Harish-Chandra modules and Galois orders revisited

TL;DR

This paper explores properties of Harish-Chandra modules and Galois algebras, focusing on transfer properties, invariants, and the construction of irreducible modules, linking various approaches and introducing infinite rank generalized Weyl algebras.

Contribution

It provides new insights into transfer properties of Harish-Chandra modules, links different approaches to Galois rings, and introduces the concept of infinite rank generalized Weyl algebras.

Findings

Established transfer properties to spherical subalgebras.

Analyzed freeness over Harish-Chandra subalgebras.

Constructed concrete irreducible Harish-Chandra modules.

Abstract

The main subject of study of this paper are general properties of HarishChandra algebras and modules with respect wito a pair of algebra and subalgebra, with special focus on the transfer properties to a "spherical subalgebra". We also discuss general properties of Galois rings and algebras, where the former discussion is specialized, and we obtain an important link between different approaches to it in the literature. Then we focus our study into finite multiplicative invariants on the ring of differential operators on the torus and fixed rings under the action of a finite group of algebra automorphisms of generalized Weyl algebras. We study freeness over the Harish-Chandra subalgebra and the Gelfand-Kirillov Conjecture for them. Our last section construction some concrete irreducible Harish-Chandra modules. This paper also introduces the notion of an infinite rank generalized Weyl algebra.

Equations

$$X_i\lambda ={\sigma }_i(\lambda )X_i;Y_i\lambda ={\sigma }_i^{-1}(\lambda )Y_i,\lambda \in D,i=1,\dots ,n;$$

$$Y_i{X}_i=a_i,X_i{Y}_i={\sigma }_i(a_i),i=1,\dots ,n;$$

$$[X_i,X_j]=[Y_i,Y_j]=[X_i,Y_j],i\ne j,i,j=1,\dots ,n.$$

$$(†)f_{{\varphi }^{-1}m}(a)=f_m(\varphi (a)).$$

$$z.T(n)=f_n(z)T(n);$$

$$X_i.T(n)=T({\sigma }_{i}n);$$

$$Y_{i}T(n)=f_{{\sigma }_i^{-1}n}(a_i)T({\sigma }_i^{-1}n),$$

$$z\in D,X_i,Y_i\in D(a,\sigma ),i=1,\dots ,n.$$

$$X_{i}zT(n)=f_n(z)X_{i}T(n)=f_n(z)T({\sigma }_{i}n)=f_{{\sigma }_{i}n}({\sigma }_i(z))T({\sigma }_i(n))={\sigma }_i(z)X_{i}T(n),i=1,\dots ,n;$$

$$Y_{i}zT(n)=f_n(z)f_{{\sigma }_i^{-1}n}(a_i)T({\sigma }_i^{-1}n)=f_{{\sigma }_i^{-1}n}({\sigma }_i^{-1}z)f_{{\sigma }_i^{-1}n}(a_i)T({\sigma }_i^{-1}n)={\sigma }_i^{-1}(z)Y_{i}T(n),i=1,\dots ,n;$$

$$Y_i{X}_{i}T(n)=f_n(a_i)T(n)=a_{i}T(n),i=1,\dots ,n$$

$$X_i{Y}_{i}T(n)=f_{{\sigma }_i^{-1}n}(a_i)T(n)=f_n({\sigma }_i(a_i))T(n)={\sigma }_i(a_i)T(n),i=1,\dots ,n.$$

Full text

1. Introduction

Let $\mathsf{k}$ be the base field, algebraically closed with zero characteristic. All rings under discussion are $\mathsf{k}$-algebras. All our modules are left modules, unless said otherwise.

This paper study certain aspects of the theory of generalized Weyl algebras and Galois algebras and orders.The former class of algebras was introduced and studied in [2]. Many important algebras of small Gelfand-Kirillov dimension arising in noncommutative geometry are generalized Weyl algebras, such as the first Weyl algebra and its quantization; the quantum plane and the quantum sphere; $U(sl_{2}(\mathsf{k}))$ and its quantization; the Heisenberg algebra and its quantizations; quantum $2\times 2$ matrices; Witten’s and Woronowic’s deformations; Noetherian down-up algebras. For the representation theory of such algebras we refer to [2], [3], [4], [5].

Galois algebras and orders were introduced by V. Futorny and S. Ovsienko in [15] and [16], in a further refinement of the theory of Harish-Chandra categories in [7], in case we have a pair of algebra/subalgebra $U\supset\Gamma$, with $\Gamma$ commutative, and a certain embedding of $U$ in a skew monoid ring. The original motivation of this enterprise is the Gelfand-Tsetlin theory of representations of $gl_{n}$ ([20]) in the case of infinite dimensional modules ([7]). This theory has led to breakthrough in representation theory for many algebras (see discussion in [11]); in particular $U(gl_{n})$ ([16]) and its quantization ([12]), finite $W$-algebras of type $A$ ([14]), OGZ and quantum OGZ algebras of type $A$ ([21], [22]), as well as their parabolic subalgebras ([22]), an alternating analogue of $U(gl_{n})$ ([23]), invariant subrings of rings of differential operators and quantum groups ([17], [18]), quantized Coulomb branches ([34]), and rational Cherednik algebras ([26]). Other furthers aspects of the representation theory of Galois algebras were developed in [28], [33], [8], [13].

The objective of this work is to study some properties of Galois algebras and generalized Weyl algebras as an abstract algebraic object per se.

In the second section of this paper, we recall the definition of the basic objects and some results that will be used thorough the text.

In the third section we study some ring theoretical properties of Galois orders — a line of research not much explored since the original work [15]. We recall the important class of linear Galois algebras and discuss some of its properties. Then we relate Galois orders to PI-algebras, and show that under very modest hypothesis, they are never PI-algebras (Theorem 3.6), and as a consequence we obtain that many important algebras are not PI (Theorem 3.7).

In the fourth section we recall the notions of principal and rational Galois orders from [22], and we compare them with the notion of linear Galois algebras. We show that all the Galois orders in [18] and [17] are principal Galois orders (Theorem 4.9), and $A_{n}^{S_{n}}$ in particular is also rational (Theorem 4.10), which raises the question of whether all Galois orders are principal. We finish by checking the validity of the Gelfand-Kirillov Conjecture for many Galois algebras (Theorem 4.11).

In the fifth section we generalize the result of [18] that $A_{n}(\mathsf{k})^{G(m,p,n)}$ is a principal Galois order from $p=1$ to all $p|m$ (Corollary 5.11), as a particular case of a result showing that the invariants of a broad class of generalized Weyl algebras under the action of such groups are principal Galois order (Theorem 5.10). Corollary 5.11 had appeared previously in [26], but our method of proof is different and elementary. We also generalize the result from [17, Theorem 8] and show that the invariants of the $n$-th quantized Weyl algebra under the action of $G(m,p,n)$ are a Galois order as well.

Finally, in the sixth section, we show explicitly that the non-triviality of the Futorny and Ovsienko lifting [16] holds for any generalized Weyl algebra over any algebraically closed field. We construct generalized tableaux modules for each generalized Weyl algebra that is a Galois order considered in [18] (Theorem 6.3).

2. Preliminaries

Lets recall first the notion of Galois algebras and orders.

Definition 2.1**.**

([15])* A Galois algebra $U$ over $\Gamma$ is an algebra finitely generated over a commutative subalgebra $\Gamma$ given by the following data:*

1. (1)

$\Gamma\subset U$* a commutative finitely generated subalgebra which is a domain, and has $K$ as field of fractions.* 2. (2)

A Galois extension $L$ of $K$, with Galois group $G$. 3. (3)

$\mathcal{M}\subset Aut_{\mathsf{k}}\,L$* a monoid of automorphisms such that $m_{|K}=m^{\prime}_{|K}$, $m,m^{\prime}\in M$ implies $m=m^{\prime}$.* 4. (4)

$G$* acts on $\mathcal{M}$ by conjugation.* 5. (5)

We have an embedding of $U$ in the invariant skew monoid ring $\mathcal{K}=(L*\mathcal{M})^{G}$ such that $UK=KU=\mathcal{K}$.

Definition 2.2**.**

([7])* A Galois algebra over $\Gamma$ is called a right (left) Galois order over $\Gamma$ if for every right (left) finite dimensional $K$-vector subspace $W\subset\mathcal{K}$, $W\cap\Gamma$ is a finitely generated right (left) $\Gamma$-module. If $U$ is both left and right Galois order over $\Gamma$, then we say that $U$ is a Galois order over $\Gamma$.*

Definition 2.3**.**

Let $U$ be an algebra and $\Gamma$ a commutative subalgebra of $U$. We say that $\Gamma$ * is a Harish-Chandra subalgebra if for every $u\in U$, $\Gamma u\Gamma$ is finitely generated left and right $\Gamma$-module.*

We recall the following in [22]:

Proposition 2.4**.**

If all automorphisms in $\mathcal{M}$ and $G$ are induced from an automorphism of a certain finitely generated integrally closed domain $\Lambda$, with $Frac\,\Lambda=L$, $\Lambda^{G}=\Gamma$, then

• •

$\Gamma$* is a Harish-Chandra subalgebra in every Galois algebra $U$ containing it.*

• •

If moreover $U$ is a Galois order, then $\Gamma$ is a maximal commutative subalgebra.

Proof.

The first claim follows from[22, Lemma 2.4, Proposition 2.5]. The second one follows from [22, Proposition 2.14]. ∎

Assumption**.**

All Galois algebras under discussion in this paper will satisfy the conditions in Proposition 2.4.

Definition 2.5**.**

Let $U$ be a Galois order embedded in skew monoid ring $(L*\mathcal{M})^{G}$. For $u\in U,u=\sum_{m\in\mathcal{M}}\alpha_{m}m,\alpha_{m}\in L$, we define $supp\,u=\{m\in\mathcal{M}|\alpha_{m}\neq 0\}$.

Proposition 2.6**.**

Let $\Xi\subset(L*\mathcal{M})^{G}$ and let $U$ be the subalgebra generated by $\Xi$ and $\Gamma$. Then $U$ is a Galois algebra in this skew monoid ring if and only if $\bigcup_{x\in\Xi}supp\,x$ generates $\mathcal{M}$ as a monoid.

Proof.

[15, Prop. 4.1(1)], [22, Prop. 2.9]. ∎

We recall the following nice lemma from [18]:

Lemma 2.7**.**

Let $U$ be an algebra and $\Gamma\subset U$ an affine commutative subalgebra. Let $G$ be a finite group of automorphisms of $U$ such that $G(\Gamma)\subset\Gamma$. If $U$ is projective right (left) $\Gamma$-module and $\Gamma$ is projective over $\Gamma^{G}$, then $U^{G}$ is projective right (left) $\Gamma^{G}$-module.

Proof.

[18, Lemma 11]. ∎

Definition 2.8**.**

Let $U$ be a Galois order over $\Gamma$. A finitely generated module $M$ is called Gelfand-Tsetlin if $M|_{\Gamma}=\bigoplus_{m\in Specm\,\Gamma}M(m)$, where $M(m)=\{v\in M|m^{k}v=0,k>>0\}$. Equivalently, $\Gamma$ acts locally finitely on $M$ (cf. [7]).

One of the important consequences of the Galois order structure is a finiteness and nontriviality of lifting of maximal ideals of the commutative subalgebra ([16]). Namely, if $U$ is a Galois order over a commutative subalgebra $\Gamma$ then for any maximal ideal $m\in Specm\,\Gamma$ there exists finitely many isomorphism classes of irreducible $U$-modules with nonzero annihilator containing $m$. This allow us to parametrize (up to some finiteness) the irreducible $U$-modules with $\Gamma$-torsion by $Specm\,\Gamma$.

Let’s now recall the notion of generalized Weyl algebras.

Definition 2.9**.**

[[2]] Let $D$ be an algebra over $\mathsf{k}$. Let $a=(a_{1},\ldots,a_{n})$ be a n-uple of elements of $Z(D)$ which are not zero divisors. Let $\sigma=(\sigma_{1},\ldots,\sigma_{n})$ be a tuple of automorphisms of $D$ such that $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},i,j=1,\ldots,n$; $\sigma_{i}(a_{j})=a_{j}$, if $j\neq i$. The generalizd Weyl algebra of rank $n$ is the algebra with generators $D,X_{i},Y_{i}$, $i=1,\ldots,n$, and relations

[TABLE]

[TABLE]

[TABLE]

We will call the $X_{i},Y_{i}$ the GWA generators; $D$ is called the defining algebra. It is denoted by $D(a,\sigma)$.

We will only consider generalized Weyl algebras such that $D$ is an affine commutative domain. Notice that under this assumption, every generalized Weyl algebra will be a Noetherian domain — hence an Ore domain ([2]). It is also clear that a generalized Weyl algebra is a free left and right module over its defining algebra $D$.

The following elementary property is important ([2]):

Proposition 2.10**.**

The tensor product over $\mathsf{k}$ of two generalized Weyl algebras $D(a,\sigma)\otimes D(a^{\prime},\sigma^{\prime})\simeq(D\otimes D^{\prime})(a\ast a^{\prime},\sigma\ast\sigma^{\prime}),$ is again a generalized Weyl algebra, where $\ast$ is the tensor product of automorphisms, and the concatenation of $a,a^{\prime}$.

In [18], it was shown that under very mild conditions, every generalized Weyl algebra $D(a,\sigma)$ is a Galois order with Harish-Chandra subalgebra $D$. We recall this result here:

Theorem 2.11**.**

1. We have the following:
2. (1)

There is an embedding of $D(a,\sigma)$ into $D*\mathbb{Z}^{n}$, where $ye_{i}$ acts as $\sigma_{i}^{y}$, $y\in\mathbb{Z}$, $e_{i}$ the canonical basis of $\mathbb{Z}^{n}$. The embedding sends $X_{i}$ to $e_{i}$ and $Y_{i}$ to $a_{i}e_{i}^{-1}$. 3. (2)

Let $G$ be the abelian group generated by $\sigma_{1},\ldots,\sigma_{n}$: $y_{1}\sigma_{1}+\ldots+y_{n}\sigma_{n}:=\sigma_{1}^{y_{1}}\ldots\sigma_{n}^{y_{n}}$ for all integer $y_{1},\ldots,y_{n}$. If $\sigma_{1},\ldots,\sigma_{n}$ are linearly independent over $\mathbb{Z}$, then $D(a,\sigma)$ is a Galois order over $D$ in the skew group ring ${\rm{Frac}}\ D*\mathbb{Z}^{n}$ with the embedding above, and $D$ is a Harish-Chandra subalgebra.

Proof.

The first item is [18, Prop. 13]. The second item is [18, Thm. 14]. ∎

3. Galois orders and PI-algebras

We recall the following important class of Galois algebras (cf. [9]):

Definition 3.1**.**

Let $V$ be a complex $n$-dimensional vector space, such that we have $L\simeq\mathbb{C}(t_{1},\dots,t_{n},z_{1},\ldots,z_{s})$ as the field of fractions of the symmetric algebra $S(V^{*})$. Let $G$ be a complex reflection group acting on $V$ by reflections. This action can be extended to an action of $G$ on $L$ and we set $K=L^{G}$. Suppose $G$ normalizes a fixed submonoid $\mathcal{M}\subset Aut_{\mathbb{C}}\,L$ and $\Gamma$ is a polynomial subalgebra such that ${\rm{Frac}}\ \Gamma=K$. A Galois order $U$ over $\Gamma$ in $(L*\mathcal{M})^{G}$ is called linear Galois algebra.

1. (1)

If $\mathcal{M}\simeq\mathbb{Z}^{n}$ and the canonical generators $e_{1},\ldots,e_{n}$ act by shifts $e_{i}(t_{j})=t_{j}-\delta_{ij}$ fixing the $z^{\prime}s$, then we have a linear Galois algebras of shift type. 2. (2)

If $\mathcal{M}\simeq\mathbb{Z}^{n}$ or $\mathbb{N}^{n}$, $e_{i}(t_{j})=q^{\delta_{ij}}t_{j}$ fixing the $z^{\prime}s$, $0\neq q\in\mathbb{C}$ is not a root of unity, we have quantum linear Galois algebra (**[17, Section 5.4]**).

Remark 3.2**.**

In cases (1) and (2) above, $L*\mathcal{M}$ itself is trivially a Galois algebra.

We notice that most known examples of Galois algebras are linear (cf. [15], [16], [11], [9], [18], [17], [22, Sections 4, 5],)

Proposition 3.3**.**

If $U$ is a Galois algebra embedded in $\mathcal{K}$ and $S=\Gamma\setminus\{0\}$, then $S$ is a left and right denominator set in $U$ and localization by $S$ (on the left or on the right) gives us $U_{S}=\mathcal{K}$. In particular, if $\mathcal{K}$ is an Ore domain, so is $U$.

Proof.

[15, Proposition 4.2] ∎

Proposition 3.4**.**

Every linear Galois algebra of quantum or shift type is an Ore domain.

Proof.

In both cases, we now that $(L*\mathcal{M})$ is a domain, since it is isomorphic to an iterated Ore extension of $L$. Hence $\mathcal{K}=(L*\mathcal{M})^{G}$ is an Ore domain by [10]. The result follows from the above Proposition 3.3. ∎

We can now state our main result of this section. Before that, we need to recall the notion of PI-algebra (for a gentle introduction to the subject with plenty references we refer to [6]):

Definition 3.5**.**

Let $X=\{x_{1},\ldots,x_{n},\ldots\}$ be a countable set and $\mathsf{k}\langle X\rangle$ the free associative algebra on this set. An algebra $A$ is called a $PI$-algebra if there is some $0\neq f(x_{1},\ldots,x_{n})\in\mathsf{k}\langle X\rangle$ such that $f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}\in A$.

Theorem 3.6**.**

Let $U$ be a Galois order over $\Gamma$ in a invariant skew monoid ring $\mathcal{K}=\mathcal{L}^{G}$, $\mathcal{L}:=(L*\mathcal{M})$ and suppose $\mathcal{L}$ is an Ore domain which is not a skew field. Then $U$ is not a PI-algebra.

Proof.

By [10], $\mathcal{K}$ is also an Ore domain. Hence $U$ is an Ore domain by Proposition 3.3. Suppose, to the contrary, that $U$ is a PI-algebra. By Posner’s Theorem ([6, Theorem B.6.5]), central localization of $U$ gives us a skew field $\mathcal{Q}$. Since $\Gamma$ is maximal commutative, by Proposition 3.3, we must have $\mathcal{Q}=\mathcal{K}$. We will show that this leads to a contradiction. By assumption, $\mathcal{L}$ is not a skew field; hence, for a suitable $a\in\mathcal{L}$, $\mathcal{L}a$ is a proper right ideal. Now consider $I=\bigcap_{g\in G}g(\mathcal{L}a)$. This right ideal is proper: in an Ore domain the intersection of a finite number of one sided proper ideals is non-trivial; and clearly $G$-invariant. Applying Bergson and Isaac’s Theorem (cf. [29, Corollary 1.5]), we have that $I\cap\mathcal{K}$ is a proper right ideal, which is impossible as $\mathcal{K}$ should be a skew field. ∎

This approach has the advantage that the only information needed is the Galois algebra realization. Ring theoretical properties, that might hard to check, such as primitiveness or the center, in order to apply Kaplansky’s Theorem (cf. [6]), are not needed.

Theorem 3.7**.**

The following algebras are not PI-algebras:

1. (1)

Finite $W$ algebras of $A$ type $W(\pi)$; and in particular $U(gl_{n})$ (**[14, Theorem 3.6]**). 2. (2)

Level $p$ Yangians $Y_{p}(gl_{n})$ (cf. **[14]**, **[22]**). 3. (3)

Quantized OGZ algebras of type $A$, such as $U_{q}(gl_{n})$ ($q$* not a root of unity), and the quantized Heisenberg algebra ([21, Theorem 3.9]).* 4. (4)

OGZ algebras of type $A$ (cf. **[27]**, **[22, Theorem 4.6]**). 5. (5)

The parabolic subalgebras of the previous algebras (**[22, Theorem 1.2, Theorem 1.3]**). 6. (6)

The alternating analogue of $U(gl_{n})$ (**[23]**). 7. (7)

All linear Galois orders of quantum and shift types.

We expect that a slightly different approach, using the results in [34], may be used to show that all quantized Coulomb branches are not PI-algebras.

4. Principal, rational and linear Galois orders, and the Gelfand-Kirillov Conjecture

In this section we discuss the notion of principal and rational Galois orders introduced in [22], with relation to subrings of invariants, and show that all the Galois algebras in [18], [17] are principal Galois orders. We also compare the notion of rational Galois algebra with that of linear Galois algebra.

We begin recalling the notion of principal Galois orders.

Definition 4.1** ([22]).**

Let $U$ be a Galois order embedded in skew monoid ring $(L*\mathcal{M})^{G}$. For $u\in U,u=\sum_{m\in\mathcal{M}}\alpha_{m}m,\alpha_{m}\in L$, we define the evaluation map $u:L\mapsto L$, as $u(l)=\sum_{m\in\mathcal{M}}\alpha_{m}m(l)$. $U$ is a principal Galois order if for each $u\in U$, $a\in\Gamma$, $u(a)\in\Gamma$; or, in other words, $U(\Gamma)\subset\Gamma$.

Now we recall the notion of rational Galois orders, introduced in [22], their representations considered in [13]. First, a preliminary result.

Theorem 4.2**.**

Let $V$ a finite dimensional complex vetcor space, $G\subset GL(V)$ be a finite complex reflection group, $\chi$ a character of $G$ and $\Gamma_{\chi}=\{f\in S(V^{*})|g.f=\chi(g)f\}$. There exists a uniquely defined $d_{\chi}\in S(V^{*})$ such that $\Gamma_{\chi}=S(V^{*})d_{\chi}$.

Proof.

[32, Thm 2.5]. ∎

Let $V$ be a complex $n$-dimensional vector space. It acts on $S=S(V^{*})$ the following way: $t_{v}.f(x)=f(x-v),x,v\in V,f\in S(V^{*})$. We can extend this action to $L=Frac\,S$ and form the smash product $L*V$. Let $G\subset GL(V)$ be a complex reflection group. We have for $g\in G$, $gt_{v}g^{-1}=t_{g(v)}$, $v\in V$. We can then consider $(L*V)^{G}$.

Definition 4.3** ([22]).**

Let $X$ a subset of $L*V$ such that :

• •

$g(x)=x,\forall g\in G,x\in X$;

• •

For all $x\in X$ there exists a character $\chi$ such that $d_{\chi}x\in S(V^{*})*V$.

Then the subalgebra of $L*V$ generated by $S(V^{*})^{G}$ and $X$ is called a rational Galois order in $(L*\mathcal{M})^{G}$, where $\mathcal{M}\subset V$ is the submonoid generated by $\bigcup_{x\in X}supp\,x$.

The two above classes of Galois algebras are related as follows:

Theorem 4.4**.**

Every rational Galois order is a principal Galois order.

Proof.

[22, Theorem 4.2]. ∎

We now collect an easy statement.

Lemma 4.5**.**

Every Galois order $U$ which is a skew monoid ring or a generalized Weyl algebra is a principal Galois order.

Proof.

The first claim is obvious; the second one follows from Theorem 2.11(2). ∎

The following is the main proposition of this section.

Proposition 4.6**.**

If $U$ is a principal Galois order over $\Gamma$, and $U^{G}$ is a Galois ring over $\Gamma^{G}$, for some finite group of automorphisms $G$ of $U$, with $G(\Gamma)\subset\Gamma$, then $U^{G}$ is a principal Galois order over $\Gamma^{G}$.

Proof.

Call $K:=Frac(\Gamma)$. Since $U^{G}\subset U$ and the later is a principal Galois order, $U^{G}(\Gamma)\subset\Gamma$. On the other hand, by [22, Lemma 2.19], $U^{G}(K^{G})\subset K^{G}$. These two facts combined implies that $U^{G}(\Gamma^{G})\subset\Gamma^{G}$. ∎

We compare the classes of Galois orders.

Proposition 4.7**.**

Let $R=\mathbb{C}[x_{1},\ldots,x_{3}]$, and $\sigma$ be the Nagata automorphism, known to be wild ([31]). Let $R^{\prime}=R_{1}\otimes R_{2}\otimes R_{3}$, where each $R_{i}$ is a copy of $R$, $i=1,2,3$; and let $\sigma_{1}=\sigma\otimes 1\otimes 1$, $\sigma_{2}=1\otimes\sigma\otimes 1$, $\sigma_{3}=1\otimes 1\otimes\sigma$. Consider $R^{\prime}*\mathbb{Z}^{3}$, the skew group ring of $R^{\prime}$ with the group generated by $\sigma_{i},i=1,2,3$. The symmetric group $S_{3}$ acts on this ring permuting the $R_{i}$ factors and by conjugation on $\mathbb{Z}^{3}$. $U=(R^{\prime}*\mathbb{Z}^{3})^{S_{3}}$ is then an example of linear Galois algebra which is a principal Galois orders but it is not of shift or quantum type, neither rational.

Proof.

That it is a principal Galois order follows from Lemma 4.5 and Proposition 4.6. The other claims are clear. ∎

Now we proceed to show that the linear Galois orders considered in [18] and [17] are principal. We recall the objects involved.

$G_{m}\subset\mathsf{k}$ denote the cyclic group in $m$ elements, $G(m,p,n)$,$p|m$, the infinite three parameter family of irreducible complex reflection groups of Shephard and Todd. They are given as follows: $A(m,p,n)$ is the subgroup of $G_{m}^{\otimes n}$ given by elements $(h_{1},\ldots,h_{n})$ such that $(\prod_{i=1}^{n}h_{i})^{m/p}=id$; $G(m,p,n)=A(m,p,n)\rtimes S_{n}$.

$S_{n},B_{n}=C_{n},D_{n}$ are the classical Weyl groups.

The $n$-th Weyl algebra is $A_{n}(\mathsf{k})=\mathsf{k}\langle x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\rangle$ with canonical generators and relations $[x_{i},x_{j}]=[y_{i},y_{j}]=0;[y_{i},x_{j}]=\delta_{ij},i,j=1,\ldots,n$. Call $t_{i}=y_{i}x_{i}$, and consider the automorphisms of $\mathsf{k}[t_{1},\ldots,t_{n}]$, $\sigma_{i}(t_{j})=t_{j}-\delta_{ij},i,j=1,\ldots,n$; call $\mathbb{Z}^{n}$ the free abelian group generated by $\sigma_{1},\ldots,\sigma_{n}$. Then $A_{n}(\mathsf{k})$ is a Galois order in $\mathsf{k}(t_{1},\ldots,t_{n})*\mathbb{Z}^{n}$ with embedding $x_{i}\mapsto\sigma_{i}$, $y_{i}\mapsto t_{i}\sigma_{i}^{-1}$ ([15]). The Weyl algebra is also naturally identified as the ring of differential operators on the affine space.

$D(\mathbb{T}^{n})$, the ring of differential operators on the $n$-torus, is the localization $A_{n}(\mathsf{k})_{x_{1}\ldots x_{n}}$. Moreover $D(\mathbb{T}^{n})\simeq\mathsf{k}[t_{1},\ldots,t_{n}]*\mathbb{Z}^{n}$, where $t_{i}$ and $\mathbb{Z}^{n}$ are the same as in the above case of the Weyl algebra [18, Section 4.4.].

Given $0,1\neq q$ not a root of unity, we have the quantum plane $\mathsf{k}_{q}[x,y]:=\mathsf{k}\langle x,y|yx=qxy\rangle$; the first quantized Weyl algebra $A_{1}^{q}(\mathsf{k})=\mathsf{k}\langle x,y|yx-qxy=1\rangle$.

Lemma 4.8**.**

Both $\mathsf{k}_{q}[x,y]$ and $A_{1}^{q}(\mathsf{k})$ are generalized Weyl algebras.

Proof.

$\mathsf{k}_{q}[x,y]$ is a generalized Weyl algebra of rank one with $D=\mathsf{k}[h]$, $a=h$, $\sigma(h)=qh$. $A_{1}^{q}(\mathsf{k})$ is a generalized Weyl algebra of rank one with $D=\mathsf{k}[h]$, $a=h$, $\sigma(h)=q^{-1}(h-1)$, cf. [2]. ∎

The quantum affine space is $\mathcal{O}_{q}(k^{2n})=k_{q}[x_{1},y_{1}]\otimes\ldots\otimes k_{q}[x_{n},y_{n}]$ and the quantum torus $\mathcal{O}_{q}(k^{*2n})$ is its localization by $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ . The $n$-th quantized Weyl algebra is $A_{n}^{q}(\mathsf{k})=A_{1}^{q}(\mathsf{k})^{\otimes n}$.

Theorem 4.9**.**

All the invariants of the Weyl algebra $A_{n}(\mathsf{k})^{G}$, where $G$ belongs to $\{G_{m}^{\otimes n},\mathcal{A}_{n},G(m,1,n)\}$; all the invariants of the ring of differential operators on the torus $\mathcal{D}(\mathbb{T}^{n})^{W}$, where $W\in\{S_{n},B_{n}=C_{n},D_{n}\}$, the natural invariants of $\mathcal{O}_{q}(k^{2n})$ and $\mathcal{O}_{q}(k^{*2n})$ under the action of groups $G(m,p,n)$; the invariants of the $\mathbb{C}_{q}[x,y]$ and $A_{1}^{q}(\mathbb{C})$ under the action of any finite group; and $A_{n}^{q}(\mathsf{k})^{S_{n}}$, are all principal Galois orders.

Proof.

The Weyl algebra $A_{n}(\mathsf{k})$ is a generalized Weyl algebra, and $\mathcal{D}(\mathbb{T}^{n})$ is isomorphic to a skew group ring (cf. above). Hence the first two statements follows from Lemma 4.5 and Proposition 4.6 (cf.[18] Proposition 22, Corollary 24, Proposition 25, Theorem 27 and Theorem 33). The other statements are proved following the same reasoning, as the quantum algebras in question are invariants of a certain skew monoid ring or a generalized Weyl algebra (cf. Lemma 4.8), again using Lemma 4.5 and Proposition 4.6 (cf. [17] Proposition 5, Theorem 5, Theorem 6, Proposition 6, Theorem 7, Theorem 8, Proposition 7). ∎

In case of the invariants $A_{n}(\mathsf{k})^{S_{n}}$, more can be said.

Theorem 4.10**.**

$A_{n}(\mathsf{k})^{S_{n}}$* is a rational Galois order.*

Proof.

By [25, Theorem 5], $A_{n}(\mathsf{k})^{S_{n}}$ is generated by the elementary symmetric polynomials in the set of indeterminates $x_{1},\ldots,x_{n}$ and $y_{1},\ldots,y_{n}$ separetly. Hence $A_{n}(\mathsf{k})^{S_{n}}$ is generated inside $(\mathsf{k}(t_{1},\ldots,t_{n})*\mathbb{Z}^{n})^{S_{n}}$ by $\mathsf{k}[t_{1},\ldots,t_{n}]^{S_{n}}$ and the elementary symmetric polynomials in $\sigma_{i}$ and $t_{i}\sigma_{i}^{-1}$. Hence it is a rational Galois order.

∎

With these results in mind, we think the following conjecture is natural:

Conjecture 1**.**

All Galois orders are principal Galois orders.

We now discuss a generalization of the Gelfand-Kirillov Conjecture for linear Galois algebras of shift type discussed in [9, Theorem 6], that also reproves the result in [23, Theorem 8.4].

Theorem 4.11**.**

Let $U$ be a Galois algebra in $(\mathsf{k}(t_{1},\ldots,t_{n};z_{1},\ldots,z_{s})*\mathcal{M})^{G}$ such that $\mathcal{M}\simeq\mathbb{Z}^{n}$ with canonical basis acting by shifts in $t_{1},\ldots,t_{n}$ and fixing the $z_{1},\ldots,z_{s}$. If $G$ is a pseudoreflection group or $\mathsf{k}(t_{1},\ldots,t_{n};z_{1},\ldots,z_{s})^{G}$ is rational, then the Gelfand-Kirillov Conjecture holds for $U$: $Frac\,U\simeq Frac\,A_{n}(\mathsf{k}(z_{1},\ldots,z_{s}))$.

Proof.

Follows directly from [19, Theorem 6.1], and Proposition 3.3. ∎

Remark 4.12**.**

The last Theorem holds without the hypothesis in Assumption.

5. Invariants of generalized Weyl algebras

In this section, we discuss the realization of a wide class of invariants of generalized Weyl algebras as principal Galois orders. In particular, we extend the result in [18, Theorem 27] and show that for all groups $G(m,p,n)$, the invariants of the Weyl algebra are a principal Galois order.

We recall the following important result.

Theorem 5.1**.**

Let $U$ be a finitely generated left and right Noetherian $\mathsf{k}$-algebra, and $G$ a finite group of $\mathsf{k}$-algebra automorphisms. Then $U^{G}$ is a finitely generated $\mathsf{k}$ algebra.

Proof.

[30]. ∎

Definition 5.2**.**

Let $D(a,\sigma)$ be a generalized Weyl algebra of rank 1, with $\sigma$ an automorphism of infinite order. We define $D_{n}(a,\sigma)$ to be $D(a,\sigma)^{\otimes n}$. It is itself a generalized Weyl algebra (cf. Proposition 2.10) $D^{\prime}(a=(a_{1},\ldots,a_{n}),\sigma=(\sigma_{1},\ldots,\sigma_{n}))$, where $D^{\prime}=D\otimes\ldots\otimes D$ n times, $a_{i}=1\otimes\ldots\otimes a\otimes\ldots 1$, $a$ in the i-th position, and $\sigma_{i}=1\otimes\ldots\otimes\sigma\otimes\ldots 1$, $\sigma$ in the i-th position.

We observe that, under these conditions, $D_{n}(a,\sigma)$ is always a Galois order ([15], [18, Theorem 14]).

In the rest of this section, we will consider generalized Weyl algebras $D_{n}(a,\sigma)$ when the automorphism has infinite order and $D=\mathsf{k}[h]$ or $D=\mathsf{k}[h^{\pm}]$. Whenever we consider the embedding of Theorem 2.11, $D_{n}(a,\sigma)\rightarrow\mathsf{k}(h_{1},\ldots,h_{n})*\mathbb{Z}^{n}$, the canonical basis of $\mathbb{Z}^{n}$ will be denoted by $e_{1},\ldots,e_{n}$, with $e_{i}$ acting as $\sigma_{i}$.

The following proposition is essential for what follows.

Proposition 5.3**.**

Consider $D_{n}(a,\sigma)$, a generalized Weyl algebra of rank $n$ with $D^{\prime}=\mathsf{k}[h_{1},\ldots,h_{n}]$ or $\mathsf{k}[h_{1}^{\pm},\ldots,h_{n}^{\pm}]$. We have an action of $S_{n}$ by algebra automorphisms on $D_{n}(a,\sigma)$ such that $\pi(h_{i})=h_{\pi(i)}$, $\pi(X_{i})=X_{\pi(i)}$ and $\pi(Y_{i})=Y_{\pi(i)}$. Then $D_{n}(a,\sigma)^{S_{n}}$ is a principal Galois order in $(\mathsf{k}(h_{1},\ldots,h_{n})*\mathbb{Z}^{n})^{S_{n}}$ with Harish-Chandra subalgebra $\mathsf{k}[h_{1},\ldots,h_{n}]^{S_{n}}$ or $\mathsf{k}[h_{1}^{\pm},\ldots,h_{n}^{\pm}]^{S_{n}}$. $D_{n}(a,\sigma)^{S_{n}}$ is a free left and right module over the Harish-Chandra subalgebra.

Proof.

By Theorem 5.1, $D_{n}(a,\sigma)^{S_{n}}$ has a finite generating set as $\mathsf{k}$-algebra, $d_{1},\ldots,d_{s}$, to which we can adjoin $d=X_{1}+\ldots+X_{n}$ and $d^{*}=Y_{1}+\ldots Y_{n}$. By Theorem 2.11, $D_{n}(a,\sigma)^{S_{n}}$ embedds in $(\mathsf{k}(h_{1},\ldots,h_{n})*\mathbb{Z}^{n})^{S_{n}}$ with $d\mapsto e_{1}+\ldots+e_{n}$ and $d^{*}\mapsto a_{1}e_{1}^{-1}+\ldots+a_{n}e_{n}^{-1}$. Hence we can apply Proposition 2.6, and $D(a,\sigma)^{S_{n}}$ is then a Galois ring in $(\mathsf{k}(h_{1},\ldots,h_{n})*\mathbb{Z}^{n})^{S_{n}}$. Then, by Lemma 4.5 and Proposition 4.6, it is a principal Galois order. $D_{n}(a,\sigma)$ is a free $D^{\prime}$-module, and hence the Galois order is a free moduler over its Harish-Chandra subalgebra by Lemma 2.7 and [1, Corol. 4.5], as $D^{\prime}$ is a projective $D^{\prime S_{n}}$-module. ∎

Let $G_{m}=\langle g\rangle\subset k$ be the cyclic group with $m$ elements, generated by a primitive $m$-root of unity $\xi$.

Theorem 5.4**.**

Consider $G_{m}$ with an action on $D(a,\sigma)$ with $gX=\xi X$, $gY=\xi^{-1}Y$, and trivially on $D$. Then $D(a,\sigma)^{G_{m}}$ is isomorphic to $D(a_{m},\sigma^{m})$, where $a_{m}$ is $a\sigma^{-1}(a)\ldots\sigma^{-(m-1)}a$. If $X,Y$ are the GWA generators of $D(a,\sigma)$ and $X^{\prime},Y^{\prime}$ the ones from $D(a_{m},\sigma^{m})$, the mentioned isomorphism sends $X^{m}\mapsto X^{\prime}$, $Y^{m}\mapsto Y^{\prime}$.

Proof.

[24, Thm. 2.7]. ∎

Proposition 5.5**.**

Consider the diagonal action of $G_{m}^{\otimes n}$ on $D_{n}(a,\sigma)$, induced by the action above. Then $D_{n}(a,\sigma)^{G_{m}^{\otimes n}}\simeq D_{n}(a_{m},\sigma^{m})$. If $X_{i},Y_{i}$ are the GWA generators of $D_{n}(a,\sigma)$ and $X_{i}^{\prime},Y_{i}^{\prime}$ the ones from $D_{n}(a_{m},\sigma^{m})$, the isomorphism sends $X_{i}^{m}\mapsto X_{i}^{\prime}$, $Y_{i}^{m}\mapsto Y_{i}^{\prime}$.

Proof.

Immediate from the above theorem. ∎

Corollary 5.6**.**

Under the situation above, $D(a,\sigma)^{G_{m}}$ embedds into $Frac\,D*\mathcal{M}$, where $\mathbb{Z}=\langle e\rangle$ is the group of automorphisms of $D$ such that $e^{y}=\sigma^{y}$, for every integer $y$, and $\mathcal{M}=\langle e^{m}\rangle$.

Proof.

It follows from Theorem 2.11, because $D(a,\sigma)^{G_{m}}\simeq D(a_{m},\sigma^{m})$ by Theorem 5.4. ∎

Proposition 5.7**.**

Let $D_{n}(a,\sigma)$ be a generalized Weyl algebra of rank $n$. Let $\mathbb{Z}^{n}$ be the group of automorphisms of $k[h_{1},\ldots,h_{n}]$ or $k[h_{1}^{\pm},\ldots,h_{n}^{\pm}]$ such that $e_{i}(h_{j})=\sigma_{i}(h_{j})$, $e_{1},\ldots,e_{n}$ the canonical basis of $\mathbb{Z}^{n}$. Let $\mathcal{M}=\langle e_{1}^{m},\ldots,e_{n}^{m}\rangle\subset\mathbb{Z}^{n}$. Then for $G=G(m,1,n)$, $D_{n}(a,\sigma)^{G}$ is a principal Galois order in $(k(h_{1},\ldots,h_{n})*\mathcal{M})^{S_{n}}$. $D_{n}(a,\sigma)$ is a free left and right module over its Harish-Chandra subalgebra $\mathsf{k}[h_{1},\ldots,h_{n}]^{S_{n}}$ or $\mathsf{k}[h_{1}^{\pm},\ldots,h_{n}^{\pm}]^{S_{n}}$.

Proof.

Recall that $G(m,1,n)=G_{m}^{\otimes n}\rtimes S_{n}$. Hence $D_{n}(a,\sigma)^{G(m,1,n)}\simeq(D_{n}(a,\sigma)^{G_{m}^{\otimes n}})^{S_{n}}$. In this way our result follows from Propositions 5.3, 5.5, using the embedding $D(a,\sigma)\rightarrow\mathsf{k}(h_{1},\ldots,h_{n})*\mathbb{Z}^{n}$ in Theorem 2.11; then we apply Corollary 5.6. ∎

Remark 5.8**.**

We notice that this action of $G(m,1,n)$ differs, in the case of the quantum affine space and the quantum torus, from the action considered in [17].

Given this action of $G(m,1,n)$, we also have one of the subgroup $G(m,p,n)$ on $D_{n}(a,\sigma)$.

Theorem 5.9**.**

$D_{n}(a,\sigma)^{G(m,p,n)}=\bigoplus_{k=0}^{p-1}(X_{1}\ldots X_{n})^{km/p}D_{n}(a,\sigma)^{G(m,1,n)}$.

Proof.

Let $\Xi=(g,\pi)\in A(m,1,n)\rtimes S_{n}$, $g=(g_{1},\ldots,g_{n})$ be an element of $G(m,1,n)$ that, in the quotient by $G(m,p,n)$, maps to generator of $G_{p}$ (cf. [18, Lemma 29]). We have $\Xi(X_{1},\ldots X_{n})^{m/p}=\xi(X_{1},\ldots X_{n})^{m/p}$, where $\xi$ is a $p$-root of unity. Hence in $(X_{1}\ldots X_{n})^{km/p}D_{n}(a,\sigma)^{G(m,1,n)}$, $\Xi$ acts with eigenvalue $\xi^{m}$. Since $\Xi$ is an operator with order $p$, $0=\prod_{i=0}^{p-1}(\Xi-\xi^{i}I)$ anihilates $D_{n}(a,\sigma)^{G(m,p,n)}$. The direct sum, then, is just the eigenspace decomposition for the operator $\Xi$. ∎

Theorem 5.10**.**

Let $D_{n}(a,\sigma)$ be a generalized Weyl algebra of rank $n$. Let $\mathbb{Z}^{n}$ be the group of automorphisms of $k[h_{1},\ldots,h_{n}]$ or $k[h_{1}^{\pm},\ldots,h_{n}^{\pm}]$ such that $e_{i}(h_{j})=\sigma_{i}(h_{j})$, $e_{1},\ldots,e_{n}$ the canonical basis of $\mathbb{Z}^{n}$. Let $\mathfrak{M}=\langle e_{1}^{m},\ldots,e_{n}^{m},(e_{1}e_{2}\ldots e_{n})^{m/p}\rangle\subset\mathbb{Z}^{n}$. Then for $G=G(m,p,n)$, $D_{n}(a,\sigma)^{G}$ is a principal Galois order in $(k(h_{1},\ldots,h_{n})*\mathfrak{M})^{S_{n}}$. The subalgebra $k[h_{1},\ldots,h_{n}]^{S_{n}}$ or $k[h_{1}^{\pm},\ldots,h_{n}^{\pm}]^{S_{n}}$ is the Harish-Chandra subalgebra and the principal Galois order is a free left and right module over it.

Proof.

By Propopostion 5.7 and Theorem 5.9. Being explict: $D_{n}(a,\sigma)^{G}$ is generated by by $X_{i}^{m},Y_{i}^{m},(X_{1}\ldots X_{n})^{m/p}$ over $D$, by Theorem 5.9, and the embedding that realizes the invariant algebra as a Galois algebra is $X_{i}^{m}\mapsto e_{i}^{m}$, $Y_{i}\mapsto a_{im}e_{i}^{-m}$, $(X_{1}\ldots X_{n})^{m/p}\mapsto(e_{1}e_{2}\ldots e_{n})^{m/p}$, $a_{im}=a_{i}\sigma_{i}^{-1}(a_{i})\ldots\sigma_{i}^{-(m-1)}(a_{i})$, $i=1,\ldots,n$. ∎

Now we can prove our generalization of [18, Theorem 27].

Corollary 5.11**.**

The invariants of the Weyl algebra $A_{n}(\mathsf{k})^{G(m,p,n)}$ are principal Galois order in $(k(h_{1},\ldots,h_{n})*\mathfrak{M})^{S_{n}}$, where $e_{i}(h_{j})=h_{j}-\delta_{ij}$, we have $\mathfrak{M}=\langle e_{1}^{m},\ldots,e_{n}^{m},(e_{1}e_{2}\ldots e_{n})^{m/p}\rangle\subset\mathbb{Z}^{n}$ and $h_{i}=\partial_{i}x_{i}$. The Harish-Chandra subalgebra is $\mathsf{k}[h_{1},\ldots,h_{n}]^{S_{n}}$ and $A_{n}(\mathsf{k})^{G(m,p,n)}$ is a free left and right module over it.

Proof.

A particular case of the above theorem, since $A_{1}(\mathsf{k})=\mathsf{k}[h](h,\sigma)$, where $\sigma(h)=h-1$. ∎

This has a consequence for the skew field of fractions of a certain invariant skew group ring, related to the Gelfand-Kirillov Conjecture.

Corollary 5.12**.**

Let $\mathbb{Z}^{n}$ be the group of automorphisms of $k(h_{1},\ldots,h_{n})$, $e_{i}(h_{j})=h_{j}-\delta_{ij}$, $e_{i}$ a canonical basis. Let $\mathfrak{M}=\langle e_{1}^{m},\ldots,e_{n}^{m},(e_{1},\ldots,e_{n})^{m/p}\rangle\subset\mathbb{Z}^{n}$ , $p|m$. Then $Frac\,(k(h_{1},\ldots,h_{n})*\mathfrak{M})^{S_{n}}\simeq Frac\,A_{n}(\mathsf{k})$.

Proof.

By Corollary 5.11 and Proposition 3.3, we have that $Frac\,(k(h_{1},\ldots,h_{n})*\mathfrak{M})^{S_{n}}\simeq Frac\,A_{n}(\mathsf{k})^{G(m,p,n)}$; the later skew field is isomorphic to $Frac\,A_{n}(\mathsf{k})$ by [19, Theorem 1.1]. ∎

Finally, we also have a generalization of the result in [17, Theorem 8] that $(A_{n}^{q})^{S_{n}}$ is a Galois order.

Corollary 5.13**.**

For $G=G(m,p,n)$, $(A_{n}^{q})^{G}$ is a Galois order over polynomial Harish-Chandra subalgebra, and it is a free left and right module over this subalgebra.

Proof.

A simple consequence of Lemma 4.8 and Theorem 5.10.

∎

6. Weight modules for generalized Weyl algebras

In this section $D$ will be an affine commutative domain over an algebraically closed field $\mathsf{k}$ of any characteristic, $D(a,\sigma)$ a completely arbitrary generalized Weyl algebra of rank $n$.

Definition 6.1**.**

A finitely generated $D(a,\sigma)$ module $M$ is called a weight module if $M|_{D}=\bigoplus_{m\in Specm\,D}M(m)$, where $M(m)=\{v\in M|mv=0\}$. That is, $D$ acts diagonally on $M$.

One of the main properties of Galois order theory [16] is that, if $U$ is a Galois order over a commutative $\Gamma$, then every maximal ideal $m$ of it lift to an irreducible Gelfand-Tsetlin module (cf. Definition 2.8) $M$ such that $M(m)\neq 0$ (and moreover the number of such liftings is finite [16]).

For every $m\in Specm\,D$, the liftng result mentioned above holds for irreducible weight modules for $D(a,\sigma)$ for all generalized Weyl algebras. This is an easy consequence of the fact that a generalized Weyl algebra is free module over its novel subalgebra. The novelty here is that we will show this lifting explicty.

Given a maximal ideal of $D$, $m$, denote by $f_{m}:D/m\rightarrow\mathsf{k}$ the canonical isomorphism. Let $\phi$ be any automorphism of $D$. Then it is easy to see that:

[TABLE]

In the classical case of Gelfand-Tsetlin modules for $gl_{n}$ the irreducible finite dimensional ones are parametrized up to finitness by the Gelfand-Tsetlin tableaux. These are nothing but elements of the maximal spectrum of of the Gelfand-Tsetlin subalgebra, since it is polynomial (cf. [7]). Motivated by this and the construction of generic Gelfand-Tsetlin modules in [7], we have the following construction.

Fix a maximal ideal $m$ of $D$. Consider the symbols $T(\theta(m))$, where $\theta$ runs throught all the elements of the group of automorphisms of $D$ generated the $\sigma_{i}$, $i=1,\ldots,n$. We call those $T(.)$ generalized tableaux. Let $T_{m}$ be the vector space with basis the generalized tableaux.

For each basis element $T(n)\in T_{m}$, where $n$ is an ideal of the form $\theta(m)$, define the following linear actions of $D(a,\sigma)$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proposition 6.2**.**

With the above actions, we have a representation of the generalized Weyl algebra $D(a,\sigma)$ on $T_{m}$

Proof.

We need to check that the operators defined above satisfy the defining relations of the generalized Weyl algebras (cf. Definition 2.9). This is easy to do, by repeated application of $(\dagger)$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The other defining relations are trivial to check. ∎

Theorem 6.3**.**

For each $m\in Specm\,D$, there exists an irreducible $D(a,\sigma)$-module $T_{m}$ which has a weight-space decomposition with respect to $D$, lifting $m$ — that is, with non-trivial $m$-weight space.

Proof.

Immediate by the above Proposition. The fact that the module is irreducible is clear by construction. ∎

In particular, if $D(a,\sigma)$ is a Galois order, Theorem 6.3 gives us a family of irreducible Gelfand-Tsetlin modules, refining the results in [18].