta}$

for all $\alpha,\beta\in\mathbb{Z}^{\mathbb{I}}$. Then, it follows easily that

(4.3) $\displaystyle s\,u=u\,\widetilde{\mu}(s)\quad\forall
u\in(U_{\mathfrak{q}})_{\mu},\,s\in U_{\mathfrak{q}}^{0}$

and

(4.4)
$\displaystyle\widetilde{\nu}\circ\widetilde{\mu}=\widetilde{\nu+\mu}\quad\forall\nu,\mu\in\mathbb{Z}^{\mathbb{I}}.$

The multiplication of $U_{\mathfrak{q}}$ induces a linear isomorphism

(4.5) $\displaystyle U_{\mathfrak{q}}^{-}\otimes U_{\mathfrak{q}}^{0}\otimes
U_{\mathfrak{q}}^{+}\longrightarrow U_{\mathfrak{q}}$

where

$\displaystyle U_{\mathfrak{q}}^{+}=\Bbbk\langle E_{i}\mid
i\in\mathbb{I}\rangle\simeq\mathfrak{B}_{\mathfrak{q}},\quad
U_{\mathfrak{q}}^{0}=\Bbbk\langle K_{i},L_{i}\mid i\in\mathbb{I}\rangle,\quad
U_{\mathfrak{q}}^{-}=\Bbbk\langle F_{i}\mid i\in\mathbb{I}\rangle$

are subalgebras of $U_{\mathfrak{q}}$. We notice that $U_{\mathfrak{q}}^{0}$
identifies with the groups algebra
$\Bbbk(\mathbb{Z}^{\mathbb{I}}\times\mathbb{Z}^{\mathbb{I}})$ for any matrix
$\mathfrak{q}$.

######  Definition 4.1.

Let $p:U_{\mathfrak{q}}\longrightarrow u_{\mathfrak{q}}$ be a
$\mathbb{Z}^{\mathbb{I}}$-graded algebra projection and set
$u_{\mathfrak{q}}^{\pm,0}:=p(U_{\mathfrak{q}}^{\pm,0})$. We call
$u_{\mathfrak{q}}$ small quantum group if the multiplication
$u_{\mathfrak{q}}^{-}\otimes u_{\mathfrak{q}}^{0}\otimes
u_{\mathfrak{q}}^{+}\longrightarrow u_{\mathfrak{q}}$ induces a linear
isomorphism and $u_{\mathfrak{q}}^{\pm}\simeq U_{\mathfrak{q}}^{\pm}$.

We say they are “small” because $u_{\mathfrak{q}}^{\pm}$ is finite-dimensional
but we do not make any assumption on the size of $u_{\mathfrak{q}}^{0}$. Thus,
$U_{\mathfrak{q}}$ is even a small quantum group.

######  Example 4.2.

Let $\mathfrak{q}$ be as in Example 3.1. Then $U_{\mathfrak{q}}/\langle
K_{i}-L_{i}^{-1},K_{i}^{2\operatorname{{\mathrm{ord}}}q}-1\mid
i\in\mathbb{I}\rangle$ is an small quantum group. Moreover, if
$\operatorname{{\mathrm{ord}}}q$ is an odd prime, not $3$ if $\mathfrak{g}$
has a component of type $G_{2}$, then $u_{q}(\mathfrak{g})\simeq
U_{\mathfrak{q}}/\langle
K_{i}-L_{i}^{-1},K_{i}^{2\operatorname{{\mathrm{ord}}}q}-1\mid
i\in\mathbb{I}\rangle$.

######  Example 4.3.

Let us explain how to construct the small quantum group $u_{\mathfrak{q}}$ of
Figure 2. By [23, Corollary 5.9], there is a skew Hopf pairing $\eta$ between
$U^{+}_{\mathfrak{q}}\\#\Bbbk\langle K_{i}\mid i\in\mathbb{I}\rangle$ and
$U^{-}_{\mathfrak{q}}\\#\Bbbk\langle L_{i}\mid i\in\mathbb{I}\rangle$, and the
corresponding Drinfeld double is isomorphic to $U_{\mathfrak{q}}$. Let
$\Gamma=\overline{\langle K_{i}\mid i\in\mathbb{I}\rangle}$ be a group
quotient and set $g_{i}=\overline{K_{i}}$, $i\in\mathbb{I}$. Suppose the
character $\chi_{i}:\Gamma\longrightarrow\Bbbk^{\times}$,
$\chi_{i}(g_{j})=q_{ij}$ for all $i,j\in\mathbb{I}$, is well-defined and set
$\widetilde{\Gamma}=\langle\chi_{i}\mid i\in\mathbb{I}\rangle$. Then $\langle
L_{i}\mid i\in\mathbb{I}\rangle\longrightarrow\widetilde{\Gamma}$,
$L_{i}\mapsto\chi_{i}$, is a group quotient. Moreover, $\eta$ induces a
pairing between $U^{+}_{\mathfrak{q}}\\#\Bbbk\Gamma$ and
$U^{-}_{\mathfrak{q}}\\#\Bbbk\widetilde{\Gamma}$, and the corresponding
Drinfeld double is $u_{\mathfrak{q}}$.

######  Example 4.4.

The braided Drinfeld doubles introduced in [32, §3.2] are small quantum
groups.

###  4.1. Lusztig isomorphisms

In [23, Theorem 6.11], Heckenberger constructs certain algebra isomorphisms

(4.6) $\displaystyle
T_{i}=T_{i}^{(\sigma_{i}^{\mathfrak{q}})^{*}\mathfrak{q}}:U_{(\sigma_{i}^{\mathfrak{q}})^{*}\mathfrak{q}}\longrightarrow
U_{\mathfrak{q}}$

for all $i\in\mathbb{I}$. They emulate some properties of the Lusztig
automorphisms of small quantum groups. We emphasize that these isomorphisms
depends on the matrix defining the Drinfeld double of the domain, but we will
omit this in the notation when no confusion can arise. We do not need the
precise definition of these functions. We just recall some useful properties
for us.

Let $w:w^{-*}\mathfrak{q}\rightarrow\mathfrak{q}$ be a morphism in
$\mathcal{W}$. We choose a reduced expression
$w=1^{\mathfrak{q}}\sigma_{i_{k}}\cdots\sigma_{i_{1}}$ and denote

(4.7) $\displaystyle T_{w}=T_{i_{k}}\cdots
T_{i_{1}}:U_{w^{-*}\mathfrak{q}}\longrightarrow U_{\mathfrak{q}}.$

This isomorphism depends on our choosed expression for $w$. However, if
$w=1^{\mathfrak{q}}\sigma_{j_{k}}\cdots\sigma_{j_{1}}$ is another reduced
expression, there exists
$\underline{a}=(a)_{i\in\mathbb{I}}\in(\Bbbk^{\times})^{\mathbb{I}}$ such that

(4.8) $\displaystyle T_{i_{k}}\cdots T_{i_{1}}=T_{j_{k}}\cdots
T_{j_{1}}\varphi_{\underline{a}}$

where $\varphi_{\underline{a}}$ is the algebra automorphism of
$U_{w^{-*}\mathfrak{q}}$ given by

$\displaystyle\varphi_{\underline{a}}(K_{i})=K_{i},\quad\varphi_{\underline{a}}(L_{i})=L_{i},\quad\varphi_{\underline{a}}(E_{i})=a_{i}E_{i}\quad\mbox{and}\quad\varphi_{\underline{a}}(F_{i})=a_{i}^{-1}F_{i}\quad\forall
i\in\mathbb{I}.$

Indeed, both reduced expression of $w$ can be transformed each other using
only the Coxeter type relations by [25, Theorem 5]. Then, (4.8) follows using
[23, Theorem 6.19 and Proposition 6.8 $(ii)$].

These isomorphisms permute the weight spaces according in the following way:

(4.9) $\displaystyle
T_{w}\left((U_{w^{-*}\mathfrak{q}})_{\alpha}\right)=(U_{\mathfrak{q}})_{w\alpha}$

for all $\alpha\in\mathbb{Z}^{\mathbb{I}}$, cf. [26, Proposition 4.2]. They
also have a good behavior on the middle subalgebras
$U^{0}_{w^{-*}\mathfrak{q}}=U^{0}_{\mathfrak{q}}$. Explicitly,

(4.10) $\displaystyle T_{w}(K_{\alpha}L_{\beta})=K_{w\alpha}L_{w\beta}$

for all $\alpha,\beta\in\mathbb{Z}^{\mathbb{I}}$ because $T_{i}^{\pm
1}(K_{j})=K_{\sigma_{i}^{\mathfrak{q}}(\alpha_{j})}$ and $T_{i}^{\pm
1}(L_{j})=L_{\sigma_{i}^{\mathfrak{q}}(\alpha_{j})}$ by definition [23, Lemma
6.6]. It follows that

(4.11) $\displaystyle T_{w}\circ\widetilde{\mu}\circ
T_{w}^{-1}{}_{|U^{0}_{\mathfrak{q}}}=\widetilde{w(\mu)}$

for all $\mu\in\mathbb{Z}^{\mathbb{I}}$; keep in mind that $\widetilde{\mu}$
depends on $w^{-*}\mathfrak{q}$ and $\widetilde{w(\mu)}$ depends on
$\mathfrak{q}$.

For the longest element $w_{0}$, there is a permutation $f$ of $\mathbb{I}$
such that
$w_{0}^{-*}\mathfrak{q}(\alpha_{i},\alpha_{j})=f^{*}\mathfrak{q}(\alpha_{i},\alpha_{j})=\mathfrak{q}(\alpha_{f(i)},\alpha_{f(j)})$.
Also, there is
$\underline{b}=(b)_{i\in\mathbb{I}}\in(\Bbbk^{\times})^{\mathbb{I}}$ such that

(4.12) $\displaystyle
T_{w_{0}}=\phi_{1}\,\varphi_{f}\,\varphi_{\underline{b}}$

by [23, Corollary 6.21]; where $\phi_{1}$ is the algebra automorphism of
$U_{\mathfrak{q}}$ given by

$\displaystyle\phi_{1}(K_{i})=K_{i}^{-1},\quad\phi_{1}(L_{i})=L_{i}^{-1},\quad\phi_{1}(E_{i})=F_{i}L_{i}^{-1}\quad\mbox{and}\quad\varphi_{1}(F_{i})=K_{i}^{-1}E_{i}$

for all $i\in\mathbb{I}$ and $\varphi_{f}:U_{\mathfrak{q}}\longrightarrow
U_{f^{*}\mathfrak{q}}$ is the algebra isomorphism given by

$\displaystyle\varphi_{f}(K_{i})=K_{f(i)},\quad\varphi_{f}(L_{i})=L_{f(i)},\quad\varphi_{f}(E_{i})=E_{f(i)},\quad\mbox{and}\quad\varphi_{f}(F_{i})=F_{f(i)}$

for all $i\in\mathbb{I}$.

Let $\tau$ be the algebra antiautomorphism of $U_{\mathfrak{q}}$ defined by

(4.13)
$\displaystyle\tau(K_{i})=K_{i},\quad\tau(L_{i})=L_{i},\quad\tau(E_{i})=F_{i}\quad\mbox{and}\quad\tau(F_{i})=E_{i}$

for all $i\in\mathbb{I}$, see [23, Proposition 4.9 $(7)$]. Notice that
$\tau^{2}={\mathrm{id}}$.

The generators of the PBW basis of the Nichols algebras can be constructed
using the Lusztig isomorphisms as follows. Let
$w_{0}=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{n}}$ be a reduced
expression of the longest element in ${}^{\mathfrak{q}}\mathcal{W}$ and recall
from (3.8) that
$\beta_{\nu}=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{\nu-1}}(\alpha_{i_{\nu}})$,
$1\leq\nu\leq n$, are the positive roots of $\mathfrak{q}$. We set

(4.14) $\displaystyle E_{\beta_{\nu}}=T_{i_{1}}\cdots
T_{i_{\nu-1}}(E_{i_{\nu}})\in(U^{+}_{\mathfrak{q}})_{\beta_{\nu}}\quad\mbox{and}\quad
F_{\beta_{\nu}}=T_{i_{1}}\cdots
T_{i_{\nu-1}}(F_{i_{\nu}})\in(U^{-}_{\mathfrak{q}})_{-\beta_{\nu}},$

for all $1\leq\nu\leq n$; by an abuse of notation $E_{i_{\nu}}$ and
$F_{i_{\nu}}$ denote the generators of
$U_{(\sigma_{i_{\nu-1}}\cdots\sigma_{i_{1}})^{*}\mathfrak{q}}$. These elements
depend on the reduced expression of $w_{0}$.

By [26, Theorem 4.9], we know that

(4.15) $\displaystyle\begin{split}\left\\{E_{\beta_{f(1)}}^{m_{1}}\cdots
E_{\beta_{f(n)}}^{m_{n}}\right.&\left.\mid 0\leq
m_{\nu}<b^{\mathfrak{q}}(\beta_{\nu}),\,1\leq\nu\leq
n\right\\}\quad\mbox{and}\\\ &\left\\{F_{\beta_{f(1)}}^{m_{1}}\cdots
F_{\beta_{f(n)}}^{m_{n}}\mid 0\leq
m_{\nu}<b^{\mathfrak{q}}(\beta_{\nu}),\,1\leq\nu\leq n\right\\}\end{split}$

are linear basis of $U_{\mathfrak{q}}^{+}$ and $U_{\mathfrak{q}}^{-}$ for any
permutation $f$ of $\mathbb{I}$. It is immediate that

(4.16)
$\displaystyle{\mathrm{ch}}\,U^{-}_{\mathfrak{q}}=\prod_{\beta\in\Delta_{+}^{\mathfrak{q}}}\frac{1-e^{-b^{\mathfrak{q}}(\beta)\beta}}{1-e^{-\beta}}=\prod_{\beta\in\Delta_{+}^{\mathfrak{q}}}\left(1+e^{-\beta}+\cdots+e^{(1-b^{\mathfrak{q}}(\beta))\beta}\right).$

We point out that the weight space of minimum degree of $U^{-}_{\mathfrak{q}}$
is one-dimensional and spanned by

(4.17) $\displaystyle
F^{\mathfrak{q}}_{top}=F_{\beta_{1}}^{b^{\mathfrak{q}}(\beta_{1})-1}\cdots
F_{\beta_{n}}^{b^{\mathfrak{q}}(\beta_{n})-1}.$

######  Example 4.5.

Keeping the notation of Example 3.4,
$T_{1}^{\mathfrak{p}}:U_{\mathfrak{p}}\rightarrow U_{\mathfrak{q}}$ is defined
by

$\displaystyle T_{1}^{\mathfrak{p}}(E_{1})$
$\displaystyle=F_{1}L_{1}^{-1},\quad T_{1}^{\mathfrak{p}}(E_{2})=E_{12},\quad
T_{1}^{\mathfrak{p}}(K_{\alpha}L_{\beta})=K_{\sigma_{1}^{\mathfrak{p}}(\alpha)}L_{\sigma_{1}^{\mathfrak{p}}(\alpha)},$
$\displaystyle T_{1}^{\mathfrak{p}}(F_{1})$
$\displaystyle=K_{1}^{-1}E_{1},\quad
T_{1}^{\mathfrak{p}}(F_{2})=\frac{1}{q-1}(F_{1}F_{2}+F_{2}F_{1}).$

###  4.2. Parabolic subalgebras

We fix $i\in\mathbb{I}$ and denote $U_{\mathfrak{q}}(\alpha_{i})$ and
$U_{\mathfrak{q}}(-\alpha_{i})$ the subalgebras of $U_{\mathfrak{q}}$
generated by $E_{i}$ and $F_{i}$, respectively. We set

(4.18) $\displaystyle
P_{\mathfrak{q}}(\alpha_{i})=U_{\mathfrak{q}}(-\alpha_{i})U^{0}_{\mathfrak{q}}U^{+}_{\mathfrak{q}}.$

This is a subalgebra of $U_{\mathfrak{q}}$ thanks to the defining relations.

By the definition of the Lusztig isomorphisms, the restriction
$T_{i}:P_{\sigma_{i}^{*}\mathfrak{q}}(\alpha_{i})\longrightarrow
P_{\mathfrak{q}}(\alpha_{i})$ is an isomorphism and we can decompose
$P_{\mathfrak{q}}(\alpha_{i})$ as

(4.19) $\displaystyle
P_{\mathfrak{q}}(\alpha_{i})=U_{\mathfrak{q}}(\alpha_{i})\,U^{0}_{\mathfrak{q}}\,T_{i}(U^{+}_{(\sigma_{i}^{\mathfrak{q}})^{*}\mathfrak{q}}).$

Indeed, $T_{i}(E_{i})=F_{i}L_{i}^{-1}$,
$T_{i}(E_{j})=\operatorname{ad}_{c}^{-c_{ij}^{\mathfrak{q}}}E_{i}(E_{j})$ and
$T_{i}(F_{i})=K_{i}^{-1}E_{i}$ [23, Lemma 6.6].

###  4.3. Some definitions

We introduce some elements which will be key in the analysis of the
representations of $U_{\mathfrak{q}}$.

######  Definition 4.6.

Let $\beta\in\Delta^{\mathfrak{q}}$ and $n\in\mathbb{N}$. We set

$\displaystyle[\beta;n]=(n)_{q_{\beta}^{-1}}K_{\beta}-(n)_{q_{\beta}}L_{\beta}\quad\mbox{and}\quad[n;\beta]=(n)_{q_{\beta}^{-1}}L_{\beta}-(n)_{q_{\beta}}K_{\beta}.$

It follows from the defining relations that

(4.20) $\displaystyle
E_{i}F^{n}_{i}=F_{i}^{n}E_{i}+F_{i}^{n-1}[\alpha_{i};n]\quad\mbox{and}\quad
F_{i}E^{n}_{i}=E_{i}^{n}F_{i}+E_{i}^{n-1}[n;\alpha_{i}].$

for all $i\in\mathbb{I}$. Moreover, once we have fixed a PBW basis as in
(4.15), we can apply the corresponding Lusztig isomorphisms to the above
identities and obtain that

$\displaystyle
E_{\beta}F^{n}_{\beta}=F_{\beta}^{n}E_{\beta}+F_{\beta}^{n-1}[\beta;n]\quad\mbox{and}\quad
F_{\beta}E^{n}_{\beta}=E_{\beta}^{n}F_{\beta}+E_{\beta}^{n-1}[n;\beta]$

for all $\beta\in\Delta^{\mathfrak{q}}_{+}$.

######  Definition 4.7.

Given $\beta\in\Delta^{\mathfrak{q}}$, $\mu\in\mathbb{Z}^{\mathbb{I}}$ and a
$U_{\mathfrak{q}}^{0}$-algebra $\mathbf{A}$ with structural map
$\pi:U_{\mathfrak{q}}^{0}\longrightarrow\mathbf{A}$, we define
$t_{\beta}^{\pi}(\mu)$ as the unique
$t\in\\{1,...,b^{\mathfrak{q}}(\beta)-1\\}$ such that

$\displaystyle 1=q_{\beta}^{1-t}\pi\tilde{\mu}(K_{\beta}L_{\beta}^{-1}),$

if it exists, and otherwise $t^{\pi}_{\beta}(\mu)=0$.

Equivalently, we see can say that, modulo $b^{\mathfrak{q}}(\beta)$,
$t_{\beta}^{\pi}(\mu)$ is the unique $t\in\\{1,...,b^{\mathfrak{q}}(\beta)\\}$
such that $\pi\tilde{\mu}([\beta;t])=0$. In fact,
$\pi\tilde{\mu}([\beta;b^{\mathfrak{q}}(\beta)])=0$ and

(4.21)
$\displaystyle[\beta;t]=(t)_{q_{\beta}}L_{\beta}\left(q_{\beta}^{1-t}K_{\beta}L_{\beta}^{-1}-1\right).$

We also observe that

(4.22)
$\displaystyle[t;\beta]=(t)_{q_{\beta}^{-1}}L_{\beta}\left(1-q_{\beta}^{t-1}K_{\beta}L_{\beta}^{-1}\right).$

Given a $U_{\mathfrak{q}}^{0}$-algebra $\mathbf{A}$ with structural map
$\pi:U_{\mathfrak{q}}^{0}\longrightarrow\mathbf{A}$ and a morphism
$w\in\mathcal{W}^{\mathfrak{q}}$, we denote $\mathbf{A}[w]$ the
$U_{\mathfrak{q}}^{0}$-algebra with structural map
$\pi[w]:U_{\mathfrak{q}}^{0}\longrightarrow\mathbf{A}$ defined by

(4.23)
$\displaystyle\pi[w](K_{\alpha}L_{\alpha})=\pi(K_{w^{-1}\alpha}L_{w^{-1}\beta})$

for all $\alpha,\beta\in\mathbb{Z}^{\mathbb{I}}$. We highlight that
$\pi[w]=\pi\circ T_{w}^{-1}{}_{|U_{\mathfrak{q}}^{0}}$ for any Lusztig
isomorphism $T_{w}:U_{\mathfrak{q}}\longrightarrow U_{w^{*}\mathfrak{q}}$
associated to $w$ by (4.10).

If $\beta=w\alpha_{i}\in\Delta^{w^{*}\mathfrak{q}}$ for some $i\in\mathbb{I}$,
it holds that

(4.24) $\displaystyle
t_{w\alpha_{i}}^{\pi[w]}(w\mu)=t_{\alpha_{i}}^{\pi}(\mu).$

In fact, we have that

$\displaystyle\pi[w]\widetilde{w\mu}([\beta,t])$
$\displaystyle=\pi[w]\left((t)_{w^{*}\mathfrak{q}(\beta,\beta)^{-1}}w^{*}\mathfrak{q}(\beta,w\mu)K_{\beta}-(t)_{w^{*}\mathfrak{q}(\beta,\beta)}w^{*}\mathfrak{q}(w\mu,-\beta)L_{\beta}\right)$
(4.25)
$\displaystyle=\pi\left((t)_{q_{\alpha_{i}}^{-1}}\mathfrak{q}(\alpha_{i},\mu)K_{\alpha_{i}}-(t)_{q_{\alpha_{i}}}\mathfrak{q}(\mu,-\alpha_{i})L_{\alpha_{i}}\right)$
$\displaystyle=\pi\tilde{\mu}[\alpha_{i},t].$

As in [26, Definition 2.16], we define the group homomorphism
$\rho^{\mathfrak{q}}:\mathbb{Z}^{\mathbb{I}}\longrightarrow\Bbbk^{\times}$ by
$\rho^{\mathfrak{q}}(\alpha_{i})=\mathfrak{q}(\alpha_{i},\alpha_{i})$ for all
$i\in\mathbb{I}$.

######  Definition 4.8.

Given $\beta\in\Delta^{\mathfrak{q}}$, $\mu\in\mathbb{Z}^{\mathbb{I}}$ and a
$U_{\mathfrak{q}}^{0}$-algebra $\mathbf{A}$ with structural map
$\pi:U_{\mathfrak{q}}^{0}\longrightarrow\mathbf{A}$, we define
$n_{\beta}^{\pi}(\mu)$ as the unique
$n\in\\{1,...,b^{\mathfrak{q}}(\beta)-1\\}$ such that

$\displaystyle
q_{\beta}^{n}=\rho^{\mathfrak{q}}(\beta)\,\pi\widetilde{\mu}(K_{\beta}L_{\beta}^{-1}),$

if it exists, and otherwise $n^{\pi}_{\beta}(\mu)=0$.

The above numbers are related in the following way.

######  Lemma 4.9.

If $\beta=w\alpha_{i}\in\Delta^{\mathfrak{q}}$ for some $i\in\mathbb{I}$, then
$n_{\beta}^{\pi}(\mu)=t_{\beta}^{\pi}(\mu\langle w\rangle)$.

###### Proof.

We first claim that

(4.26)
$\displaystyle\rho^{\mathfrak{q}}(w\alpha_{i})=\mathfrak{q}(w\alpha_{i},w\alpha_{i})\,\mathfrak{q}(0\langle
w\rangle,w\alpha_{i})\,\mathfrak{q}(w\alpha_{i},0\langle w\rangle).$

We prove it by induction on the length of $w$. If $w=1^{\mathfrak{q}}$, then
$n_{\alpha_{i}}^{\pi}(\mu)=t_{\alpha_{i}}^{\pi}(\mu)$ by (4.21). We now assume
the equality holds for all bicharacters and morphisms in $\mathcal{W}$ of
length $r$. Thus, if $w=\sigma_{j}w_{1}$ with $j\in\mathbb{I}$ and
$\ell(\sigma_{j}w_{1})=1+\ell(w_{1})=1+r$, similar to (3.12), one can check
that

$\displaystyle
0\langle\sigma_{j}w_{1}\rangle=-\sum_{\gamma\in\Delta_{+}^{\sigma_{j}^{-*}(\mathfrak{q})}:w_{1}^{-1}\gamma\in\Delta_{-}^{(\sigma_{j}w_{1})^{-*}\mathfrak{q}}}(b^{\mathfrak{q}}(\sigma_{j}\gamma)-1)\sigma_{j}\gamma+(b^{\mathfrak{q}}(\alpha_{j})-1)\sigma_{j}\alpha_{j}.$

Therefore

$\displaystyle\mathfrak{q}(\sigma_{j}w_{1}\alpha_{i},\sigma_{j}w_{1}\alpha_{i})$
$\displaystyle\,\mathfrak{q}(0\langle\sigma_{j}w_{1}\rangle,\sigma_{j}w_{1}\alpha_{i})\,\mathfrak{q}(\sigma_{j}w_{1}\alpha_{i},0\langle\sigma_{j}w_{1}\rangle)=$
$\displaystyle=\sigma_{j}^{-*}(\mathfrak{q})(w_{1}\alpha_{i},w_{1}\alpha_{i})\,\sigma_{j}^{-*}(\mathfrak{q})(0\langle
w_{1}\rangle,w_{1}\alpha_{i})\,\sigma_{j}^{-*}(\mathfrak{q})(w_{1}\alpha_{i},0\langle
w_{1}\rangle)\times$
$\displaystyle\quad\quad\quad\sigma_{j}^{-*}(\mathfrak{q})(\alpha_{j},w_{1}\alpha_{i})^{b^{\sigma_{j}^{-*}(\mathfrak{q})}(\alpha_{j})-1}\,\sigma_{j}^{-*}(\mathfrak{q})(w_{1}\alpha_{i},\alpha_{j})^{b^{\sigma_{j}^{-*}(\mathfrak{q})}(\alpha_{j})-1}$
$\displaystyle\overset{(\star)}{=}\rho^{\sigma_{j}^{-*}(\mathfrak{q})}(w_{1}\alpha_{i})\,\frac{\rho^{\mathfrak{q}}(\sigma_{j}w_{1}\alpha_{i})}{\rho^{\sigma_{j}^{-*}(\mathfrak{q})}(w_{1}\alpha_{i})}=\rho^{\mathfrak{q}}(\sigma_{j}w_{1}\alpha_{i});$

$(\star)$ follows from the inductive hypothesis and [26, Lemma 2.17]. This
concludes the induction and our claim holds.

In consequence, we have that

$\displaystyle q_{\beta}\pi\widetilde{\mu\langle
w\rangle}(K_{\beta}L_{\beta}^{-1})$
$\displaystyle=\mathfrak{q}(\beta,\beta)\mathfrak{q}(0\langle
w\rangle,\beta)\mathfrak{q}(\beta,0\langle
w\rangle)\,\pi\widetilde{\mu}(K_{\beta}L_{\beta}^{-1})$
$\displaystyle=\rho^{\mathfrak{q}}(\beta)\,\pi\widetilde{\mu}(K_{\beta}L_{\beta}^{-1})$

which implies the lemma. ∎

##  5\. Andersen–Jantzen–Soergel categories

In [1, Section 2], Andersen, Jantzen and Soergel have defined certain
categories of modules for algebras fulfilling most of the more remarkable
features of the small quantum groups at roots if unity. We will call them AJS
categories. They consider any $\mathbb{Z}^{\mathbb{I}}$-graded $\Bbbk$-algebra
$U$ endowed with a triangular decomposition

$\displaystyle U^{-}\otimes U^{0}\otimes U^{+}\longrightarrow U,$

i.e. this is a $\Bbbk$-linear isomorphism induced by the multiplication and
$U^{-}$, $U^{0}$ and $U^{+}$ are $\mathbb{Z}^{\mathbb{I}}$-graded subalgebras
satisfying the following properties, cf. [1, §1.1 and §2.1]:

(5.1) $\displaystyle U^{0}\subset U_{0},\quad(U^{\pm})_{0}=\Bbbk;$ (5.2)
$\displaystyle(U^{\pm})_{\nu}\neq 0\Rightarrow\pm\nu\geq 0;$ (5.3)
$\displaystyle\mbox{$\sup(U^{\pm})$ is finite};$ (5.4)
$\displaystyle\mbox{Each $(U^{\pm})_{\nu}$, and hence $U^{\pm}$, is finite-
dimensional over $\Bbbk$}.$

They also assume that $U^{0}$ is commutative and the existence of a group
homomorphism $\mathbb{Z}^{\mathbb{I}}\longrightarrow\operatorname{Aut}_{\Bbbk-
alg}(U^{0})$, $\mu\mapsto\widetilde{\mu}$, such that

(5.5) $\displaystyle s\,u=u\,\widetilde{\mu}(s)\quad\forall u\in
U_{\mu},\,s\in U^{0}.$

From (5.1), we deduce that there are augmentation maps
$U^{\pm}\longrightarrow(U^{\pm})_{0}=\Bbbk$. We denote both of them
$\varepsilon$.

######  Example 5.1.

A small quantum group satisfies all the previous properties.

We fix a Noetherian commutative $U^{0}$-algebra $\mathbf{A}$ with structural
map $\pi:U^{0}\longrightarrow\mathbf{A}$. We now present the AJS category
$\mathcal{C}_{\mathbf{A}}$, cf. [1, 2.3]. An object of
$\mathcal{C}_{\mathbf{A}}$ is a $\mathbb{Z}^{\mathbb{I}}$-graded
$U\otimes\mathbf{A}$-module $M$, or equivalently a left $U$-module and right
$\mathbf{A}$-module, such that

(5.6) $M$ is finitely generated over $\mathbf{A}$; (5.7)
$M_{\mu}\mathbf{A}\subset M_{\mu}$, (5.8) $U_{\nu}M_{\mu}\subset M_{\nu+\mu}$,
(5.9) $s\,m=m\,\pi\widetilde{\mu}(s)$,

for all $\mu,\nu\in\mathbb{Z}^{\mathbb{I}}$, $m\in M_{\mu}$ and $s\in U^{0}$.
The last compatibility means that the $U^{0}$-action is determined by the
$\mathbf{A}$-action and the $\mathbb{Z}^{\mathbb{I}}$-grading. The morphisms
between two objects are the morphisms of $\mathbb{Z}^{\mathbb{I}}$-graded
$U\otimes\mathbf{A}$-modules.

The authors also defined categories $\mathcal{C}^{\prime}_{\mathbf{A}}$ and
$\mathcal{C}^{\prime\prime}_{\mathbf{A}}$ in similar fashion by replacing $U$
with $U^{0}U^{+}$ and $U^{0}$, respectively. There are obvious induced
functors
$\mathcal{C}^{\prime\prime}_{\mathbf{A}}\longrightarrow\mathcal{C}^{\prime}_{\mathbf{A}}$
and $\mathcal{C}^{\prime}_{\mathbf{A}}\longrightarrow\mathcal{C}_{\mathbf{A}}$
which are left adjoints of the forgetful functors. Moreover, the categories
$\mathcal{C}_{\mathbf{A}}$ and $\mathcal{C}^{\prime}_{\mathbf{A}}$ have enough
projectives [1, Lemma 2.7].

We next summarize the most important attributes of the AJS categories.

###  5.1. Verma modules

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$. We denote $\mathbf{A}^{\mu}$ the free
right $\mathbf{A}$-module of rank one concentrated in degree $\mu$ and
generated by the symbol
$\boldsymbol{|}\mu\boldsymbol{\rangle}=\boldsymbol{|}\mu\boldsymbol{\rangle}_{\mathbf{A}}$.

We consider $\mathbf{A}^{\mu}$ as an object of
$\mathcal{C}^{\prime}_{\mathbf{A}}$ with left $U^{+}$-action given by the
augmentation map and left $U^{0}$-action determined by (5.9). Explicitly,

(5.10) $\displaystyle
su\cdot\boldsymbol{|}\mu\boldsymbol{\rangle}=\varepsilon(u)\,\pi\widetilde{\mu}(s)\,\boldsymbol{|}\mu\boldsymbol{\rangle}\quad\forall
s\in U^{0},\,u\in U^{+}.$

We call Verma modules to the induced modules

(5.11) $\displaystyle
Z_{\mathbf{A}}(\mu)=U\otimes_{U^{0}U^{+}}\mathbf{A}^{\mu}$

with $U$ and $\mathbf{A}$ acting by left and right multiplication on the left
and right factor, respectively. It is isomorphic to $U^{-}\otimes\mathbf{A}$
as $U^{-}\otimes\mathbf{A}$-module and its weight spaces are

(5.12) $\displaystyle
Z_{\mathbf{A}}(\mu)_{\beta}=(U^{-})_{\beta-\mu}\otimes\boldsymbol{|}\mu\boldsymbol{\rangle}$

for all $\beta\in\mathbb{Z}^{\mathbb{I}}$. Therefore

(5.13)
$\displaystyle{\mathrm{ch}}\,Z_{\mathbf{A}}(\mu)=e^{\mu}\,{\mathrm{ch}}\,U^{-}.$

By an abuse of notation, we also denote
$\boldsymbol{|}\mu\boldsymbol{\rangle}$ the generator
$1\otimes\boldsymbol{|}\mu\boldsymbol{\rangle}$ of $Z_{\mathbf{k}}(\mu)$.

A $Z$-filtration of a module in $\mathcal{C}_{\mathbf{A}}$ is a filtration
whose subquotient are isomorphic to Verma modules. In [1, §2.11-§2.16] is
proved that projective modules admit $Z$-filtrations. Several other properties
of these modules are proved in [1, Sections 2-4].

###  5.2. Simple modules

Assume that $\mathbf{A}=\mathbf{k}$ is a field. Then $Z_{\mathbf{k}}(\mu)$ has
a unique simple quotient denoted $L_{\mathbf{k}}(\mu)$ [1, §4.1]. This object
is characterized as the unique (up to isomorphism) simple highest-weight
module $L$ in $\mathcal{C}_{\mathbf{k}}$, that is $L$ is generated by some
$v\in L_{\mu}$ with $(U^{+})_{\nu}v=0$ for all $\nu>0$. We say that $v$ is
highest-weight vector of weight $\mu$. Moreover, each simple module in
$\mathcal{C}_{\mathbf{k}}$ is isomorphic to a unique simple highest-weight
module. This characterization of the simple modules implies that their
characters are linearly independent.

We notice that all modules have composition series of finite length. For
$M\in\mathcal{C}_{\mathbf{k}}$, $[M:L_{\mathbf{k}}(\lambda)]$ denotes the
number of composition factors isomorphic to $L_{\mathbf{k}}(\lambda)$. Two
important properties of the Verma modules are

$\displaystyle[Z_{\mathbf{k}}(\mu):L_{\mathbf{k}}(\mu)]=1\quad\mbox{and}\quad[Z_{\mathbf{k}}(\mu):L_{\mathbf{k}}(\lambda)]\neq
0\Rightarrow\lambda\leq\mu.$

The next lemma is standard. It says that we can read the composition factors
of a module from its character. In particular, modules with equal characters
have the same composition factors.

######  Lemma 5.2.

Let $M\in\mathcal{C}_{\mathbf{k}}$. It holds that
${\mathrm{ch}}\,M=\sum_{\lambda}a_{\lambda}\,{\mathrm{ch}}\,L_{\mathbf{k}}(\lambda)$
if and only if $a_{\lambda}=[M:L_{\mathbf{k}}(\lambda)]$ for all
$\lambda\in\mathbb{Z}^{\mathbb{I}}$. ∎

######  Example 5.3.

Let $U=U_{\mathfrak{q}}$ be a Drinfeld double, $\mathbf{k}=\Bbbk$ and $\mu=0$.
Then
$K_{\alpha}L_{\beta}\cdot\boldsymbol{|}0\boldsymbol{\rangle}=\pi(K_{\alpha}L_{\beta})\boldsymbol{|}0\boldsymbol{\rangle}$
and hence $Z_{\Bbbk}(0)=\mathcal{M}^{\mathfrak{q}}(\pi)$ is the Verma module
of [26, Definition 5.1] and $L_{\Bbbk}(0)=L^{\mathfrak{q}}(\pi)$ its quotient
as in [26, $(5.7)$].

######  Example 5.4.

$L_{\mathbf{k}}(\mu)=\mathbf{k}^{\mu}$ is one-dimensional if and only if

(5.14) $\displaystyle\pi\tilde{\mu}(K_{i}L_{i}^{-1})=1\quad\forall
i\in\mathbb{I}.$

Indeed, using (4.1) and (5.10), $\mathbf{k}^{\mu}$ is the simple quotient of
$Z_{\mathbf{k}}(\mu)$ if and only if

$\displaystyle
0=(K_{i}-L_{i})\cdot\boldsymbol{|}\mu\boldsymbol{\rangle}=\left(\mathfrak{q}(\alpha_{i},\mu)\pi(K_{i})-\frac{1}{\mathfrak{q}(\mu,\alpha_{i})}\pi(L_{i})\right)\boldsymbol{|}\mu\boldsymbol{\rangle}$

which is equivalent to our claim.

###  5.3. Blocks

Let $\sim_{b}$ denote the smallest equivalence relation in
$\mathbb{Z}^{\mathbb{I}}$ such that $\lambda\sim_{b}\mu$ if
$\operatorname{Hom}_{\mathcal{C}_{\mathbf{A}}}(Z_{\mathbf{A}}(\lambda),Z_{\mathbf{A}}(\mu))\neq
0$ or
$\operatorname{{\mathrm{Ext}}}^{1}_{\mathcal{C}_{\mathbf{k}}}(Z_{\mathbf{A}}(\lambda),Z_{\mathbf{A}}(\mu))\neq
0$. The equivalence classes of $\sim_{b}$ are called blocks [1, §6.9]. In case
$\mathbf{A}=\mathbf{k}$ is a field, this definition coincides with the usual
definition of blocks via simple modules [1, Lemma 6.12]. Namely, $\lambda$ and
$\mu$ belong to the same block if $L_{\mathbf{k}}(\lambda)$ and
$L_{\mathbf{k}}(\mu)$ have a non trivial extension.

Let $\mathcal{D}_{\mathbf{A}}$ denote the full subcategory of
$\mathcal{C}_{\mathbf{A}}$ of all objects admitting a $Z$-filtration. If $b$
is a block, $\mathcal{D}_{\mathbf{A}}(b)$ is the full subcategory of all
object admitting a $Z$-filtration whose factors are $Z_{\mathbf{A}}(\mu)$ with
$\mu\in b$. Likewise $\mathcal{C}_{\mathbf{A}}(b)$ denotes the full
subcategory of all objects in $\mathcal{C}_{\mathbf{A}}$ which are the
homomorphic image of an object in $\mathcal{D}_{\mathbf{A}}(b)$. The abelian
categories $\mathcal{D}_{\mathbf{A}}$ and $\mathcal{C}_{\mathbf{A}}$ decompose
into the sum $\oplus_{b}\mathcal{D}_{\mathbf{A}}(b)$ and
$\oplus_{b}\mathcal{C}_{\mathbf{A}}(b)$. These and other properties are proved
in [1, 6.10]. We also will call blocks to the subcategories
$\mathcal{D}_{\mathbf{A}}(b)$ and $\mathcal{C}_{\mathbf{A}}(b)$. The principal
block, denoted $\mathcal{C}_{\mathbf{k}}(0)$, is the block containing
$L_{\mathbf{k}}(0)$.

###  5.4. Duals

Let $M\in\mathcal{C}_{\mathbf{A}}$. Then
$M^{\tau}=\operatorname{Hom}_{\mathbf{A}}(M,\mathbf{A})$ is an object in
$\mathcal{C}_{\mathbf{A}}$ with $U$-action

$\displaystyle(uf)(m)=f(\tau(u)m),$

for all $m\in M$ and $f\in M^{\tau}$, and homogeneous components

$\displaystyle(M^{\tau})_{\lambda}=\left\\{f\in\operatorname{Hom}_{\mathbf{A}}(M,\mathbf{A})\mid
f(M_{\mu})=0\,\forall\mu\neq\lambda\right\\}$

for all $\lambda\in\mathbb{Z}^{\mathbb{I}}$. If $M$ is $\mathbf{A}$-free, then

(5.15) $\displaystyle{\mathrm{ch}}(M^{\tau})={\mathrm{ch}}(M).$

From the characterization of the simple objects in $\mathcal{C}_{\mathbf{k}}$,
$\mathbf{k}$ is a field, we deduce that

$\displaystyle L_{\mathbf{k}}(\mu)^{\tau}\simeq L_{\mathbf{k}}(\mu).$

for all $\mu\in\mathbb{Z}^{\mathbb{I}}$.

###  5.5. AJS categories versus usual module categories

Assume that $U^{0}$ is finite-dimensional. By forgetting the right
$\mathbf{A}$-action, we obtain a fully faithful exact functor from the AJS
categories $\mathcal{C}_{\mathbf{A}}$ to the category ${}_{U}\mathcal{G}$ of
finite-dimensional $\mathbb{Z}^{\mathbb{I}}$-graded $U$-modules. Roughly
speaking, the next proposition says that we know all the simple objects and
the blocks of the category ${}_{U}\mathcal{G}$ if we know the simple module
$L_{\Bbbk}(0)$ and the principal block $\mathcal{C}_{\Bbbk}(0)$ for all the
algebra maps $\pi:U^{0}\longrightarrow\Bbbk$.

######  Proposition 5.5.

Every block in ${}_{U}\mathcal{G}$ is equivalent as an abelian category to the
principal block of $\mathcal{C}_{\Bbbk}$ for some algebra map
$\pi:U^{0}\longrightarrow\Bbbk$.

###### Proof.

One can construct Verma and simple modules in ${}_{U}\mathcal{G}$ like in the
AJS categories, see for instance [16]. Given an algebra map
$\pi:U^{0}\longrightarrow\Bbbk$, let us denote of $\Delta(\pi)$ and $L(\pi)$
the corresponding Verma module and simple modules in ${}_{U}\mathcal{G}$.
Namely, $L(\pi)$ coincides with the image of $L_{\Bbbk}(0)$ under the
forgetful functor. We claim that every simple module in ${}_{U}\mathcal{G}$
belonging to the same block of $L(\pi)$ is isomorphic to the image of
$L_{\Bbbk}(\mu)$ under the forgetful functor for some $\mu$ in the principal
block of $\mathcal{C}_{\Bbbk}$. In fact, suppose that $L(\mu)$ is a simple
module in ${}_{U}\mathcal{G}$ and

$\displaystyle 0\longrightarrow L(\mu)\longrightarrow M\longrightarrow
L(\pi)\longrightarrow 0$

is a non trivial extension. Let $m_{\mu}$ be a highest-weight vector in
$L(\mu)$ and $m\in M$ such that its image in $L(\pi)$ is a highest-weight
vector. Then $m$ generates $M$ and there is $u_{\mu}\in U_{\mu}$ such that
$m_{\mu}=u_{\mu}\cdot m$. Let $s\in U^{0}$. Hence

$\displaystyle s\cdot m_{\mu}=su_{\mu}\cdot m\overset{\eqref{property
D}}{=}u_{\mu}\widetilde{\mu}(s)\cdot m\overset{\eqref{eq:compatibilidad sm
ma}}{=}(\pi\widetilde{\mu})(s)\,m_{\mu}.$

This implies that $L(\mu)$ is isomorphic to the image of $L_{\Bbbk}(\mu)$
under the forgetful functor. Therefore every simple module of
${}_{U}\mathcal{G}$ belonging to the same block of $L(\pi)$ also is in the
image of the principal block, and the proposition follows. ∎

###  5.6. AJS categories and quotients

Let $p:U\longrightarrow\overline{U}$ be a $\mathbb{Z}^{\mathbb{I}}$-graded
algebra projection and set $\overline{U}^{\pm,0}:=p(U^{\pm,0})$. Assume that
$\overline{U}^{-}\otimes\overline{U}^{0}\otimes\overline{U}^{+}\longrightarrow\overline{U}$
induces a linear isomorphism, $\overline{U}^{\pm}\simeq U^{\pm}$ and
$\widetilde{\mu}$ induces an algebra automorphism on $\overline{U}^{0}$. Thus,
given an algebra map $\overline{\pi}:\overline{U}\longrightarrow\mathbf{A}$ we
can consider the corresponding AJS category which we denote
$\overline{\mathcal{C}}_{\mathbf{A}}$. We write
$\overline{Z}_{\mathbf{A}}(\mu)$ and $\overline{L}_{\mathbf{k}}(\mu)$ for the
Verma and simple modules in $\overline{\mathcal{C}}_{\mathbf{A}}$ and
$\overline{\mathcal{C}}_{\mathbf{k}}$, respectively. If
$\pi=\overline{\pi}\circ p:U\longrightarrow\mathbf{A}$, we get an obvious
functor
$\operatorname{Inf}_{\overline{U}}^{U}:\overline{\mathcal{C}}_{\mathbf{A}}\longrightarrow\mathcal{C}_{\mathbf{A}}$
such that
$\operatorname{Inf}_{\overline{U}}^{U}(\overline{Z}_{\mathbf{A}}(\mu))\simeq
Z_{\mathbf{A}}(\mu)$ and
$\operatorname{Inf}_{\overline{U}}^{U}(\overline{L}_{\mathbf{k}}(\mu))\simeq
L_{\mathbf{k}}(\mu)$ for all $\mu\in\mathbb{Z}^{\mathbb{I}}$, by the
assumptions on $p$. Then
${\mathrm{ch}}\overline{Z}_{\mathbf{k}}(\mu)={\mathrm{ch}}Z_{\mathbf{k}}(\mu)$
and
${\mathrm{ch}}\overline{L}_{\mathbf{k}}(\mu)={\mathrm{ch}}L_{\mathbf{k}}(\mu)$
and therefore

(5.16)
$\displaystyle\left[\overline{Z}_{\mathbf{k}}(\mu):\overline{L}_{\mathbf{k}}(\lambda)\right]=[Z_{\mathbf{k}}(\mu):L_{\mathbf{k}}(\lambda)]$

for all $\mu,\lambda\in\mathbb{Z}^{\mathbb{I}}$ by Lemma 5.2.

Moreover, if $\widetilde{\mu}$ does not descend to an algebra automorphism on
$\overline{U}^{0}$, we have an analogous to (5.16) by considering the category
${}_{\overline{U}}\mathcal{G}$ instead of $\overline{\mathcal{C}}_{\Bbbk}$.

##  6\. Twisted Verma modules

Through this section we fix a matrix
$\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}}\in(\Bbbk^{\times})^{\mathbb{I}\times\mathbb{I}}$
with finite-dimensional Nichols algebra $\mathfrak{B}_{\mathfrak{q}}$, where
$\mathbb{I}=\mathbb{I}_{\theta}$. Let $\mathcal{X}$ be the $\mathcal{G}$-orbit
of $\mathfrak{q}$ and $\mathcal{W}$ its Weyl groupoid. Let $U_{\mathfrak{q}}$
be the Hopf algebra introduced in Section 4. In the sequel $\mathbf{A}$
denotes a Noetherian commutative ring and $\mathbf{k}$ denotes a field. We
assume that both are algebras over $U^{0}$ and, by abuse of notation, we
denote their structural maps
$\pi:U_{\mathfrak{q}}^{0}\longrightarrow\mathbf{A}$ and
$\pi:U_{\mathfrak{q}}^{0}\longrightarrow\mathbf{k}$. We denote
$\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$,
$\mathcal{C}^{\prime\mathfrak{q}}_{\mathbf{A}}$ and
$\mathcal{C}^{\prime\prime\mathfrak{q}}_{\mathbf{A}}$ the corresponding AJS
categories; and similarly over the field $\mathbf{k}$. We point out that the
results regarding to simple modules of the following subsections only hold for
the AJS categories over the field $\mathbf{k}$, cf. §5.2, and the results
being valid over $\mathbf{A}$ obviously hold over $\mathbf{k}$.

In this section we construct and study different Verma modules for
$U_{\mathfrak{q}}$ using the Lusztig isomorphisms of §4.1. To do this, we
mimic the ideas from [1] where the authors use the Lusztig automorphisms [34].
Although our demonstrations are almost identical to [1], it is worthwhile to
do thoroughly again them as we have morphisms between possibly different
algebras parameterized by $\mathcal{X}$ and permuted by the action of the Weyl
groupoid.

###  6.1. $w$-Verma modules

Let $w:w^{-*}\mathfrak{q}\rightarrow\mathfrak{q}$ be a morphism in
$\mathcal{W}$ with $w=1^{\mathfrak{q}}\sigma_{i_{k}}\cdots\sigma_{i_{1}}$ a
reduced expression. We consider the Lusztig isomorphism

(6.1) $\displaystyle T_{w}=T_{i_{k}}\cdots
T_{i_{1}}:U_{w^{-*}\mathfrak{q}}\longrightarrow U_{\mathfrak{q}}.$

Thus, the triangular decomposition of $U_{w^{-*}\mathfrak{q}}$ induces a new
triangular decomposition on $U_{\mathfrak{q}}$. Explicitly,

(6.2) $\displaystyle T_{w}(U^{-}_{w^{-*}\mathfrak{q}})\otimes
U^{0}_{\mathfrak{q}}\otimes T_{w}(U^{+}_{w^{-*}\mathfrak{q}})\longrightarrow
U_{\mathfrak{q}}$

since $T_{w}(U^{0}_{w^{-*}\mathfrak{q}})=U^{0}_{\mathfrak{q}}$.

Given $\mu\in\mathbb{Z}^{\mathbb{I}}$, we conisder $\mathbf{A}^{\mu}$ as a
$U^{0}_{\mathfrak{q}}\,T_{w}(U^{+}_{w^{-*}\mathfrak{q}})$-module with action

(6.3) $\displaystyle
su\cdot\boldsymbol{|}\mu\boldsymbol{\rangle}=\varepsilon(u)\,\pi(\widetilde{\mu}(s))\,\boldsymbol{|}\mu\boldsymbol{\rangle}\quad\forall
s\in U_{\mathfrak{q}}^{0},\,u\in T_{w}(U^{+}_{w^{-*}\chi})$

and imitating [1, §4.3], we introduce the module

(6.4) $\displaystyle
Z^{w}_{\mathbf{A}}(\mu)=U_{\mathfrak{q}}\otimes_{U_{\mathfrak{q}}^{0}T_{w}(U^{+}_{w^{-*}\mathfrak{q}})}\mathbf{A}^{\mu}$

which belongs to $\mathcal{C}^{\mathfrak{q}}_{\mathbf{A}}$. We call it
$w$-Verma module. Of course,
$Z_{\mathbf{A}}^{1^{\mathfrak{q}}}(\mu)=Z_{\mathbf{A}}(\mu)$. We notice that
$Z^{w}_{\mathbf{A}}(\mu)$ does not depend on the reduced expression of $w$. In
fact, if $\tilde{T}_{w}$ is the Lusztig isomorphism associated to other
reduced expression of $w$, then
$T_{w}(U^{\pm}_{w^{-*}\mathfrak{q}})=\tilde{T}_{w}(U^{\pm}_{w^{-*}\mathfrak{q}})$
by (4.8). Hence the decomposition (6.2) does not depend on the reduced
expression. Moreover, (6.3) defines the same module for both expressions, and
a posteriori isomorphic Verma modules.

######  Lemma 6.1.

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$ and $w\in{}^{\mathfrak{q}}\mathcal{W}$.
Then

$\displaystyle{\mathrm{ch}}Z_{\mathbf{A}}^{w}(\mu)=e^{\mu-(w(\varrho^{w^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}})}{\mathrm{ch}}U^{-}_{\mathfrak{q}}.$

###### Proof.

By the definition, we see that
${\mathrm{ch}}\,Z_{\mathbf{A}}^{w}(\mu)=e^{\mu}\,{\mathrm{ch}}\,T_{w}(U^{-}_{w^{-*}\mathfrak{q}})$.
On the other hand,
$T_{w}(U^{-}_{w^{-*}\mathfrak{q}})\simeq(U_{w^{-*}\mathfrak{q}}^{-})[w]$ as
$\mathbb{Z}^{\mathbb{I}}$-graded object by (4.9). Let us write
$\Delta_{+}^{\mathfrak{q}}=R_{1}\cup R_{2}$ with

$\displaystyle
R_{1}=\left\\{\beta\in\Delta_{+}^{\mathfrak{q}}\,:\,w^{-1}\beta\in\Delta_{-}^{w^{-*}\mathfrak{q}}\right\\}\quad\mbox{and}\quad
R_{2}=\left\\{\beta\in\Delta_{+}^{\mathfrak{q}}\,:\,w^{-1}\beta\in\Delta_{+}^{w^{-*}\mathfrak{q}}\right\\}.$

Therefore

(6.5)
$\displaystyle\begin{split}{\mathrm{ch}}\,T_{w}(U^{-}_{w^{-*}\mathfrak{q}})\overset{\eqref{eq:w
ch}}{=}w({\mathrm{ch}}\,U^{-}_{w^{-*}\mathfrak{q}})&\overset{\eqref{eq:ch
U-}}{=}\prod_{\gamma\in\Delta_{+}^{w^{-*}\mathfrak{q}}}\frac{1-e^{-b^{w^{-*}\mathfrak{q}}(\gamma)w\gamma}}{1-e^{-w\gamma}}\\\
&=\prod_{\beta\in
R_{1}}\frac{1-e^{b^{\mathfrak{q}}(\beta)\beta}}{1-e^{\beta}}\cdot\prod_{\beta\in
R_{2}}\frac{1-e^{-b^{\mathfrak{q}}(\beta)\beta}}{1-e^{-\beta}}\\\
&=\prod_{\beta\in
R_{1}}e^{(b^{\mathfrak{q}}(\beta)-1)\beta}\cdot\prod_{\beta\in\Delta_{+}^{\mathfrak{q}}}\frac{1-e^{-b^{\mathfrak{q}}(\beta)\beta}}{1-e^{-\beta}}\\\
&\overset{\eqref{eq:w varrho -
varrho}}{=}e^{\varrho^{\mathfrak{q}}-w(\varrho^{w^{-*}\mathfrak{q}})}{\mathrm{ch}}U^{-}_{\mathfrak{q}}\end{split}$

and the lemma follows. ∎

As a consequence, we have that

(6.6)
$\displaystyle{\mathrm{ch}}Z_{\mathbf{A}}(\mu)={\mathrm{ch}}Z_{\mathbf{A}}^{w}(\mu\langle
w\rangle).$

for all $\mu\in\mathbb{Z}^{\mathbb{I}}$ and $w\in{}^{\mathfrak{q}}\mathcal{W}$
with $\mu\langle
w\rangle=\mu-(w(\varrho^{w^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}})$, recall
Definition 3.6. Moreover, we next see that the Hom spaces between these
$w$-Verma modules are free of rank one over $\mathbf{A}$, cf. [1, Lemma 4.7].

######  Proposition 6.2.

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$ and $x,w\in{}^{\mathfrak{q}}\mathcal{W}$.
Then

$\displaystyle\operatorname{Hom}_{\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}}(Z_{\mathbf{A}}^{x}(\mu\langle
x\rangle),Z_{\mathbf{A}}^{w}(\mu\langle
w\rangle))\simeq\mathbf{A}\quad\mbox{and}\quad\operatorname{Hom}_{\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}}(Z_{\mathbf{A}}^{x}(\mu\langle
x\rangle),Z_{\mathbf{A}}^{w}(\mu\langle w\rangle)^{\tau})\simeq\mathbf{A}.$

###### Proof.

Let us abbreviate $M^{x}=Z_{\mathbf{A}}^{x}(\mu\langle x\rangle)$. We have
that $(M^{x})_{\mu\langle x\rangle}$ is $\mathbf{A}$-free of rank $1$ and
$\mu\langle x\rangle+x\beta$ is not a weight for any
$\beta\in\Delta_{+}^{x^{-*}\mathfrak{q}}$ because
${\mathrm{ch}}\,M^{x}=e^{\mu\langle
x\rangle}{\mathrm{ch}}\,T_{x}(U^{-}_{x^{-*}\mathfrak{q}})$. By (6.6), these
claims are also true for $M^{w}$. Thus $(M^{w})_{\mu\langle
x\rangle}=v\mathbf{A}$ and $T_{x}(U^{+}_{x^{-*}\mathfrak{q}})v=0$. Therefore,
there is a morphism $M^{x}\rightarrow M^{w}$, induced by
$\boldsymbol{|}\mu\langle x\rangle\boldsymbol{\rangle}\mapsto v$ and any other
morphism has to be a multiple of it. This shows the first isomorphism. The
second one follows similarly by using (5.15). ∎

######  Corollary 6.3.

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$ and $w\in{}^{\mathfrak{q}}\mathcal{W}$.
Then $Z^{w}_{\mathbf{A}}(\mu\langle w\rangle)$ and $Z_{\mathbf{A}}(\mu)$
belong to the same block.

###### Proof.

It follows as [1, Lemma 6.11] using the above proposition. ∎

###  6.2. Twisted simple modules

Let $w\in\mathcal{W}$. The new triangular decomposition (6.2) on
$U_{\mathfrak{q}}$ satisfies (5.1)–(5.5) with respect to the $w$-twisted
$\mathbb{Z}^{\mathbb{I}}$-grading on $U_{\mathfrak{q}}$ and the partial order
$\leq^{w}$; (5.5) holds thanks to (4.11). Similar to [1, §4.3], this
observation ensures that the $w$-Verma modules satisfy most of the properties
of the usual Verma modules. For instance, we can construct the $w$-Verma
modules via induced functors, they have simple head, and the projectives
modules admit $Z^{w}$-filtrations.

Therefore, in the case of the field $\mathbf{k}$, the $w$-Verma module
$Z^{w}_{\mathbf{k}}(\mu)$ has a unique simple quotient which we denote
$L^{w}_{\mathbf{k}}(\mu)$ for all $\mu\in\mathbb{Z}^{\mathbb{I}}$. As the
simple modules in $\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ are determined by
their highest-weights, $w$ induces a bijection in $\mathbb{Z}^{\mathbb{I}}$,
$\mu\leftrightarrow\mu_{w}$, such that

(6.7) $\displaystyle L^{w}_{\mathbf{k}}(\mu)\simeq L_{\mathbf{k}}(\mu_{w}).$

Notice that if $w_{0}\in{}^{\mathfrak{q}}\mathcal{W}$ is the longest element,
then $\mu_{w_{0}}$ is the lowest-weight of $L_{\mathbf{k}}(\mu)$ by (4.12);
i.e. $(U^{-})_{\nu}\cdot L_{\mathbf{k}}(\mu)_{\mu_{w_{0}}}=0$ for all $\nu<0$.

We next give more information about the $w$-Verma modules over a the field
$\mathbf{k}$.

######  Lemma 6.4 ([1, Lemma 4.8]).

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$ and $w\in{}^{\mathfrak{q}}\mathcal{W}$.
Then the socle of $Z_{\mathbf{k}}^{w}(\mu)$ is a simple module in
$\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$. Furthermore, the element
$T_{w}(F_{top}^{w^{-*}\mathfrak{q}})\boldsymbol{|}\mu\boldsymbol{\rangle}$
generates the socle and spans the homogeneous component of weight
$\mu-w(\beta_{top}^{w^{-*}\mathfrak{q}})$.

###### Proof.

The socle of $T_{w}(U^{-}_{w^{-*}\mathfrak{q}})$ as a module over itself is
spanned by $T_{w}(F_{top}^{w^{-*}\mathfrak{q}})$ whose degree is
$-w(\beta_{top}^{w^{-*}\mathfrak{q}})$. Then any simple submodule of
$Z_{\mathbf{k}}^{w}(\mu)$ in $\mathcal{C}^{\mathfrak{q}}_{\mathbf{k}}$ should
contains
$T_{w}(F_{top}^{w^{-*}\mathfrak{q}})\boldsymbol{|}\mu\boldsymbol{\rangle}$ and
the lemma follows. ∎

Morphisms as in the lemma below exists by Proposition 6.2. Recall that $w_{0}$
denotes the longest element in ${}^{\mathfrak{q}}\mathcal{W}$.

######  Lemma 6.5 ([1, Lemma 4.9]).

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$,

$\displaystyle\Phi:Z_{\mathbf{k}}(\mu)\longrightarrow
Z_{\mathbf{k}}^{w_{0}}(\mu\langle
w_{0}\rangle)\quad\mbox{and}\quad\Phi^{\prime}:Z_{\mathbf{k}}^{w_{0}}(\mu\langle
w_{0}\rangle)\longrightarrow Z_{\mathbf{k}}(\mu)$

be non-zero morphisms in $\mathcal{C}^{\mathfrak{q}}_{\mathbf{k}}$. Then

$\displaystyle
L_{\mathbf{k}}(\mu)\simeq\operatorname{soc}Z_{\mathbf{k}}^{w_{0}}(\mu\langle
w_{0}\rangle)={\mathrm{Im}}\Phi\quad\mbox{and}\quad
L_{\mathbf{k}}^{w_{0}}(\mu\langle
w_{0}\rangle)\simeq\operatorname{soc}Z_{\mathbf{k}}(\mu)={\mathrm{Im}}\Phi^{\prime}.$

###### Proof.

As $\Phi^{\prime}$ is graded, $\Phi^{\prime}$ sends $\boldsymbol{|}\mu\langle
w_{0}\rangle\boldsymbol{\rangle}$ to the generator of the socle of
$Z_{\mathbf{k}}(\mu)$ by Lemma 6.4. This shows the second isomorphism and the
second one follows in a similar way. ∎

######  Example 6.6.

$L^{w_{0}}_{\mathbf{k}}(\mu\langle w_{0}\rangle)=\mathbf{k}^{\mu\langle
w_{0}\rangle}$ is one-dimensional if and only if

(6.8) $\displaystyle\pi\widetilde{\mu\langle
w_{0}\rangle}(K_{i}L_{i}^{-1})=1\quad\forall i\in\mathbb{I}.$

Indeed, this follows as Example 5.3 using (6.3) and (6.4).

###  6.3. Highest weight theory

The proofs of the next results run as in [1].

######  Lemma 6.7 ([1, Lemma 4.10]).

For all $\mu\in\mathbb{Z}^{\mathbb{I}}$, we have that
$Z_{\mathbf{A}}^{w_{0}}(\mu\langle w_{0}\rangle)\simeq
Z_{\mathbf{A}}(\mu)^{\tau}$ and
$Z_{\mathbf{A}}(\mu+\beta_{top}^{\mathfrak{q}})\simeq
Z_{\mathbf{A}}^{w_{0}}(\mu)^{\tau}$. ∎

######  Proposition 6.8 ([1, Propositions 11 and 4.12]).

Let $\lambda,\mu\in\mathbb{Z}^{\theta}$. Then

$\displaystyle\operatorname{{\mathrm{Ext}}}^{n}_{\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}}\left(Z_{\mathbf{A}}(\lambda),Z_{\mathbf{A}}^{w_{0}}(\mu)\right)=\begin{cases}\mathbf{A},&\mbox{if
$n=0$ and $\mu=\lambda\langle w_{0}\rangle$}\\\
0,&\mbox{otherwise.}\end{cases}$

∎

A by-product of the above is that the category
$\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ satisfies all the axioms of the
definition of highest weight category in [14, §3.2] except for the axiom (2)
since $\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ has infinitely many simple
objects.

######  Theorem 6.9.

$\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ is a highest weight category with
infinitely many simple modules $L_{\mathbf{k}}(\lambda)$,
$\lambda\in\mathbb{Z}^{\mathbb{I}}$. The standard and costandard modules are
$Z_{\mathbf{k}}(\lambda)$ and $Z_{\mathbf{k}}(\lambda)^{\tau}$,
$\lambda\in\mathbb{Z}^{\mathbb{I}}$. ∎

Another interesting consequence is the so-called BGG Reciprocity [1,
Proposition 4.15]. Let
$P_{\mathbf{k}}(\lambda)\in\mathcal{C}^{\mathfrak{q}}_{\mathbf{k}}$ be the
projective cover of $L_{\mathbf{k}}(\lambda)$,
$\lambda\in\mathbb{Z}^{\mathbb{I}}$. Recall $P_{\mathbf{k}}(\lambda)$ admits a
$Z$-filtration [1, Lemma 2.16]. Given $\mu\in\mathbb{Z}^{\mathbb{I}}$, we
denote $[P_{\mathbf{k}}(\lambda):Z_{\mathbf{k}}(\mu)]$ the number of
subquotients isomorphic to $Z_{\mathbf{k}}(\mu)$ in a $Z$-filtration of
$P_{\mathbf{k}}(\lambda)$.

######  Theorem 6.10 (BGG Reciprocity).

Let $\lambda,\mu\in\mathbb{Z}^{\mathbb{I}}$. Then

$\displaystyle[P_{\mathbf{k}}(\lambda):Z_{\mathbf{k}}(\mu)]=[Z_{\mathbf{k}}(\mu):L_{\mathbf{k}}(\lambda)].$

###### Proof.

We know that
$[Z_{\mathbf{k}}(\mu):L_{\mathbf{k}}(\lambda)]=\dim\operatorname{Hom}_{\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}}(P_{\mathbf{k}}(\lambda),Z_{\mathbf{k}}(\mu))$
[1, 4.15 $(2)$]. Using Proposition 6.8 we can deduce that
$[P_{\mathbf{k}}(\lambda):Z_{\mathbf{k}}(\mu)]=\dim\operatorname{Hom}_{\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}}\left(P_{\mathbf{k}}(\lambda),Z_{\mathbf{k}}^{w_{0}}(\lambda\langle
w_{0}\rangle)\right)$ which is equal to
$[Z_{\mathbf{k}}^{w_{0}}(\lambda\langle
w_{0}\rangle):L_{\mathbf{k}}(\lambda)]$. Since
${\mathrm{ch}}\,Z_{\mathbf{k}}(\lambda)={\mathrm{ch}}\,Z_{\mathbf{k}}^{w_{0}}(\lambda\langle
w_{0}\rangle)$, Lemma 5.2 implies the equality in the statement. ∎

##  7\. Morphisms between twisted Verma modules

We keep the notation of the previous section. Here we construct generators of
the Hom spaces between twisted Verma modules. Recall they are
$\mathbf{A}$-free of rank one by Proposition 6.2. We will proceed in an
inductive fashion starting from morphisms between a sort of parabolic Verma
modules.

###  7.1. Parabolic Verma modules

We fix $i\in\mathbb{I}$ and write $\sigma_{i}=1^{\mathfrak{q}}\sigma_{i}$. We
also fix $\mu\in\mathbb{Z}^{\mathbb{I}}$. To shorten notation, we write
$\mu^{\prime}=\mu\langle\sigma_{i}\rangle=\mu-(b^{\mathfrak{q}}(\alpha_{i})-1)\alpha_{i}$.

Recall $P_{\mathfrak{q}}(\alpha_{i})$ from (4.18)-(4.19). By construction
$P_{\mathfrak{q}}(\alpha_{i})$ has a triangular decomposition satisfying
(5.1)-(5.5). We denote
${}_{\alpha_{i}}\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$ the corresponding AJS
category. The parabolic Verma module and the parabolic $\sigma_{i}$-Verma
module are

(7.1)
$\displaystyle\Psi_{\mathbf{A}}(\mu)=P_{\mathfrak{q}}(\alpha_{i})\otimes_{U_{\mathfrak{q}}^{0}U_{\mathfrak{q}}^{+}}\mathbf{A}^{\mu}\quad\mbox{and}\quad\Psi^{\prime}_{\mathbf{A}}(\mu^{\prime})=P_{\mathfrak{q}}(\alpha_{i})\otimes_{U^{0}T_{i}(U^{+}_{\sigma_{i}^{*}\mathfrak{q}})}\mathbf{A}^{\mu^{\prime}},$

respectively. These are object of
${}_{\alpha_{i}}\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$.

Clearly, the elements $F_{i}^{t}\boldsymbol{|}\mu\boldsymbol{\rangle}$, $0\leq
t<b^{\mathfrak{q}}(\alpha_{i})$, form a basis of $\Psi_{\mathbf{A}}(\mu)$. By
(4.20), the action of $P_{\mathfrak{q}}(\alpha_{i})$ is given by

(7.2) $\displaystyle E_{i}\cdot
F_{i}^{t}\boldsymbol{|}\mu\boldsymbol{\rangle}$
$\displaystyle=\pi\tilde{\mu}[\alpha_{i};t]\,F_{i}^{t-1}\boldsymbol{|}\mu\boldsymbol{\rangle},$

$E_{j}\cdot F_{i}^{t}\boldsymbol{|}\mu\boldsymbol{\rangle}=0$ for all
$j\in\mathbb{I}\setminus\\{i\\}$ and $F_{i}\cdot
F_{i}^{t}\boldsymbol{|}\mu\boldsymbol{\rangle}=F_{i}^{t+1}\boldsymbol{|}\mu\boldsymbol{\rangle}$.
The weight of $F_{i}^{t}\boldsymbol{|}\mu\boldsymbol{\rangle}$ is
$\mu-t\alpha_{i}$.

In turn, the elements
$E_{i}^{t}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}$, $0\leq
t<b^{\mathfrak{q}}(\alpha_{i})$, form a basis of
$\Psi^{\prime}_{A}(\mu^{\prime})$. By (4.20),

(7.3) $\displaystyle F_{i}\cdot
E_{i}^{t}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}$
$\displaystyle=\pi\widetilde{\mu^{\prime}}[t;\alpha_{i}]\,E_{i}^{t-1}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}.$

These are vectors of weights
$\mu^{\prime}+t\alpha_{i}=\mu-(b^{\mathfrak{q}}(\alpha_{i})-1-t)\alpha_{i}$,
respectively. This implies that $E_{j}\cdot
E_{i}^{t}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}=0$ for all
$j\in\mathbb{I}\setminus\\{i\\}$. Of course, $E_{i}\cdot
E_{i}^{t}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}=E_{i}^{t+1}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}$.

Therefore there exists in
${}_{\alpha_{i}}\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$ a morphism

(7.4) $\displaystyle
f_{i}:\Psi_{\mathbf{A}}(\mu)\longrightarrow\Psi^{\prime}_{\mathbf{A}}(\mu^{\prime})$

such that
$f_{i}(\boldsymbol{|}\mu\boldsymbol{\rangle})=E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-1}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}$.
Indeed, this is the morphism induced by the fact that $E_{j}\cdot
E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-1}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}=0$
for all $j\in\mathbb{I}$. Moreover, any morphism from $\Psi_{\mathbf{A}}(\mu)$
to $\Psi^{\prime}_{\mathbf{A}}(\mu^{\prime})$ is a $\mathbf{A}$-multiple of
$f_{i}$ since the weight spaces are of rank one over $\mathbf{A}$. We can
compute explicitly this morphism as follows:

(7.5) $\displaystyle f_{i}(F_{i}^{n}\boldsymbol{|}\mu\boldsymbol{\rangle})$
$\displaystyle=F_{i}^{n}\cdot
E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-1}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}=\prod_{t=1}^{n}\pi\widetilde{\mu^{\prime}}([b^{\mathfrak{q}}(\alpha_{i})-t;\alpha_{i}])\,E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-1-n}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}$

for all $0\leq n<b^{\mathfrak{q}}(\alpha_{i})$.

Analogously, there exists

(7.6) $\displaystyle
f_{i}^{\prime}:\Psi^{\prime}_{\mathbf{A}}(\mu^{\prime})\longrightarrow\Psi_{\mathbf{A}}(\mu)$

given by

(7.7) $\displaystyle
f_{i}^{\prime}(E_{i}^{n}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle})$
$\displaystyle=E_{i}^{n}\cdot
F_{i}^{b^{\mathfrak{q}}(\alpha_{i})-1}\boldsymbol{|}\mu\boldsymbol{\rangle}=\prod_{t=1}^{n}\pi\widetilde{\mu}([\alpha_{i};b^{\mathfrak{q}}(\alpha_{i})-t])\,F_{i}^{b^{\mathfrak{q}}(\alpha_{i})-1-n}\boldsymbol{|}\mu\boldsymbol{\rangle}.$

######  Lemma 7.1.

If $\pi\widetilde{\mu}([\alpha_{i};t])$ is a unit in $\mathbf{A}$ for all
$1\leq t\leq b^{\mathfrak{q}}(\alpha_{i})-1$, then $f_{i}$ and
$f_{i}^{\prime}$ are isomorphisms.

###### Proof.

The claim for $f_{i}^{\prime}$ follows directly from (7.7) by the hypothesis.
By the formula (7.5), $f_{i}$ is an isomorphism if
$\pi\widetilde{\mu^{\prime}}[b^{\mathfrak{q}}(\alpha_{i})-t;\alpha_{i}]$ is a
unit for all $1\leq t\leq b^{\mathfrak{q}}(\alpha_{i})-1$. We have that
$q_{ii}^{-t-1}\pi\widetilde{\mu^{\prime}}(K_{i}L_{i}^{-1})=q_{ii}^{1-t}\pi\tilde{\mu}(K_{i}L_{i}^{-1})$
and hence (4.22) implies that

(7.8)
$\displaystyle\pi\widetilde{\mu^{\prime}}[b^{\mathfrak{q}}(\alpha_{i})-t;\alpha_{i}]=(b^{\mathfrak{q}}(\alpha_{i})-t)_{q_{ii}^{-1}}\pi\widetilde{\mu^{\prime}}(L_{i})\left(1-q_{ii}^{1-t}\pi\tilde{\mu}(K_{i}L_{i}^{-1})\right).$

Since $(b^{\mathfrak{q}}(\alpha_{i})-t)_{q_{ii}^{-1}}$ and
$\pi\widetilde{\mu^{\prime}}(L_{i})$ are units, $f_{i}$ is an isomorphism by
the hypothesis and (4.21). ∎

We next restrict ourselves to the case of the field $\mathbf{k}$. To simplify
notation, we write

$\displaystyle t_{i}=t^{\pi}_{\alpha_{i}}(\mu),$

recall Definition 4.7. By Lemma 7.1, $f_{i}$ and $f_{i}^{\prime}$ are
isomorphisms in ${}_{\alpha_{i}}\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ if
$t_{i}=0$.

######  Lemma 7.2.

Suppose $t_{i}\neq 0$. Then

$\displaystyle\operatorname{{\mathrm{Ker}}}f_{i}={\mathrm{Im}}f_{i}^{\prime}=\langle\,F_{i}^{n}\boldsymbol{|}\mu\boldsymbol{\rangle}\mid
n\geq
t_{i}\rangle\quad\mbox{and}\quad{\mathrm{Im}}f_{i}=\operatorname{{\mathrm{Ker}}}f_{i}^{\prime}=\langle
E_{i}^{n}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}\mid n\geq
b^{\mathfrak{q}}(\alpha_{i})-t_{i}\rangle.$

###### Proof.

Using (7.5), (7.8) and (4.22), we see that
$\operatorname{{\mathrm{Ker}}}f_{i}=\langle\,F_{i}^{n}\boldsymbol{|}\mu\boldsymbol{\rangle}\mid
n\geq t_{i}\rangle$ and ${\mathrm{Im}}f_{i}=\langle
E_{i}^{n}\boldsymbol{|}\mu\boldsymbol{\rangle}\mid n\geq
b^{\mathfrak{q}}(\alpha_{i})-t_{i}\rangle$.

For $f_{i}^{\prime}$, we first observe that
$b^{\mathfrak{q}}(\alpha_{i})-t_{i}$ is the minimum natural number $s_{i}$
such that $q_{ii}^{1+s_{i}}\pi\widetilde{\mu}(K_{i}L_{i}^{-1})-1=0$. Indeed,
$q_{ii}^{1+s_{i}}\pi\widetilde{\mu}(K_{i}L_{i}^{-1})-1=0=q_{ii}^{1-t_{i}}\pi\tilde{\mu}(K_{i}L_{i}^{-1})-1$
implies $q_{ii}^{t_{i}+s_{i}}=1$, recall (4.21). Hence
$b^{\mathfrak{q}}(\alpha_{i})=\operatorname{{\mathrm{ord}}}q_{ii}\leq
t_{i}+s_{i}$. On the other hand,
$\pi\tilde{\mu}([\alpha_{i};b^{\mathfrak{q}}(\alpha_{i})-s_{i}])=0$ and hence
$t_{i}\leq b^{\mathfrak{q}}(\alpha_{i})-s_{i}$. Therefore
$b^{\mathfrak{q}}(\alpha_{i})=t_{i}+s_{i}$.

Finally, the equalities for ${\mathrm{Im}}f_{i}^{\prime}$ and
$\operatorname{{\mathrm{Ker}}}f_{i}^{\prime}$ follow from (7.7) since
$\pi\widetilde{\mu}([\alpha_{i};b^{\mathfrak{q}}(\alpha_{i})-t])=(b^{\mathfrak{q}}(\alpha_{i})-t)_{q_{ii}}\pi\widetilde{\mu}(L_{i})(q_{ii}^{1+t}\pi\widetilde{\mu}(K_{i}L_{i}^{-1})-1)$.
∎

As a consequence of the above lemma we see that
$F_{i}^{t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}$ is a highest-weight
vector, i.e. $E_{j}\cdot F_{i}^{t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}=0$
for all $j\in\mathbb{I}$, since the weights of
$\operatorname{{\mathrm{Ker}}}f_{i}$ are $\mu-n\alpha_{i}$ with $n\geq t_{i}$.
Therefore there exists a morphism

(7.9) $\displaystyle g_{i}:\Psi_{\mathbf{k}}(\mu-
t_{i}\alpha_{i})\longrightarrow\Psi_{\mathbf{k}}(\mu)$

in ${}_{\alpha_{i}}\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ such that
$\boldsymbol{|}\mu-t_{i}\alpha_{i}\boldsymbol{\rangle}\mapsto
F_{i}^{t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}$. Clearly,
${\mathrm{Im}}g_{i}=\operatorname{{\mathrm{Ker}}}f_{i}$.

######  Lemma 7.3.

There exists a long exact sequence

$\displaystyle\cdots\Psi_{\mathbf{k}}(\mu-(b^{\mathfrak{q}}(\alpha_{i})+t_{i})\alpha_{i})\longrightarrow\Psi_{\mathbf{k}}(\mu-b^{\mathfrak{q}}(\alpha_{i})\alpha_{i})\longrightarrow\Psi_{\mathbf{k}}(\mu-
t_{i}\alpha_{i})\overset{g_{i}}{\longrightarrow}\Psi_{\mathbf{k}}(\mu)\longrightarrow\cdots$

###### Proof.

The kernel of $g_{i}$ is generated by
$F_{i}^{b^{\mathfrak{q}}(\alpha_{i})-t_{i}}\boldsymbol{|}\mu-
t_{i}\alpha_{i}\boldsymbol{\rangle}$ since $g_{i}(F_{i}^{n}\boldsymbol{|}\mu-
t_{i}\alpha_{i}\boldsymbol{\rangle})=F_{i}^{n+t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}$.
Hence this generator is a highest-weight vector. Therefore we have a morphism
$\Psi_{\mathbf{k}}(\mu-b^{\mathfrak{q}}(\alpha_{i})\alpha_{i})\longrightarrow\Psi_{\mathbf{k}}(\mu-
t_{i}\alpha_{i})$ and its kernel is generated by
$F_{i}^{t_{i}}\boldsymbol{|}\mu-b^{\mathfrak{q}}(\alpha_{i})\alpha_{i}\boldsymbol{\rangle}$.
We can repeat the arguments with $\mu-b^{\mathfrak{q}}(\alpha_{i})\alpha_{i}$
instead of $\mu$ in order to construct the desired long exact sequence. ∎

######  Remark 7.4.

In a similar way, we can see that $T_{i}(E_{j})\cdot
E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-t_{i}}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}=0$
for all $j\in\mathbb{I}$. Therefore there exists a morphism
$g_{i}^{\prime}:\Psi_{\mathbf{k}}^{\prime}(\mu^{\prime}+(b^{\mathfrak{q}}(\alpha_{i})-t_{i})\alpha_{i})\longrightarrow\Psi_{\mathbf{k}}^{\prime}(\mu^{\prime})$
in ${}_{\alpha_{i}}\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ such that
$\boldsymbol{|}\mu^{\prime}+(b^{\mathfrak{q}}(\alpha_{i})-t_{i})\alpha_{i}\boldsymbol{\rangle}\mapsto
E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-t_{i}}\boldsymbol{|}\mu^{\prime}\boldsymbol{\rangle}$
and ${\mathrm{Im}}g_{i}^{\prime}=\operatorname{{\mathrm{Ker}}}f_{i}^{\prime}$.

###  7.2. Morphisms between Verma modules twisted by a simple reflection

Here we lift the morphisms of the above subsection to morphisms between actual
Verma modules. We continue with the same fixed elements $i\in\mathbb{I}$ and
$\mu\in\mathbb{Z}^{\mathbb{I}}$. Recall $t_{i}=t^{\pi}_{\alpha_{i}}(\mu)$.

We can construct the Verma modules inducing from the parabolic ones. Namely,
we have the next isomorphisms in $\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$:

(7.10) $\displaystyle Z_{\mathbf{A}}(\mu)$ $\displaystyle\simeq
U_{\mathfrak{q}}\otimes_{P_{\mathfrak{q}}(\alpha_{i})}(P_{\mathfrak{q}}(\alpha_{i})\otimes_{U_{\mathfrak{q}}^{0}U_{\mathfrak{q}}^{+}}\mathbf{A}^{\mu})\simeq
U_{\mathfrak{q}}\otimes_{P_{\mathfrak{q}}(\alpha_{i})}\Psi_{\mathbf{A}}(\mu)\quad\mbox{and}\quad$
(7.11) $\displaystyle Z^{\sigma_{i}}_{\mathbf{A}}(\mu)$ $\displaystyle\simeq
U_{\mathfrak{q}}\otimes_{P_{\mathfrak{q}}(\alpha_{i})}(P_{\mathfrak{q}}(\alpha_{i})\otimes_{U_{\mathfrak{q}}^{0}T_{i}(U^{+}_{\sigma_{i}^{*}\mathfrak{q}})}\mathbf{A}^{\mu})\simeq
U_{\mathfrak{q}}\otimes_{P_{\mathfrak{q}}(\alpha_{i})}\Psi^{\prime}_{\mathbf{A}}(\mu).$

These isomorphisms allow us to lift $f_{i}$ and $f_{i}^{\prime}$ to morphisms
in $\mathcal{C}^{\mathfrak{q}}_{\mathbf{A}}$.

######  Lemma 7.5.

The morphisms

(7.12) $\displaystyle\varphi=1\otimes f_{i}:Z_{\mathbf{A}}(\mu)\longrightarrow
Z^{\sigma_{i}}_{\mathbf{A}}(\mu\langle\sigma_{i}\rangle)\quad\mbox{and}\quad\varphi^{\prime}=1\otimes
f_{i}^{\prime}:Z^{\sigma_{i}}_{\mathbf{A}}(\mu\langle\sigma_{i}\rangle)\longrightarrow
Z_{\mathbf{A}}(\mu)$

are generators of the respective Hom spaces as $\mathbf{A}$-modules. Also,

(7.13) $\displaystyle\operatorname{{\mathrm{Ker}}}\varphi\simeq
U_{\mathfrak{q}}\otimes_{P_{\mathfrak{q}}(\alpha_{i})}\operatorname{{\mathrm{Ker}}}f_{i}\quad\mbox{and}\quad\operatorname{{\mathrm{Ker}}}\varphi^{\prime}\simeq
U_{\mathfrak{q}}\otimes_{P_{\mathfrak{q}}(\alpha_{i})}\operatorname{{\mathrm{Ker}}}f_{i}^{\prime}.$

###### Proof.

On the space of weight $\mu$, $\varphi$ is an isomorphism by construction.
Then $\varphi$ is a generator of
$\operatorname{Hom}_{\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}}(Z_{\mathbf{A}}(\mu),Z^{\sigma_{i}}_{\mathbf{A}}(\mu\langle\sigma_{i}\rangle))$
by Proposition 6.2. By the PBW basis (4.15), $U_{\mathfrak{q}}$ is free over
$P_{\mathfrak{q}}(\alpha_{i})$ and hence
$\operatorname{{\mathrm{Ker}}}\varphi\simeq
U_{\mathfrak{q}}\otimes_{P_{\mathfrak{q}}(\alpha_{i})}\operatorname{{\mathrm{Ker}}}f_{i}$.
The proof for $\varphi^{\prime}$ is analogous. ∎

From the above considerations we can extend immediately to $\varphi$ and
$\varphi^{\prime}$ Lemmas 7.1-7.3, cf. [1, Lemmas 5.8 and 5.9].

######  Lemma 7.6.

Assume that $\pi\tilde{\mu}([\alpha_{i};t])$ is a unit in $\mathbf{A}$ for all
$1\leq t\leq b^{\mathfrak{q}}(\alpha_{i})-1$. Then $\varphi$ and
$\varphi^{\prime}$ are isomorphisms in
$\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$. ∎

######  Lemma 7.7.

If $t_{i}=0$, then $L_{\mathbf{k}}(\mu)\simeq
L_{\mathbf{k}}^{\sigma_{i}}(\mu\langle\sigma_{i}\rangle)$ in
$\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$. ∎

######  Lemma 7.8.

Suppose $t_{i}\neq 0$. Then

$\displaystyle\operatorname{{\mathrm{Ker}}}\varphi={\mathrm{Im}}\varphi^{\prime}=U_{\mathfrak{q}}\cdot
F_{i}^{t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}\quad\mbox{and}\quad{\mathrm{Im}}\varphi=\operatorname{{\mathrm{Ker}}}\varphi^{\prime}=U_{\mathfrak{q}}\cdot
E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}.$

∎

Moreover, if $t_{i}\neq 0$, we have that

(7.14)
$\displaystyle{\mathrm{ch}}\operatorname{{\mathrm{Ker}}}\varphi=e^{\mu}\left(e^{-t_{i}\alpha_{i}}+\cdots+e^{(1-b^{\mathfrak{q}}(\alpha_{i}))\alpha_{i}}\right)\prod_{\gamma\in\Delta_{+}^{\mathfrak{q}}\setminus\\{\alpha_{i}\\}}\frac{1-e^{-b^{\mathfrak{q}}(\gamma)\gamma}}{1-e^{-\gamma}}.$

because of the PBW basis (4.15) and (7.13).

The morphism $\psi$ below is induced by $g_{i}$ of (7.9).

######  Lemma 7.9.

Suppose $t_{i}\neq 0$. Then
$F_{i}^{t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}$ is a highest-weight
vector of $Z_{\mathbf{k}}(\mu)$ and hence there exists a morphism

$\displaystyle\psi:Z_{\mathbf{k}}(\mu-t_{i}\alpha_{i})\longrightarrow
Z_{\mathbf{k}}(\mu)$

in $\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ such that $\boldsymbol{|}\mu-
t_{i}\alpha_{i}\boldsymbol{\rangle}\mapsto
F_{i}^{t_{i}}\boldsymbol{|}\mu\boldsymbol{\rangle}$. Moreover,
${\mathrm{Im}}\psi=\operatorname{{\mathrm{Ker}}}\varphi$ and there is a long
exact sequence

$\displaystyle\cdots
Z_{\mathbf{k}}(\mu-(b^{\mathfrak{q}}(\alpha_{i})+t_{i})\alpha_{i})\longrightarrow
Z_{\mathbf{k}}(\mu-b^{\mathfrak{q}}(\alpha_{i})\alpha_{i})\longrightarrow
Z_{\mathbf{k}}(\mu-
t_{i}\alpha_{i})\overset{\psi}{\longrightarrow}Z_{\mathbf{k}}(\mu)\longrightarrow\cdots$

∎

######  Remark 7.10.

There is a morphism
$\psi^{\prime}:Z_{\mathbf{k}}^{\sigma_{i}}(\mu-(t_{i}-1)\alpha_{i})\longrightarrow
Z_{\mathbf{k}}^{\sigma_{i}}(\mu\langle\sigma_{i}\rangle)$ in
$\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$, induced by $\psi^{\prime}$ of
Remark 7.4, such that
$\boldsymbol{|}\mu-(t_{i}-1)\alpha_{i}\boldsymbol{\rangle}\mapsto
E_{i}^{b^{\mathfrak{q}}(\alpha_{i})-t_{i}}\boldsymbol{|}\mu\langle\sigma_{i}\rangle\boldsymbol{\rangle}$
and
${\mathrm{Im}}\psi^{\prime}=\operatorname{{\mathrm{Ker}}}\varphi^{\prime}$.

From the long exact sequence, we see that

$\displaystyle{\mathrm{ch}}(\operatorname{{\mathrm{Ker}}}\varphi)=\sum_{\ell\geq
0}{\mathrm{ch}}Z_{\mathbf{k}}(\mu-(\ell
b^{\mathfrak{q}}(\alpha_{i})-t_{i})\alpha_{i})-\sum_{\ell\geq
1}{\mathrm{ch}}Z_{\mathbf{k}}(\mu-\ell
b^{\mathfrak{q}}(\alpha_{i})\alpha_{i}).$

As another consequence we obtain the next isomorphisms between simple modules.

######  Proposition 7.11.

Suppose $t_{i}\neq 0$. In $\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$, it holds
that

$\displaystyle L_{\mathbf{k}}(\mu)\simeq
L^{\sigma_{i}}_{\mathbf{k}}(\mu-(t_{i}-1)\alpha_{i})\quad\mbox{and}\quad
L_{\mathbf{k}}(\mu-t_{i}\alpha_{i})\simeq
L_{\mathbf{k}}^{\sigma_{i}}(\mu\langle\sigma_{i}\rangle).$

###### Proof.

Since ${\mathrm{Im}}\varphi={\mathrm{Im}}\psi^{\prime}$ and
${\mathrm{Im}}\psi={\mathrm{Im}}\varphi^{\prime}$, the respective domains must
to have isomorphic heads, which are the isomorphisms in the statement. ∎

######  Remark 7.12.

Maps similar to $\varphi$ were constructed in [26, Lemma 5.8]. Those maps are
just linear morphism between Verma modules in different categories. By
considering the $\sigma_{i}$-Verma module, we obtain morphisms between objects
in the same category.

###  7.3. Twisting the categories

Let $w:\mathfrak{q}\longrightarrow w^{*}\mathfrak{q}$ be a morphism in
$\mathcal{W}$ and $w=\sigma_{i_{k}}\cdots\sigma_{i_{1}}1^{\mathfrak{q}}$ a
reduced expression. Fix the Lusztig isomorphism

$\displaystyle T_{w}=T_{i_{1}}\cdots T_{i_{k}}:U_{\mathfrak{q}}\longrightarrow
U_{w^{*}\mathfrak{q}}.$

Recall the $U_{\mathfrak{q}}^{0}$-algebra $\mathbf{A}[w]$ from (4.23) with
structural map

(7.15) $\displaystyle\pi[w]=\pi\circ
T_{w}^{-1}{}_{|U_{\mathfrak{q}}^{0}}:U_{\mathfrak{q}}^{0}\longrightarrow\mathbf{A}.$

This does not depend on the presentation of $w$ by (4.10). Also,
$\mathbf{A}[w][x]=\mathbf{A}[xw]$ for $x:w^{*}\mathfrak{q}\longrightarrow
x^{*}(w^{*}\mathfrak{q})=(wx)^{*}\mathfrak{q}$.

We have equivalence of categories
${}^{w}F^{\mathfrak{q}}_{\mathbf{A}}:\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}\longrightarrow\mathcal{C}^{w^{*}\mathfrak{q}}_{\mathbf{A}[w]}$
given by: if $M\in\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$, then
$F_{w}(M)=M[w]$ is an object of
$\mathcal{C}^{w^{*}\mathfrak{q}}_{\mathbf{A}[w]}$ with the action of
$U_{w^{*}\mathfrak{q}}$ twisted by $T_{w}^{-1}$, that is

$\displaystyle u\cdot_{T_{w}^{-1}}m=T_{w}^{-1}(u)m\quad\forall m\in
M[w],\,u\in U_{w^{*}\mathfrak{q}}.$

Indeed $M[w]$ satisfies (5.8), since

$\displaystyle(U_{w^{*}\mathfrak{q}})_{\alpha}\cdot_{T_{w}^{-1}}M[w]_{\mu}$
$\displaystyle\overset{\eqref{eq:Tw U alpha es U w
alpha}}{=}(U_{\mathfrak{q}})_{w^{-1}\alpha}\,M_{w^{-1}\mu}\subset
M_{w^{-1}\alpha+w^{-1}\mu}=M_{w^{-1}(\alpha+\mu)}=M[w]_{\alpha+\mu}.$

It also satisfies (5.9), since for $m\in M[w]_{\mu}=M_{w^{-1}\mu}$ and $s\in
U^{0}_{w^{*}\mathfrak{q}}$ we have that

$\displaystyle
s\cdot_{T_{w}^{-1}}m=T_{w}^{-1}(s)m=m(\pi\circ\widetilde{w^{-1}(\mu)}\circ
T_{w}^{-1})(s)\overset{\eqref{eq: mu conjugado por Tw}}{=}m(\pi\circ
T_{w}^{-1}\circ\widetilde{\mu})(s)=m\pi[w](\widetilde{\mu}(s))$

Clearly, ${}^{w}F^{\mathfrak{q}}_{\mathbf{A}}$ depends on the reduced
expression of $w$. Its inverse
${}^{w}G^{\mathfrak{q}}_{\mathbf{A}}:\mathcal{C}^{w^{*}\mathfrak{q}}_{\mathbf{A}[w]}\longrightarrow\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$
is given by ${}^{w}G^{\mathfrak{q}}_{\mathbf{A}}(M)=M[w^{-1}]$ with the action
of $U_{\mathfrak{q}}$ twisted by $T_{w}$. When no confusion can arise, we will
write simply $M[w]$ and $M[w^{-1}]$ instead of
${}^{w}F^{\mathfrak{q}}_{\mathbf{A}}(M)$ and
${}^{w}G^{\mathfrak{q}}_{\mathbf{A}}(M)$, respectively.

The Verma modules of both categories are related as follows.

######  Lemma 7.13.

Let $x\in{}^{\mathfrak{q}}\mathcal{W}$, $w\in\mathcal{W}^{\mathfrak{q}}$ and
$\mu\in\mathbb{Z}^{\mathbb{I}}$. Then

(7.16) $\displaystyle Z_{\mathbf{A}}^{x}(\mu)[w]\simeq
Z_{\mathbf{A}[w]}^{wx}(w\mu)$

in the category $\mathcal{C}_{\mathbf{A}[w]}^{w^{*}\mathfrak{q}}$. Moreover,
for the field $\mathbf{k}$, it holds that

(7.17) $\displaystyle L_{\mathbf{k}}^{x}(\mu)[w]\simeq
L_{\mathbf{k}[w]}^{wx}(w\mu).$

###### Proof.

We can repeat the arguments in [1, 4.4$(2)$] but we must be thorough with the
categories where we are considering the Verma modules.

We first observe that $Z_{\mathbf{A}}^{x}(\mu)$ has not weights of the form
$\mu+x\beta$ for all $\beta\in\Delta_{+}^{x^{-*}\mathfrak{q}}$ because
${\mathrm{ch}}\,Z_{\mathbf{A}}^{x}(\mu)=e^{\mu}{\mathrm{ch}}\,T_{x}(U^{-}_{x^{-*}\mathfrak{q}})$.
Therefore $Z_{\mathbf{A}}^{x}(\mu)[w]$ has not weights of the form
$w\mu+wx\beta$ for all $\beta\in\Delta_{+}^{x^{-*}\mathfrak{q}}$. Also, the
space of weight $w\mu$ of $Z_{\mathbf{A}}^{x}(\mu)[w]$ is $\mathbf{A}$-spanned
by $v_{1}=\boldsymbol{|}\mu\boldsymbol{\rangle}_{\mathbf{A}}$. In particular,
we have that $T_{wx}(U^{+}_{x^{-*}\mathfrak{q}})$ annuls $v_{1}$.

On the other hand, we are thinking of $Z_{\mathbf{A}[w]}^{wx}(w\mu)$ as an
object in $\mathcal{C}_{\mathbf{A}[w]}^{w^{*}\mathfrak{q}}$. Thus, it is
constructed using the triangular decomposition of $U_{w^{*}\mathfrak{q}}$
induced by $T_{wx}:U_{x^{-*}\mathfrak{q}}\rightarrow U_{w^{*}\mathfrak{q}}$.
This means that its generator
$v_{2}=\boldsymbol{|}w\mu\boldsymbol{\rangle}_{\mathbf{A}[w]}$ is annuled by
$T_{wx}(U^{+}_{x^{-*}\mathfrak{q}})$. Therefore there exists a morphism
$f:Z_{\mathbf{A}[w]}^{wx}(w\mu)\longrightarrow Z_{\mathbf{A}}^{x}(\mu)[w]$
given by $v_{2}\mapsto v_{1}$.

In a similar fashion, we get a morphism
$Z_{\mathbf{A}}^{x}(\mu)\longrightarrow Z_{\mathbf{A}[w]}^{wx}(w\mu)[w^{-1}]$
in $\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$ such that $v_{1}\mapsto v_{2}$.
Then, we can transform it into a morphism
$g:Z_{\mathbf{A}}^{x}(\mu)[w]\longrightarrow Z_{\mathbf{A}[w]}^{wx}(w\mu)$ in
$\mathcal{C}_{\mathbf{A}[w]}^{w^{*}\mathfrak{q}}$ such that $v_{1}\mapsto
v_{2}$. Clearly, $g\circ f={\mathrm{id}}$ and $g[w^{-1}]\circ
f[w^{-1}]={\mathrm{id}}$. This proves the isomorphisms between the Verma
modules. Therefore, in the case of the field $\mathbf{k}$, we deduce that
their heads also are isomorphic. ∎

###  7.4. Morphisms between twisted Verma modules

Here we extend the results of §7.2 to any morphism in the Weyl groupoid using
the functors of §7.3 and following the ideas of [1, §5.10].

We fix $w\in{}^{\mathfrak{q}}\mathcal{W}$ and $\beta\in\Delta^{\mathfrak{q}}$
such that $\beta=w\alpha_{i}$ for fixed
$\alpha_{i}\in\Pi^{w^{-*}\mathfrak{q}}$ and $i\in\mathbb{I}$. We also fix
$\mu\in\mathbb{Z}^{\mathbb{I}}$. We set $t_{\beta}=t_{\beta}^{\pi}(\mu)$,
recall Definition 4.7. We will use the functor
${}^{w}F_{\mathbf{A}[w^{-1}]}^{w^{-*}\mathfrak{q}}:\mathcal{C}_{\mathbf{A}[w^{-1}]}^{w^{-*}\mathfrak{q}}\longrightarrow\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$.
For abbreviation, we will denote $M[w]$ and $f[w]$ the image of objects and
morphisms through this functor.

Using ${}^{w}F_{\mathbf{A}[w^{-1}]}^{w^{-*}\mathfrak{q}}$, Lemma 7.13 implies
that

(7.18) $\displaystyle\begin{split}Z_{\mathbf{A}[w^{-1}]}(w^{-1}\mu)[w]&\simeq
Z_{\mathbf{A}}^{w}(\mu)\quad\mbox{and}\\\
&Z_{\mathbf{A}[w^{-1}]}^{\sigma_{i}}((w^{-1}\mu)^{\prime})[w]\simeq
Z_{\mathbf{A}}^{w\sigma_{i}}(\mu-(b^{\mathfrak{q}}(\beta)-1)\beta)\end{split}$

for
$(w^{-1}\mu)^{\prime}=(w^{-1}\mu)\langle\sigma_{i}\rangle=w^{-1}\mu-(b^{w^{-*}\mathfrak{q}}(\alpha_{i})-1)\alpha_{i}$;
for the first isomorphism take $x=1^{w^{-*}\mathfrak{q}}$ and for the second
one $x=\sigma_{i}^{(\sigma_{i}w)^{-*}\mathfrak{q}}$.

On the other hand, let $\varphi$ and $\varphi^{\prime}$ be the morphisms in
the category $\mathcal{C}_{\mathbf{A}[w^{-1}]}^{w^{-*}\mathfrak{q}}$ between
$Z_{\mathbf{A}[w^{-1}]}(w^{-1}\mu)$ and
$Z_{\mathbf{A}[w^{-1}]}^{\sigma_{i}}((w^{-1}\mu)^{\prime})$ given in (7.12).
We apply to them the functor
${}^{w}F_{\mathbf{A}[w^{-1}]}^{w^{-*}\mathfrak{q}}$ and obtain the morphisms

(7.19)
$\displaystyle\begin{split}\varphi[w]&:Z_{\mathbf{A}}^{w}(\mu)\longrightarrow
Z^{w\sigma_{i}}_{\mathbf{A}}(\mu-(b^{\mathfrak{q}}(\beta)-1)\beta)\quad\mbox{and}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\varphi^{\prime}[w]:Z^{w\sigma_{i}}_{\mathbf{A}}(\mu-(b^{\mathfrak{q}}(\beta)-1)\beta)\longrightarrow
Z_{\mathbf{A}}^{w}(\mu)\quad\mbox{in
$\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$}.\end{split}$

We are ready to extend the results from §7.2.

######  Lemma 7.14.

If $\pi\tilde{\mu}([\beta;t])$ is a unit for all $1\leq t\leq
b^{\mathfrak{q}}(\beta)-1$, then $\varphi[w]$ and $\varphi^{\prime}[w]$ are
isomorphisms in $\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$.

###### Proof.

By Lemma 7.6, $\varphi$ and $\varphi^{\prime}$ are isomorphisms if
$\pi[w^{-1}]\widetilde{w^{-1}\mu}([\alpha_{i};t])$ is a unit for all $1\leq
t\leq b^{w^{-*}\mathfrak{q}}(\alpha_{i})-1=b^{\mathfrak{q}}(\beta)-1$. As
$\pi[w^{-1}]\widetilde{w^{-1}\mu}([\alpha_{i};t])=\pi\tilde{\mu}([\beta;t])$
by (4.25), the lemma follows from the hypothesis. ∎

In the case of the field $\mathbf{k}$, we immediately deduce an isomorphism
between the heads of the Verma modules.

######  Lemma 7.15.

If $t_{\beta}=0$, then $L_{\mathbf{k}}^{w}(\mu)\simeq
L_{\mathbf{k}}^{w\sigma_{i}}(\mu-(b^{\mathfrak{q}}(\beta)-1)\beta)$ in
$\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$. ∎

Suppose now $t_{\beta}\neq 0$. Since
$t_{\alpha_{i}}^{\pi[w^{-1}]}(w^{-1}\mu)=t_{\beta}$ by (4.24), we can consider
the morphisms $\psi$ and $\psi^{\prime}$ between
$Z_{\mathbf{k}[w^{-1}]}(w^{-1}\mu)$ and
$Z_{\mathbf{k}[w^{-1}]}^{\sigma_{i}}((w^{-1}\mu)^{\prime})$ in the category
$\mathcal{C}_{\mathbf{k}[w^{-1}]}^{w^{-*}\mathfrak{q}}$ given by Lemma 7.9. By
applying the functor ${}^{w}F_{\mathbf{k}[w^{-1}]}^{w^{-*}\mathfrak{q}}$ to
this lemma, we obtain the following.

######  Lemma 7.16.

Suppose $t_{\beta}\neq 0$. Then morphisms

$\displaystyle\psi[w]:Z_{\mathbf{k}}^{w}(\mu-t_{\beta}\beta)$
$\displaystyle\longrightarrow Z_{\mathbf{k}}^{w}(\mu)\quad\mbox{and}$
$\displaystyle\psi^{\prime}[w]:Z_{\mathbf{k}}^{w\sigma_{i}}(\mu-(t_{\beta}-1)\beta)\longrightarrow
Z_{\mathbf{k}}^{w\sigma_{i}}(\mu-(b^{\mathfrak{q}}(\beta)-1)\beta)$

in $\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$ satisfy that

$\displaystyle\operatorname{{\mathrm{Ker}}}\varphi[w]={\mathrm{Im}}\varphi^{\prime}[w]={\mathrm{Im}}\psi[w]\quad\mbox{and}\quad{\mathrm{Im}}\varphi[w]=\operatorname{{\mathrm{Ker}}}\varphi^{\prime}[w]={\mathrm{Im}}\psi^{\prime}[w].$

We also have a long exact sequence

$\displaystyle\cdots
Z^{w}_{\mathbf{k}}(\mu-(b^{\mathfrak{q}}(\beta)+t_{\beta})\beta)\longrightarrow
Z^{w}_{\mathbf{k}}(\mu-b^{\mathfrak{q}}(\beta)\beta)\longrightarrow
Z^{w}_{\mathbf{k}}(\mu-t_{\beta}\beta)\longrightarrow
Z^{w}_{\mathbf{k}}(\mu)\longrightarrow\cdots$

∎

Similar to Proposition 7.11, we deduce an isomorphism between the heads of the
$w$-Verma modules above.

######  Proposition 7.17.

Suppose $t_{\beta}\neq 0$. In $\mathcal{C}_{\mathbf{k}}^{\mathfrak{q}}$, it
holds that

$\displaystyle L_{\mathbf{k}}^{w}(\mu)\simeq
L_{\mathbf{k}}^{w\sigma_{i}}(\mu-(t_{\beta}-1)\beta)\quad\mbox{and}\quad
L_{\mathbf{k}}^{w}(\mu-t_{\beta}\beta)\simeq
L_{\mathbf{k}}^{w\sigma_{i}}(\mu-(b^{\mathfrak{q}}(\beta)-1)\beta).$

∎

######  Remark 7.18.

Using iteratively the prior lemma, we can calculate
$\mu_{w}\in\mathbb{Z}^{\mathbb{I}}$ such that
$L_{\mathbf{k}}^{w}(\mu)=L_{\mathbf{k}}(\mu_{w})$, recall (6.7).

Let us make some comments about the kernel of $\varphi[w]$ in the case of the
field $\mathbf{k}$. First, using (7.14) and
$\operatorname{{\mathrm{Ker}}}(\varphi[w])=\operatorname{{\mathrm{Ker}}}(\varphi)[w]$,
we have that

$\displaystyle{\mathrm{ch}}\operatorname{{\mathrm{Ker}}}(\varphi[w])=e^{\mu}\left(e^{-t_{\beta}\beta}+\cdots+\right.$
$\displaystyle\left.e^{(1-b^{\mathfrak{q}}(\beta))\beta}\right)\times$ (7.20)
$\displaystyle\prod_{\gamma\in\Delta^{\mathfrak{q}}_{+}\setminus\\{\beta\\}:w^{-1}\gamma\in\Delta^{w^{-*}\mathfrak{q}}_{+}}$
$\displaystyle\left(1+e^{-\gamma}+\cdots+e^{(1-b^{\mathfrak{q}}(\gamma))\gamma}\right)\times$
$\displaystyle\prod_{\gamma\in\Delta^{\mathfrak{q}}_{+}\setminus\\{\beta\\}:w^{-1}\gamma\in\Delta^{w^{-*}\mathfrak{q}}_{-}}\left(1+e^{\gamma}+\cdots+e^{(b^{\mathfrak{q}}(\gamma)-1)\gamma}\right).$

Then, if $\beta=w\alpha_{i}\in\Delta^{\mathfrak{q}}_{+}$, it follows that

(7.21)
$\displaystyle\mu-(w(\varrho^{w^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}})\mbox{
is not a weight of }\operatorname{{\mathrm{Ker}}}(\varphi[w])$

since
$\mu-(w(\varrho^{w^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}})=\mu+\sum_{\gamma\in\Delta_{+}^{\mathfrak{q}}\,:\,w^{-1}\gamma\in\Delta_{-}^{w^{-*}\mathfrak{q}}}(b^{\mathfrak{q}}(\gamma)-1)\gamma$
by (3.12). Lastly, we claim that

(7.22) $\displaystyle
T_{w}(F_{i})^{t_{\beta}}\boldsymbol{|}\mu\boldsymbol{\rangle}_{\mathbf{k}}\mbox{
is a $U_{\mathfrak{q}}$-generator of
}\operatorname{{\mathrm{Ker}}}(\varphi[w])$

where $T_{w}:U_{w^{-*}\mathfrak{q}}\longrightarrow U_{\mathfrak{q}}$ is a
Lusztig isomorphism associated to $w$. In fact, $\psi(\boldsymbol{|}w^{-1}\mu-
t_{\beta}\alpha_{i}\boldsymbol{\rangle}_{\mathbf{k}[w^{-1}]})=F_{i}^{t_{\beta}}\boldsymbol{|}w^{-1}\mu\boldsymbol{\rangle}_{\mathbf{k}[w^{-1}]}$
by Lemma 7.9. From the proof of Lemma 7.13, we see that
$\psi[w](\boldsymbol{|}\mu-
t_{\beta}\beta\boldsymbol{\rangle}_{\mathbf{k}})=g(F_{i}^{t_{\beta}}\boldsymbol{|}w^{-1}\mu\boldsymbol{\rangle}_{\mathbf{k}[w^{-1}]})=g(T_{w}(F_{i}^{t_{\beta}})\cdot_{T_{w}^{-1}}\boldsymbol{|}w^{-1}\mu\boldsymbol{\rangle}_{\mathbf{k}[w^{-1}]})=T_{w}(F_{i}^{t_{\beta}})\boldsymbol{|}\mu\boldsymbol{\rangle}_{\mathbf{k}}$.

####  7.4.1. A generator

We end this subsection by constructing a generator of the Hom spaces between
twisted Verma modules as in Proposition 6.2. We follow the strategy in [1,
§5.13].

We fix $x:x^{-*}\mathfrak{q}\rightarrow\mathfrak{q}$ and a reduced expression
$x^{-1}w=1^{x^{-*}\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{r}}$. We set

$\displaystyle x_{s}=1^{\mathfrak{q}}x\sigma_{i_{1}}\cdots\sigma_{i_{s-1}}$

for $1\leq s\leq r+1$. Then

$\displaystyle\mu\langle x_{s+1}\rangle=\mu\langle
x_{s}\rangle-(b^{\mathfrak{q}}(x_{s}\alpha_{i_{s}})-1)x_{s}\alpha_{i_{s}}$

for all $1\leq s\leq r$ by (3.13). Notice $x_{1}=x$ and $x_{r+1}=w$. We set
$\varphi_{s}=\varphi[x_{s}]$ the morphism in
$\mathcal{C}_{\mathbf{A}}^{\mathfrak{q}}$ given by (7.19) for $\mu\langle
x_{s}\rangle$. Explicitly,

$\displaystyle\varphi_{s}:Z_{\mathbf{A}}^{x_{s}}(\mu\langle
x_{s}\rangle)\longrightarrow Z_{\mathbf{A}}^{x_{s+1}}(\mu\langle
x_{s+1}\rangle)$

for all $1\leq s\leq r$. We get a morphism

$\displaystyle\varphi_{r}\cdots\varphi_{1}:Z_{\mathbf{A}}^{x}(\mu\langle
x\rangle)\longrightarrow Z_{\mathbf{A}}^{w}(\mu\langle w\rangle).$

######  Proposition 7.19.

Suppose that $\pi\widetilde{\mu\langle x_{s}\rangle}([x_{s}\alpha_{i_{s}};t])$
is a unit for all $1\leq t\leq b^{\mathfrak{q}}(x_{s}\alpha_{i_{s}})-1$ or
$x_{s}\alpha_{i_{s}}\in\Delta_{+}^{\mathfrak{q}}$, for all $1\leq s\leq r$.
Then $\varphi_{r}\cdots\varphi_{1}$ induces an $\mathbf{A}$-isomorphism
between the spaces of weight $\mu$ and therefore
$\varphi_{r}\cdots\varphi_{1}$ is an $\mathbf{A}$-generator of the
corresponding $\operatorname{Hom}$ space.

###### Proof.

The spaces of weight $\mu$ are free of rank $1$ over $\mathbf{A}$ by (6.6) and
then the first assertion implies the second one. Also, it is enough to prove
it for each $s$ and assuming that $\mathbf{A}$ is a field. Thus, under the
first supposition, $\varphi_{s}$ is an isomorphism by Lemma 7.14. Under the
second one, $\mu$ is not a weight of
$\operatorname{{\mathrm{Ker}}}\varphi_{s}$ by (7.21) as $\mu=\mu\langle
x_{s}\rangle-(x_{s}(\varrho^{x_{s}^{-*}\mathfrak{q}})-\varrho^{\mathfrak{q}})$
and hence $\varphi_{s}$ is an isomorphism on the spaces of weight $\mu$. ∎

We will use the following corollary with $x=1^{\mathfrak{q}}$ and
$w=w_{0}\in{}^{\mathfrak{q}}\mathcal{W}$ the longest element to prove the
linkage principle in the next section.

######  Corollary 7.20.

If $\ell(w)=\ell(x)+\ell(x^{-1}w)$, then $\varphi_{r}\cdots\varphi_{1}$ is an
$\mathbf{A}$-generator of the corresponding Hom space.

###### Proof.

We have that $x_{s}\alpha_{i_{s}}\in\Delta_{+}^{\mathfrak{q}}$ by [25, Lemma 8
$(i)$] for all $1\leq s\leq r$. ∎

##  8\. The linkage principle

We are ready to prove our main results. We follow here the ideas in [1, §6].
We keep the notation of the above sections and restrict ourselves to the case
of the field $\mathbf{k}$. Recall Definition 4.8

######  Definition 8.1.

Let $\beta\in\Delta_{+}^{\mathfrak{q}}$ and $\mu\in\mathbb{Z}^{\mathbb{I}}$.
We set

$\displaystyle\beta\downarrow\mu=\mu-n_{\beta}^{\pi}(\mu)\,\beta.$

We say that $\lambda\in\mathbb{Z}^{\mathbb{I}}$ is strongly linked to $\mu$ if
and only if there exist $\beta_{1},...,\beta_{r}\in\Delta_{+}^{\mathfrak{q}}$
such that $\lambda=\beta_{r}\downarrow\cdots\beta_{1}\downarrow\mu$. We denote

$\displaystyle{}^{\downarrow}\mu=\left\\{\lambda\in\mathbb{Z}^{\mathbb{I}}\mid\lambda\,\mbox{is
strongly linked
to}\,\mu\,\mbox{and}\,\mu-\beta_{top}^{\mathfrak{q}}\leq\lambda\leq\mu\right\\}.$

Lastly, being linked is the smallest equivalence relation in
$\mathbb{Z}^{\mathbb{I}}$ such that $\lambda$ and $\mu$ are linked if
$\lambda$ is strongly linked to $\mu$ or vice versa. We denote
$[\mu]_{\operatorname{link}}$ the equivalence class of
$\mu\in\mathbb{Z}^{\mathbb{I}}$.

######  Theorem 8.2.

If $L_{\mathbf{k}}(\lambda)$ is a composition factor of $Z_{\mathbf{k}}(\mu)$,
then $\lambda=\mu$ or $L_{\mathbf{k}}(\lambda)$ is a composition factor of
$Z_{\mathbf{k}}(\beta\downarrow\mu)$ for some
$\beta\in\Delta_{+}^{\mathfrak{q}}$. Moreover, $\lambda\in{}^{\downarrow}\mu$.

$Z_{\mathbf{k}}(\mu)$$Z^{w_{s}}_{\mathbf{k}}(\mu\langle
w_{s}\rangle)$$Z^{w_{s+1}}_{\mathbf{k}}(\mu\langle
w_{s+1}\rangle)$$\operatorname{{\mathrm{Ker}}}\varphi_{s}$$Z^{w_{s}}_{\mathbf{k}}(\mu\langle
w_{s}\rangle-t_{s}\beta_{s})$$Z_{\mathbf{k}}(\mu-
t_{s}\beta_{s})$$\varphi_{1}$$\varphi_{s-1}$$\varphi_{s}$$\psi$$\tilde{\varphi}_{1}$$\tilde{\varphi}_{s-1}$
Figure 4. Let $w_{0}=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{n}}$ be a
reduced expression of the longest element in ${}^{\mathfrak{q}}\mathcal{W}$.
We set $w_{s}=1^{\mathfrak{q}}\sigma_{i_{1}}\cdots\sigma_{i_{s-1}}$ and
$\beta_{s}=w_{s}\alpha_{i_{s}}$, $1\leq s\leq n+1$. The morphisms
$\varphi_{i}$ are constructed as in §7.4.1 with $x=1^{\mathfrak{q}}$ and
$w=w_{0}$. The morphism $\psi$ is given by Lemma 7.16 with
$t_{s}=t_{\beta_{s}}^{\pi}(\mu\langle w_{s}\rangle)$. Moreover,
$t_{s}=n_{\beta_{s}}^{\pi}(\mu)$ by Lemma 4.9. Then the compositions
$\varphi_{s}\cdots\varphi_{1}$ are generators of the corresponding Hom spaces
by Corollary 7.20 and $\Phi=\varphi_{n}\cdots\varphi_{1}$ satisfies Lemma 6.5.
The morphisms $\tilde{\varphi}_{i}$ are constructed analogously starting from
$\mu-t_{s}\beta_{s}$ instead of $\mu$.

###### Proof.

We use the notation of Figure 4. Then $w_{0}=w_{n+1}$ and $\mu\langle
w_{n+1}\rangle=\mu-\beta^{\mathfrak{q}}_{top}$. Therefore
$\Phi=\varphi_{n}\cdots\varphi_{1}:Z_{\mathbf{k}}(\mu)\longrightarrow
Z^{w_{0}}_{\mathbf{k}}(\mu-\beta^{\mathfrak{q}}_{top})$ is non-zero by
Corollary 7.20 and hence
$L_{\mathbf{k}}(\mu)\simeq{\mathrm{Im}}\Phi\simeq\operatorname{soc}Z_{\mathbf{k}}^{w_{0}}(\mu-\beta^{\mathfrak{q}}_{top})$
by Lemma 6.5.

Now, suppose $\lambda\neq\mu$. Then $L_{\mathbf{k}}(\lambda)$ is a composition
factor of $\operatorname{{\mathrm{Ker}}}\Phi$ and hence of
$\operatorname{{\mathrm{Ker}}}\varphi_{s}$ for some $s$. By Lemma 7.16,
$\operatorname{{\mathrm{Ker}}}\varphi_{s}$ is a homomorphic image of
$Z^{w_{s}}_{\mathbf{k}}(\mu\langle w_{s}\rangle-t_{s}\beta_{s})$. Therefore
$L_{\mathbf{k}}(\lambda)$ is also a composition factor of
$Z^{w_{s}}_{\mathbf{k}}(\mu\langle w_{s}\rangle-t_{s}\beta_{s})$. By Lemma
4.9, we have that $t_{s}=n_{\beta_{s}}^{\pi}(\mu)$ and hence

$\displaystyle\mu\langle w_{s}\rangle-t_{s}\beta_{s}=(\mu-
t_{s}\beta_{s})\langle w_{s}\rangle=(\beta_{s}\downarrow\mu)\langle
w_{s}\rangle.$

Thus, ${\mathrm{ch}}Z^{w_{s}}_{\mathbf{k}}(\mu\langle w_{s}\rangle-
t_{s}\beta_{s})={\mathrm{ch}}Z_{\mathbf{k}}(\beta_{s}\downarrow\mu)$ by (6.6),
and hence $L_{\mathbf{k}}(\lambda)$ is also a composition factor of
$Z_{\mathbf{k}}(\beta_{s}\downarrow\mu)$. This shows the first part of the
statement. For the second one, we repeat the reasoning with
$\beta_{s}\downarrow\mu$ instead of $\mu$, and so on. This procedure will end
after a finite number of steps since $\beta\downarrow\mu\leq\mu$ and
$\mu-\beta_{top}^{\mathfrak{q}}\leq\lambda<\mu$. Hence, there exist
$\beta_{s_{1}}$, …, $\beta_{s_{r}}$ such that
$\lambda=\beta_{s_{r}}\downarrow\cdots\beta_{s_{1}}\downarrow\mu$ as desired.
∎

Besides the following particular case, it is not necessarily true that
$L_{\mathbf{k}}(\lambda)$ is a composition factor of $Z_{\mathbf{k}}(\mu)$ if
$\lambda\in{}^{\downarrow}\mu$, see Example 8.8.

######  Lemma 8.3.

If $\lambda=\beta\downarrow\mu$, then $L_{\mathbf{k}}(\lambda)$ is a
composition factor of $Z_{\mathbf{k}}(\mu)$.

###### Proof.

We can assume we are in the situation of Figure 4. That is,
$\beta=\beta_{s}=w_{s}\alpha_{i_{s}}$ and $\lambda=\beta_{s}\downarrow\mu=\mu-
t_{s}\beta_{s}$. Thus, we have a projection from
$Z_{\mathbf{k}}^{w_{s}}(\mu\langle w_{s}\rangle)$ to
$\operatorname{{\mathrm{Ker}}}\varphi_{s}$. Notice that $\lambda$ is a weight
of $\operatorname{{\mathrm{Ker}}}\varphi_{s}$ by (7.20). On the other hand,
$\tilde{\varphi}_{s-1}\cdots\tilde{\varphi}_{1}$ induces an
$\mathbf{k}$-isomorphism between the spaces of weight $\lambda$ by Proposition
7.19 which is one-dimensional. Hence
$\psi\tilde{\varphi}_{s-1}\cdots\tilde{\varphi}_{1}$ is not zero on the space
of weight $\lambda$ because
${\mathrm{Im}}\psi=\operatorname{{\mathrm{Ker}}}\varphi_{s}$. This implies
that $L_{\mathbf{k}}(\lambda)$ is a composition factor of
$\operatorname{{\mathrm{Ker}}}\varphi_{s}$ and therefore so is of
$Z^{w_{s}}_{\mathbf{k}}(\mu\langle w_{s}\rangle)$. As
${\mathrm{ch}}Z^{w_{s}}_{\mathbf{k}}(\mu\langle
w_{s}\rangle)={\mathrm{ch}}Z_{\mathbf{k}}(\mu)$, the lemma follows. ∎

In general, we can assert that the strongly linked weights belong to the same
block.

######  Corollary 8.4.

Let $\lambda,\mu\in\mathbb{Z}^{\mathbb{I}}$. Then $\lambda$ and $\mu$ are
linked if and only if $L_{\mathbf{k}}(\lambda)$ and $L_{\mathbf{k}}(\mu)$
belong to the same block.

###### Proof.

We first prove that being linked implies the belonging to the same block. To
this end, it is enough to consider the case $\lambda=\beta\downarrow\mu$ for
some $\beta\in\Delta_{+}^{\mathfrak{q}}$ which follows from Lemma 8.3.

For the reciprocal, as above, it is enough to consider the existence of a non-
trivial extension of the form $0\longrightarrow
L_{\mathbf{k}}(\lambda)\longrightarrow M\longrightarrow
L_{\mathbf{k}}(\mu)\longrightarrow 0$. Therefore $M$ is a quotient of
$Z_{\mathbf{k}}(\mu)$ and hence $\lambda\in{}^{\downarrow}\mu$ by Theorem 8.2.
∎

###  8.1. Typical weights

For $\beta\in\Delta_{+}^{\mathfrak{q}}$ and $\mu\in\mathbb{Z}^{\mathbb{I}}$,
we introduce

(8.1)
$\displaystyle\begin{split}\mathfrak{P}_{\mathbf{k}}^{\mathfrak{q}}(\beta,\mu)&=\prod_{1\leq
t<b^{\mathfrak{q}}(\beta)}\left(q_{\beta}^{t}-\rho^{\mathfrak{q}}(\beta)\,\pi\widetilde{\mu}(K_{\beta}L_{\beta}^{-1})\right)\quad\mbox{and}\\\
\vskip 3.0pt plus 1.0pt minus
1.0pt\cr&\mathfrak{P}_{\mathbf{k}}^{\mathfrak{q}}(\mu)=\prod_{\beta\in\Delta_{+}^{\mathfrak{q}}}\mathfrak{P}_{\mathbf{k}}^{\mathfrak{q}}(\beta,\mu)\end{split}$

A weight $\mu$ is called typical if
$\mathfrak{P}_{\mathbf{k}}^{\mathfrak{q}}(\mu)\neq 0$. Otherwise, it is called
atypical and the numbers of positive roots for which
$\mathfrak{P}_{\mathbf{k}}^{\mathfrak{q}}(\beta,\mu)=0$ is its degree of
atypicality; if it is $\ell$ we say that $\mu$ is $\ell$-atypical. This
terminology is borrowed from [27], see also [39, 41].

######  Corollary 8.5 ([1, §6.3]).

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$. The following are equivalent:

  1. (1)

$\mu$ is typical.

  2. (2)

$Z_{\mathbf{k}}(\mu)=L_{\mathbf{k}}(\mu)$ is simple.

  3. (3)

$Z_{\mathbf{k}}(\mu)=L_{\mathbf{k}}(\mu)$ is projective.

###### Proof.

If $\mu$ is typical, then $L_{\mathbf{k}}(\mu)$ is the unique composition
factor of $Z_{\mathbf{k}}(\mu)$ by Theorem 8.2. Since
$[Z_{\mathbf{k}}(\mu),L_{\mathbf{k}}(\mu)]=1$, it follows that
$Z_{\mathbf{k}}(\mu)$ is simple. Instead, if
$\mathfrak{P}_{\mathbf{k}}^{\mathfrak{q}}(\mu)=0$, then
$\operatorname{{\mathrm{Ker}}}\varphi_{s}\neq 0$ for some $s$, cf. Figure 4.
Since the morphism
$\varphi_{n}\cdots\varphi_{1}:Z_{\mathbf{k}}(\mu)\longrightarrow
Z^{w_{0}}_{\mathbf{k}}(\mu-\beta^{\mathfrak{q}}_{top})$ is non trivial,
$\operatorname{{\mathrm{Ker}}}\varphi_{s}\neq Z_{\mathbf{k}}(\mu)$ and hence
$Z_{\mathbf{k}}(\mu)$ is non simple. This proves that (1) is equivalent to
(2).

The equivalence between (2) and (3) is a consequence of Theorem 6.10. ∎

######  Remark 8.6.

The equivalence between $(1)$ and $(2)$ was proved before in [26, Proposition
5.16]; see also [41, Remark 6.25].

###  8.2. $1$-atypical weights

For weights with degree of atypicality $1$ we can compute the character of the
associated simple module similar to [1, §6.4].

######  Corollary 8.7.

Let $\mu\in\mathbb{Z}^{\mathbb{I}}$ be a $1$-atypical weight with
$\mathfrak{P}_{\mathbf{k}}^{\mathfrak{q}}(\beta,\mu)=0$ for certain
$\beta\in\Delta_{+}^{\mathfrak{q}}$. Then

$\displaystyle{\mathrm{ch}}L_{\mathbf{k}}(\mu)=e^{\mu}\quad\frac{1-e^{-n^{\pi}_{\beta}(\mu)\beta}}{1-e^{-\beta}}\prod_{\gamma\in\Delta_{+}^{\mathfrak{q}}\setminus\\{\beta\\}}\frac{1-e^{-b^{\mathfrak{q}}(\gamma)\gamma}}{1-e^{-\gamma}}.$

Moreover, there exists an exact sequence

$\displaystyle 0\longrightarrow
L_{\mathbf{k}}(\beta\downarrow\mu)\longrightarrow
Z_{\mathbf{k}}(\mu)\longrightarrow L_{\mathbf{k}}(\mu)\longrightarrow 0.$

###### Proof.

We keep the notation of Figure 4: $\beta=\beta_{s}=w_{s}\alpha_{i_{s}}$,
$\lambda=\beta_{s}\downarrow\mu=\mu-t_{s}\beta_{s}$ and
$\Phi=\varphi_{n}\cdots\varphi_{1}$. As $\mu$ is $1$-atypical, the morphisms
$\varphi_{\ell}$ are isomorphisms for all $\ell\neq s$ by Lemma 7.14 and hence
${\mathrm{Im}}\varphi_{s}\simeq{\mathrm{Im}}\Phi\simeq L_{\mathbf{k}}(\mu)$,
recall Lemma 6.5. Thus, the character formula is consequence of (7.20).

For the existence of the exact sequence, we claim that