Classification of irreducible Gelfand-Tsetlin modules of sl(3)

TL;DR

This paper classifies and explicitly constructs all irreducible Gelfand-Tsetlin modules of sl(3), detailing their realizations and structures, including those with infinite-dimensional weight spaces, using tableaux and subquotients.

Contribution

It provides the first complete classification and explicit realization of all irreducible Gelfand-Tsetlin modules of sl(3), including those with infinite-dimensional weight spaces.

Findings

All simple Gelfand-Tsetlin sl(3)-modules with infinite-dimensional weight spaces are listed.

Modules are realized using regular and derivative Gelfand-Tsetlin tableaux.

Simple modules are expressed as subquotients of localized Gelfand-Tsetlin E_{21}-injective modules.

Abstract

We provide a classification and an explicit realization of all irreducible Gelfand-Tsetlin modules of the complex Lie algebra sl(3). The realization of these modules uses regular and derivative Gelfand-Tsetlin tableaux. In particular, we list all simple Gelfand-Tsetlin sl(3)-modules with infinite-dimensional weight spaces. Also, we express all simple Gelfand-Tsetlin sl(3)-modules as subquotionets of localized Gelfand-Tsetlin E_{21}-injective modules.

Equations

$$\mathfrak{h}=\mathrm{Span}\{h_i=E_{ii}-E_{i+1,i+1}|i=1,\dots ,n-1\}.$$

$${\mathrm{gl}}_1\subset {\mathrm{gl}}_2\subset \dots \subset {\mathrm{gl}}_n.$$

$$c_{mk}=\sum _{(i_1,\dots ,i_k)\in \{1,\dots ,m{\}}^k}{E}_{i_1{i}_2}{E}_{i_2{i}_3}\dots E_{i_k{i}_1}.$$

$${\gamma }_{mk}(l)=\sum _{i=1}^m(l_{mi}+m-1{)}^k\prod _{j\ne i}^m\left(1-\frac{1}{l_{mi}-l_{mj}}\right).$$

$${\gamma }_{11}(l)=$$

$${\gamma }_{21}(l)=$$

$${\gamma }_{22}(l)=$$

$${\gamma }_{31}(l)=$$

$${\gamma }_{32}(l)=$$

$${\gamma }_{33}(l)=$$

$$+(l_{31}+l_{32}+l_{33}).$$

$${\gamma }_{41}(l)=$$

$${\gamma }_{42}(l)=$$

$${\gamma }_{43}(l)=$$

$$+6(l_1^2+l_2^2+l_3^2+l_4^2)+3(l_1+l_2+l_3+l_4)-19.$$

$${\gamma }_{44}(l)=$$

$$(l_1^2{l}_2+l_1^2{l}_3+l_1^2{l}_4+l_1{l}_2^2+l_1{l}_3^2+l_1{l}_4^2+l_2^2{l}_3+l_2^2{l}_4+l_2{l}_3^2+l_2{l}_4^2+l_3^2{l}_4+l_3{l}_4^2)$$

$$-10(l_1{l}_2+l_1{l}_4+l_2{l}_3+l_3{l}_4+l_1{l}_3+l_2{l}_4)-19(l_1+l_2+l_3+l_4)-120.$$

$$M{|}_{\Gamma }=\underset{\mathsf{m}\in \mathrm{Specm}\Gamma }{⨁}M(\mathsf{m}),$$

$$M(\mathsf{m})=\{v\in M|{\mathsf{m}}^{k}v=0\text{ for some }k\ge 0\}.$$

$$M(\chi )=\{v\in M|\text{ for each }\gamma \in \Gamma \text{ , }\exists k\in {\mathbb{Z}}_{\ge 0}\text{ such that }(\gamma -\chi (\gamma ){)}^{k}v=0\}.$$

$${\mathrm{Supp}}_{GT}(M):=\{\chi \in {\Gamma }^{\ast }|M(\chi )\ne 0\}.$$

$$M=\underset{\lambda \in {\mathfrak{h}}^{\ast }}{⨁}{M}_{\lambda },$$

$$M_{\lambda }:=\{v\in M|hv=\lambda (h)v\text{ for all }h\in \mathfrak{h}\}.$$

$$\mathrm{Supp}(M):=\{\lambda \in {\mathfrak{h}}^{\ast }|M_{\lambda }\ne 0\}.$$

$${\mathrm{gl}}_1\subset \dots \subset {\mathrm{gl}}_n.$$

$$v=(v_{n1},...,v_{nn}|v_{n-1,1},...,v_{n-1,n-1}|\cdots |v_{21},v_{22}|v_{11})\in {\mathbb{C}}^{\frac{n(n+1)}{2}}$$

$$\{{\gamma }_{mk}(l)=\chi (c_{mk}){\}}_{1\le k\le m\le n}$$

$$E_{k,k+1}(T(v))=-\sum _{i=1}^k\left(\frac{{\prod }_{j=1}^{k+1}(v_{ki}-v_{k+1,j})}{{\prod }_{j\ne i}^k(v_{ki}-v_{kj})}\right)T(v+{\delta }^{ki}),$$

$$E_{k+1,k}(T(v))=\sum _{i=1}^k\left(\frac{{\prod }_{j=1}^{k-1}(v_{ki}-v_{k-1,j})}{{\prod }_{j\ne i}^k(v_{ki}-v_{kj})}\right)T(v-{\delta }^{ki}),$$

$$E_{kk}(T(v))=\left(k-1+\sum _{i=1}^k{v}_{ki}-\sum _{i=1}^{k-1}{v}_{k-1,i}\right)T(v),$$

$$c_{rs}(T(v))={\gamma }_{rs}(v)T(v),$$

$$\mathrm{Supp}(M)=\{(1,1),(-1,2),(2,-1),(0,0),(-2,1),(1,-2),(-1,-1)\}.$$

$${\dim }_{\mathbb{C}}M(\mathsf{m})\le Q_n.$$

$$\mathcal{B}(T(v)):=\{T(v+z)|z\in T_n(\mathbb{Z}),z_{nk}=0,1\le k\le n\}.$$

$$\sigma (w):=(w_{n-1,{\sigma }^{-1}[n-1](1)},\dots ,w_{n-1,{\sigma }^{-1}[n-1](1)}|\dots |w_{1,{\sigma }^{-1}[1](1)}).$$

$${\epsilon }_{rs}:={\delta }^{r,1}+{\delta }^{r+1,1}+\dots +{\delta }^{s-1,1}\in T_n(\mathbb{Z}).$$

Full text

MSC 2010 Classification: 16G99, 17B10.

Keywords: Gelfand-Tsetlin modules, Gelfand-Tsetlin bases, weight modules, localization.

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Index of notations

4. 3 Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$ and $\mathfrak{sl}(n)$

5. 3.1 Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$

6. 3.2 Gelfand-Tsetlin modules of $\mathfrak{sl}(n)$

7. 4 Families of Gelfand-Tsetlin modules for $\mathfrak{gl}(n)$

8. 4.1 Gelfand-Tsetlin tableaux

9. 4.2 Gelfand-Tsetlin formulas for finite-dimensional modules

10. 4.3 Generic modules

11. 4.3.1 Gelfand-Tsetlin formulas in terms of permutations

12. 4.4 Singular Gelfand-Tsetlin modules

13. 4.4.1 Construction of $1$-singular Gelfand-Tsetlin modules

14. 5 Gelfand-Tsetlin modules of $\mathfrak{sl}(3)$ and modules of $C({\mathfrak{h}})$

15. 6 simple extensions of singular Gelfand-Tsetlin characters

16. 7 Realizations of all simple Gelfand-Tsetlin modules for $\mathfrak{sl}(3)$

17. 7.1 Structure of generic $\mathfrak{sl}(3)$-modules $V(T(v))$

18. 7.2 Realizations of all simple generic Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules

19. 7.3 Structure of singular $\mathfrak{sl}(3)$-modules $V(T(\bar{v}))$

20. 7.3.1 Singular Gelfand-Tsetlin formulas

21. 7.3.2 Explicit formulas and computations

22. 7.3.3 Submodules generated by singular tableaux

23. 7.4 The singular block containing $L(-\rho)$

24. 7.5 Realizations of all simple singular Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules

25. 8 Localization on Gelfand-Tsetlin modules

26. 8.1 Localization and twisted localization functors

27. 8.2 Localization functors in the case of $\mathfrak{sl}(3)$

28. 8.2.1 Injectivity and surjectivity of the operator $E_{21}$

29. 8.2.2 twisted localization with respect to $E_{21}$

30. 8.3 Simple Gelfand-Tsetlin modules and localization functors

1. Introduction

Let ${\mathfrak{g}}$ be a finite-dimensional simple Lie algebra over the complex numbers, and let ${\mathfrak{h}}$ be a Cartan subalgebra of ${\mathfrak{g}}$. A ${\mathfrak{g}}$-module $M$ is called a weight module if $M=\bigoplus_{\lambda\in{\mathfrak{h}}^{*}}M_{\lambda}$, where $M_{\lambda}=\{v\in M\mid hv=\lambda(h)v,\ \forall h\in{\mathfrak{h}}\}$. The space $M_{\lambda}$ is called a weight space, the set $\{\lambda\in{\mathfrak{h}}^{*}\ |\ M_{\lambda}\neq 0\}$ is called the weight support of $M$ and the dimension of $M_{\lambda}$ is called the weight mltiplicity of $\lambda$. If $\mathfrak{h}$ does not necessarily act diagonally but acts locally finitely on $M$, then we say that $M$ is a generalized weight module. It is an easy exercise to show that a simple generalized weight module is a weight module.

A weight module $M$ is torsion free provided that all root vectors of ${\mathfrak{g}}$ act injectively on $M$. If $M$ is a torsion free module then all weight multiplicities of $M$ (finite or infinite) are equal. This invariant of $M$ is called the weight degree of $M$. Furthermore, the weight support of a torsion free module $M$ coincides with a full coset $\lambda+Q$ of ${\mathfrak{h}}^{*}/Q$, where $Q$ is the root lattice of $\mathfrak{g}$ and $\lambda$ is in the weight support of $M$. On the other hand, a simple weight module may have “full support” without being torsion free, in which case the weight multiplicities are necessarily infinite. The first examples of such modules were given in [F86b]. Simple modules with full support are called dense.

A breakthrough in the theory of weight modules with finite weight multiplicities was made by Fernando, [Fe90], in 1990 who reduced the classification of all such simple modules to determining the simple torsion free modules. He also showed that the only simple Lie algebras admitting torsion free modules are those of type $A$ or $C$. The next major breakthrough was made in 2000 by Mathieu, [M00], who classified and provided a realization of all simple torsion free modules of finite degree. Previously, the case of degree $1$ was worked out in [BL87]. Important properties of the annihilators of the torsion free modules were established in [Jos92].

The study of weight modules with infinite multiplicities is still at its initial stage. A result similar to the one of Fernando reduces the classification of all such simple modules to the classification of all simple dense modules of simple Lie algebras. For the classical simple Lie algebras this reduction was obtained in [F86a] and for all exceptional simple Lie algebras except $E_{8}$ in [FOT95]. Finally, in [DMP00], the reduction in all cases, including all important classes of finite-dimensional Lie superalgebras, was completed.

The classification of all simple dense modules, possibly with infinite weight multiplicities, remains out of reach. The latter classification is present only in the case of $\mathfrak{sl}(2)$, where in the fundamental paper of Block, [B81], all simple $\mathfrak{sl}(2)$-modules are classified, and in particular, it is shown that the simple dense modules have always weight degree $1$.

One natural category of weight modules is the category of Gelfand-Tsetlin modules. More precisely, this is the full subcategory of the category of generalized weight modules consisting of modules that admit a generalized eigenbasis for the Gelfand-Tsetlin subalgebra, a maximal commutative subalgebra of the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$. Gelfand-Tsetlin modules were introduced in [DFO89], [DFO92], [DFO94] as an attempt to generalize the celebrated tableaux construction of Gelfand-Tsetlin bases of irreducible finite-dimensional representations of simple classical Lie algebras, [GT50], [Mol06], [Zh74].

Gelfand-Tsetlin subalgebras have applications that extend beyond the study of Gelfand-Tsetlin modules. For example, these subalgebras were related to the solutions of the Euler equation in [FoM78], and to the subalgebras of $U(\mathfrak{g})$ of maximal Gelfand-Kirillov dimension in [Vi91]. Gelfand-Tsetlin subalgebras were studied in [KW06a], [KW06b] in connection with classical mechanics, and also in [Gr04], [Gr07] in connection with general hypergeometric functions on the Lie group $GL(n,\mathbb{C})$.

A general theory of Gelfand-Tsetlin modules for a class of Galois algebras (for a definition see [FO10]) was developed in [FO14]. The results for these Galois algebras can be applied to the universal enveloping algebras of $\mathfrak{sl}(n)$ and $\mathfrak{gl}(n)$, and provide structural properties of the corresponding simple Gelfand-Tsetlin modules. In the generic case the characters of the Gelfand-Tsetlin subalgebra parametrize such simple modules. However, in the nongeneric case, i.e. in the singular case and $n>2$, we may have more than one isomorphism class of simple Gelfand-Tsetlin modules with a fixed character of the Gelfand-Tsetlin subalgebra. The theory of singular Gelfand-Tsetlin modules was initiated in [FGR16] where $1$-singular modules were constructed and studied in detail. Immediately after the construction of the 1-singular modules, there was an abundance of successful attempts to construct simple Gelfand-Tsetlin modules with a given singular character. For more details, we refer the reader to the following papers [EMV], [FGR16], [FGR16a], [FGR17], [FGRZ], [FGRZ1], [FK17], [FRZ16], [H17], [RZ17], [V17a], [V17b], [Za17]. In poarticular, a classification of the simple Gelfand-Tsetlin modules was recently announced in [KTWWY] and [W].

A classification of the simple $1$-singular Gelfand-Tsetlin modules was obtained in [FGR17] and leads to the classification of all simple Gelfand-Tsetlin modules of the Lie algebra $\mathfrak{sl}(3)$ (and of $\mathfrak{gl}(3)$). The latter classification is the main purpose of the present paper and it is provided via very explicit tableaux construction.

Our classification result relies on various old results on Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules obtained in [BFL95], [F86b], [F89], [F86a], [F91], combined with newer results from [FGR14] and [R13]. We remark that some technical statements in the paper on the properties of Gelfand-Tsetlin modules can be simplified using the theory recently developed, for example, in [FGR17], [FGRZ1], [RZ17]. However, for the sake of completeness and for reader’s convenience, we opted to keep the original manuscript containing detailed and explicit proofs.

The structure of the paper is as follows. In Section $2$ we set up the notation and state basic definitions and results needed in the rest of the paper. In Section $3$ we prove some general results about the Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$. Section $4$ is devoted to the description of certain “easier to study” classes of Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$, namely finite-dimensional modules, generic modules, and $1$-singular modules. In Section 5 we collect some important definitions and preliminary results that relate Gelfand-Tsetlin modules to their Gelfand-Tsetlin character. In Section $6$ we prove the main results about existence and uniqueness of simple Gelfand-Tsetlin modules of $\mathfrak{sl}(3)$. The explicit description of all simple Gelfand-Tsetlin modules for $\mathfrak{sl}(3)$ is included in Section $7$ . Finally, in Section $8$, we study localization functors on the category of Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules and prove that any simple module in this category can be obtain from an $E_{21}$-injective module using a $E_{21}$-localization functors.

Acknowledgements. D.G. gratefully acknowledges the hospitality and excellent working conditions at the São Paulo University where part of this work was completed. V.F. is supported in part by the CNPq grant (304467/2017-0) and Fapesp grant (2018/23690-6). D.G. is supported in part by Simons Collaboration Grant 358245 and by Fapesp grant (2014/09310-5). L.E.R. was supported by Fapesp grant (2018/17955-7).

2. Preliminaries

The ground field will be ${\mathbb{C}}$. In the first part of the paper we fix an integer $n\geq 2$. For $a\in{\mathbb{Z}}$, we write $\mathbb{Z}_{\geq a}$ for the set of all integers $m$ such that $m\geq a$. Similarly, we define $\mathbb{Z}_{<a}$, etc. For a Lie algebra ${\mathfrak{a}}$ by $U(\mathfrak{a})$ we denote the universal enveloping algebra of ${\mathfrak{a}}$. For a commutative ring $R$, ${\rm Specm}\,R$ will stand for the set of maximal ideals of $R$.

By $\operatorname{\mathfrak{gl}}(n)$ we denote the general linear Lie algebra consisting of all $n\times n$ complex matrices, and by $\{E_{i,j}\ |\ 1\leq i,j\leq n\}$ - the standard basis of $\operatorname{\mathfrak{gl}}(n)$ of elementary matrices. We fix the standard Cartan subalgebra of $\operatorname{\mathfrak{gl}}(n)$, the standard triangular decomposition and the corresponding basis of simple roots of $\operatorname{\mathfrak{gl}}(n)$. The weights of $\operatorname{\mathfrak{gl}}(n)$ will be written as $n$-tuples $(\lambda_{1},...,\lambda_{n})$ through the identification ${\mathfrak{h}}^{*}\to{\mathbb{C}}^{n}$.

The Lie subalgebra ${\mathfrak{g}}=\mathfrak{sl}(n)$ of $\mathfrak{gl}(n)$ is generated by $\{E_{i,i+1},\ E_{i+1,i}\ |\ 1\leq i\leq n-1\}$. The standard Cartan subalgebra of ${\mathfrak{g}}$ will be denoted by $\mathfrak{h}$, i.e.

[TABLE]

Let $\epsilon_{i}$ denote the projection of a $n\times n$ matrix onto its $(i,i)^{th}$ entry. Then a basis of simple roots of the root system $\Delta$ of ${\mathfrak{g}}$ is given by $\pi=\{\alpha_{i}=\epsilon_{i}-\epsilon_{i+1}\ |\ i=1,\dots,n-1\}$ and the corresponding positive roots are $\Delta^{+}=\{\epsilon_{i}-\epsilon_{j}=\alpha_{i}+\cdots+\alpha_{j-1}\mid i<j\}$.

2.1. Index of notations

• •

§3.1. $c_{mk}$; $i(c_{mk})$; $\Gamma$; $\gamma_{mk}(l)$; $M(\chi)$; $Supp_{GT}(M)$; $M_{\lambda}$; $\operatorname{Supp}(M)$; $\Gamma(\pi)$; $\mathcal{GT}_{\pi}(n)$.

• •

§3.2. $\mathcal{GT}_{\pi}(n)$.

• •

§4.1 $T(v)$; $T_{n}(R)$; $\chi_{v}$.

• •

§4.2 $\delta^{ij}$; $Q_{n}$; $\mathcal{B}(T(v))$; $\mathcal{GT}_{T(v)}(n)$.

• •

§4.3. $V(T(v))$; $\tilde{S}_{m}$; $\Phi_{\ell m}$; $\varepsilon_{rs}$; $e_{t}^{(+)}(w)$; $e_{t+1}^{(-)}(w)$; $e_{r,s}(w)$; $\Omega(T(w))$; $\Omega^{+}(T(w))$; $\mathcal{N}(T(w))$; $\mathcal{I}(T(w))$.

• •

§4.4. $\mathcal{F}$; $T_{n}(\mathbb{C})_{\rm gen}$; $\mathcal{H}_{ij}^{k}$; $\Phi_{\ell m}(k,t)$; $\mathcal{H}$; $\overline{\mathcal{H}}$; $\mathcal{F}_{ij}$, $V(T(\bar{v}))$; $\mathcal{V}_{\rm gen}$; $\mathcal{S}$; $\mathcal{D}^{\bar{v}}$; $\mathcal{B}(T(\bar{v}))$; $\operatorname{Tab}(w)$.

• •

§5. $g_{\lambda}(x,y)$; $C(\mathfrak{h})$.

• •

§7.3. $\Lambda^{+}(Tab(w))$; $\mathcal{A}(\operatorname{Tab}(w))$; $\mathcal{N}(\operatorname{Tab}(w))$; $\mathcal{A}(\operatorname{Tab}(w))$; $\mathcal{C}(w)$; $\mathcal{R}(w)$; $\mathcal{A}(\operatorname{Tab}(w))$; $\mathcal{N}_{(r,s,p)}(\operatorname{Tab}(w))$; $\mathcal{N}^{(1)}(\operatorname{Tab}(w))$; $\mathcal{N}^{(2)}(\operatorname{Tab}(w))$; $\widehat{\mathcal{A}}(\operatorname{Tab}(w))$; $\widehat{\mathcal{N}}(\operatorname{Tab}(w))$; $\operatorname{Tab}(\mathcal{B})$.

• •

§7.5. $M({\mathcal{B}})$; $L({\mathcal{B}})$.

• •

§8 $\mathcal{D}_{ij}$; $\mathcal{D}^{x}_{ij}$; $\mathcal{QD}_{ij}$.

• •

§8.2. $\Theta_{x}(u)$; $\Phi_{\alpha}^{x}$.

• •

§8.3. $L_{i}^{(Gj)}$; $L_{i}^{(Cj)}$.

3. Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$ and $\mathfrak{sl}(n)$

3.1. Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$

Let for $m\leqslant n$, $\mathfrak{gl}_{m}$ be the Lie subalgebra of $\operatorname{\mathfrak{gl}}(n)$ spanned by $\{E_{ij}\,|\,i,j=1,\ldots,m\}$. We have the following chain

[TABLE]

It induces the chain $U_{1}\subset$ $U_{2}\subset$ $\ldots$ $\subset U_{n}$ for the universal enveloping algebras $U_{m}=U(\operatorname{\mathfrak{gl}}_{m})$, $1\leq m\leq n$. Let $Z_{m}$ be the center of $U_{m}$. Then $Z_{m}$ is the polynomial algebra in the $m$ variables $\{c_{mk}\,|\,k=1,\ldots,m\}$,

[TABLE]

The *(standard) * Gelfand-Tsetlin subalgebra ${{\Gamma}}$ in $U$ ([DFO94, GT50]) is generated by $\bigcup\limits_{i=1}^{n}Z_{i}$.

The algebra ${{\Gamma}}$ is a polynomial algebra in $\displaystyle\frac{n(n+1)}{2}$ variables $\{c_{ij}\,|\,1\leqslant j\leqslant i\leqslant n\}$. For $i=1,\ldots,n$ denote by $S_{i}$ the $i$-th symmetric group and set $G=S_{n}\times\cdots\times S_{1}$. Let $\Lambda$ be the polynomial algebra in variables $\{l_{ij}\,|$ $1\leqslant j\leqslant i\leqslant n\}$.

Let $\imath:{{\Gamma}}{\longrightarrow}$ $\Lambda$ be the embedding given by $\imath(c_{mk})=\gamma_{mk}(l)$ where

[TABLE]

The image of $\imath$ coincides with the subalgebra of $G-$invariant polynomials in $\Lambda$which we identify with ${\Gamma}$, see [Zh74] for more details.

Remark 3.1**.**

Note that $\Gamma$ contains the standard Cartan subalgebra of $\operatorname{\mathfrak{gl}}(n)$ spanned by $E_{ii}$, $i=1,\ldots,n$. Indeed, $c_{m1}=\sum\limits_{i=1}^{m}E_{ii}$ for each $1\leq m\leq n$. Therefore, $E_{ii}$ belong to $\Gamma$ for each $1\leq i\leq n$.

Remark 3.2**.**

We should note that the polynomials $\gamma_{mk}(l)$ are symmetric of degree $k$ in variables $l_{m1},\ldots,l_{mm}$, and $\{\gamma_{m1}(l),\ldots,\gamma_{mm}(l)\}$ generate the algebra of $S_{m}$–invariant polynomials in the variables $l_{m1},\ldots l_{mm}$ (see [Zh74]).

Example 3.3**.**

Tthe polynomials $\gamma_{mk}(l)$ for $m\leq 4$ are listed below.

[TABLE]

Definition 3.4**.**

A finitely generated $U$-module $M$ is called a Gelfand-Tsetlin module (relative to ${\Gamma}$) provided that the restriction $M|_{{\Gamma}}$ is a direct sum of ${\Gamma}$-modules:

[TABLE]

where

[TABLE]

Definition 3.5**.**

An algebra homomorphism $\chi:\Gamma\rightarrow\mathbb{C}$ will be called Gelfand-Tsetlin character.

Remark 3.6**.**

For each $\mathbf{m}\in\operatorname{Specm}{\Gamma}$ we have associated a character $\chi_{\mathbf{m}}:{\Gamma}\rightarrow{\Gamma}/\mathbf{m}\sim\mathbb{C}$. In the same way, for each non-zero character $\chi:{\Gamma}\rightarrow\mathbb{C}$, $Ker(\chi)$ is a maximal ideal of ${\Gamma}$. So, we have a natural identification between characters of ${\Gamma}$ and elements of $\operatorname{Specm}{\Gamma}$. So, using Gelfand-Tsetlin characters, a Gelfand-Tsetlin module (with respect to $\Gamma$) $M$ can be decomposed as $M=\bigoplus_{\chi\in\Gamma^{*}}M(\chi)$, where

[TABLE]

Definition 3.7**.**

Given a Gelfand-Tsetlin module $M$, the Gelfand-Tsetlin support of $M$ is the set

[TABLE]

Definition 3.8**.**

A finitely generated $U$-module $M$ is called a weight module (relative to $\mathfrak{h}$) provided that the restriction $M|_{\mathfrak{h}}$ is a direct sum of simple $\mathfrak{h}$-modules:

[TABLE]

where

[TABLE]

The weight support (or simply, the support) of $M$ is

[TABLE]

Remark 3.9**.**

Any simple Gelfand-Tsetlin module $M$ over $\operatorname{\mathfrak{gl}}(n)$ is a weight module with respect to the standard Cartan subalgebra $\mathfrak{h}$ spanned by $E_{ii}$, $i=1,\ldots,n$, see Remark 3.1. Moreover, ${\Gamma}$ is diagonalizable on any finite-dimensional simple module. On the other hand, a simple weight module $M$ need not to be Gelfand-Tsetlin, unless the $\mathfrak{h}$-weight multiplicities of $M$ are finite. The latter is true since in this case $\Gamma$ has a common eigenvector in every non-zero weight space. In particular, every highest weight module or, more general, every module from the category $\mathcal{O}$ is Gelfand-Tsetlin.

The definition of a Gelfand-Tsetlin module depends on the choice of the Gelfand-Tsetlin subalgebra $\Gamma$. One can easily define a family of Gelfand-Tsetlin subalgebras of $\operatorname{\mathfrak{gl}}(n)$ as follows. Let $\pi=\{\beta_{1},\ldots,\beta_{n-1}\}$ be a base of the root system of $\operatorname{\mathfrak{gl}}(n)$, where, $\beta_{k}=\epsilon_{i_{k}}-\epsilon_{i_{k+1}}$, $k=1,...,n-1$. Let $\operatorname{\mathfrak{gl}}_{k}$ be the subalgebra of $\operatorname{\mathfrak{gl}}(n)$ spanned by $E_{ij}$, $i,j\in\{i_{1},...,i_{k}\}$. In particular, $\operatorname{\mathfrak{gl}}_{k}\simeq\operatorname{\mathfrak{gl}}(k)$ and $\beta_{1},\ldots,\beta_{k}$ are the simple roots of $\operatorname{\mathfrak{gl}}_{k}$. Then we have a chain of embeddings

[TABLE]

Let $Z_{i}$ be the center of $U(\operatorname{\mathfrak{gl}}_{i})$ and $\Gamma(\pi)$ is the subalgebra generated by $Z_{i}$, $i=1,\ldots,n$. We will call $\Gamma(\pi)$ a Gelfand-Tsetlin subalgebra associated with $\pi$.

Each subalgebra $\Gamma(\pi)$ gives rise to a category of Gelfand-Tsetlin modules which we denote by $\mathcal{GT}_{\pi}(n)$. Let $\pi$ and $\pi^{\prime}$ be different bases of the root system. Then $\pi$ and $\pi^{\prime}$ are conjugate under the action of the Weyl group of $\operatorname{\mathfrak{gl}}(n)$. Hence $\Gamma(\pi)$ and $\Gamma(\pi^{\prime})$ are also conjugate which leads to an equivalence of the categories $\mathcal{GT}_{\pi}(n)$ and $\mathcal{GT}_{\pi^{\prime}}(n)$.

Example 3.10**.**

Two bases $\pi$ and $\pi^{\prime}$ may define the same Gelfand-Tsetlin subalgebra. Indeed, take for example the bases $\pi=\{\epsilon_{1}-\epsilon_{2},\epsilon_{2}-\epsilon_{3}\}$ and $\pi^{\prime}=\{\epsilon_{2}-\epsilon_{1},\epsilon_{1}-\epsilon_{3}\}$ of root system of $\operatorname{\mathfrak{gl}}(3)$. Then $\Gamma(\pi)=\Gamma(\pi^{\prime})$. One easily checks that $\operatorname{\mathfrak{gl}}(3)$ has three distinct Gelfand-Tsetlin subalgebras and they are parameterized of the $\operatorname{\mathfrak{gl}}_{2}$-part of the chain.

3.2. Gelfand-Tsetlin modules of $\mathfrak{sl}(n)$

Let ${\Gamma}$ be a Gelfand-Tsetlin subalgebra of $\operatorname{\mathfrak{gl}}(n)$. Consider the natural projection $\tau:\operatorname{\mathfrak{gl}}(n)\rightarrow\mathfrak{sl}(n)$, $\tau(A)=A-\frac{{\rm tr}(A)}{n}I_{n}$, which extends to an epimorphism $\bar{\tau}:U(\operatorname{\mathfrak{gl}}(n))\rightarrow U(\mathfrak{sl}(n))$. Then the image $\bar{\tau}({\Gamma})$ of ${\Gamma}$ is called the (standard) Gelfand-Tsetlin subalgebra of $\mathfrak{sl}(n)$. It is a maximal commutative subalgebra of $U(\mathfrak{sl}(n))$ isomorphic to a polynomial ring in $\frac{n(n+1)}{2}-1$ generators. With a small abuse of notation, by $\mathcal{GT}(n)$ we denote the category of all Gelfand-Tsetlin $\mathfrak{sl}(n)$-modules relative to ${\Gamma}$.

4. Families of Gelfand-Tsetlin modules for $\mathfrak{gl}(n)$

4.1. Gelfand-Tsetlin tableaux

The simple finite dimensional modules are the first natural examples of Gelfand-Tsetlin modules. In this case, an eigenbasis for the action of the generators of $\Gamma$ (1) is given by the so-called Gelfand-Tsetlin tableaux, following the original work of Gelfand and Tsetlin. In particular for every simple finite dimensional module $M$, $\dim(M(\chi))=1$ whenever $\chi\in\operatorname{Supp}_{GT}(M):$. In order to describe the Gelfand Tsetlin tableaux, we first fix some notation.

Definition 4.1**.**

Fix a vector $v=(v_{ij})_{i\leq j}^{n}\in\mathbb{C}^{\frac{n(n+1)}{2}}$.

• (i)

By $T(v)$ we will denote the following array with complex entries $\{v_{ij}\}$

*$v_{n1}$ $v_{n2}$          $\cdots$           ** $v_{n,n-1}$ *$v_{nn}$

• $v_{n-1,1}$**              $\cdots$               ** *$v_{n-1,n-1}$

$\cdots$*         $\cdots$         *$\cdots$

*$v_{21}$ *$v_{22}$

**$v_{11}$

Such an array will be called a Gelfand-Tsetlin tableau of height $n$.

• (ii)

Throughout the paper, for any ring $R$, $T_{n}(R)$ will stand for the space of the Gelfand-Tsetlin tableaux of height $n$ with entries in $R$. We will identify $T_{n}(\mathbb{C})$ with the set $\mathbb{C}^{\frac{n(n+1)}{2}}$ in the following way: to

[TABLE]

we associate a tableau $T(v)\in T_{n}(\mathbb{C})$ as above.

Remark 4.2**.**

There is a natural correspondence between the set $\Gamma^{*}$ of characters $\chi:\Gamma\to\mathbb{C}$ and the set of Gelfand-Tsetlin tableaux of height $n$. In fact, to obtain a Gelfand-Tsetlin tableau $T(l)$ from a character $\chi$ we find a solution $(l_{ij})$ of the system of equations

[TABLE]

Conversely, for every Gelfand-Tsetlin tableau $T(v)$ with entries $\{v_{ij}\;|\;1\leq j\leq i\leq n\}$, we associate $\chi_{v}\in\Gamma^{*}$ by defining $\chi_{v}(c_{mk})=\gamma_{mk}(v)$. It is clear that each tableau defines such a character uniquely. On the other hand, a tableau is defined by a character up to a permutation of the rows, i.e. an element of $S_{n}\times S_{n-1}\times\cdots\times S_{1}$.

4.2. Gelfand-Tsetlin formulas for finite-dimensional modules

In this subsection we recall the classical result of I. Gelfand and M. Tsetlin, [GT50].

Definition 4.3**.**

A Gelfand-Tsetlin tableau of height $n$ is called standard if $v_{ki}-v_{k-1,i}\in\mathbb{Z}_{\geq 0}$ and $v_{k-1,i}-v_{k,i+1}\in\mathbb{Z}_{>0}$ for all $1\leq i\leq k\leq n-1$.

Note that, for the sake of convenience, the second condition in the definition above is slightly different from the original condition in [GT50].

Theorem 4.4** (Gelfand-Tsetlin, [GT50]).**

Let $L(\lambda)$ be the simple finite dimensional module over $\mathfrak{gl}(n)$ of highest weight $\lambda=(\lambda_{1},\ldots,\lambda_{n})$. Then there exist a basis of $L(\lambda)$ parameterized by the set of all standard tableaux $T(v)=T(v_{ij})$ with fixed top row $v_{nj}=\lambda_{j}-j+1$, $j=1,\ldots,n$. Moreover, the action of the generators of $\mathfrak{gl}(n)$ on $L(\lambda)$ is given by the Gelfand-Tsetlin formulas:

[TABLE]

[TABLE]

[TABLE]

where $\delta^{ij}\in T_{n}({\mathbb{Z}})$ is defined by $(\delta^{ij})_{ij}=1$ and all other $(\delta^{ij})_{k\ell}$ are zero. If the new tableau $T(v\pm\delta^{ki})$ is not standard, then the corresponding summand of $E_{k,k+1}(T(v))$ or $E_{k+1,k}(T(v))$ is zero by definition.

We call the formulas above the Gelfand-Tsetlin formulas for $\mathfrak{gl}(n)$.

Set $e_{kk}(v):=\left(k-1+\sum\limits_{i=1}^{k}v_{ki}-\sum\limits_{i=1}^{k-1}v_{k-1,i}\right)$. We note that $E_{kk}(T(v))=e_{kk}(v)T(v)$ is well defined for any Gelfand-Tsetlin tableau $T(v)$. .

Definition 4.5**.**

Let $T(v)$ be a Gelfand-Tsetlin tableau. Then we call $(e_{11}(v),\ldots,e_{nn}(v))$ (respectively, $(e_{11}(v)-e_{22}(v),\ldots,e_{n-1,n-1}(v)-e_{nn}(v))$) the * $\mathfrak{gl}(n)$-weight (respectively, the * $\mathfrak{sl}(n)$-weight*) of the tableau $T(v)$.*

The formulas for the action of the generators of $\operatorname{\mathfrak{gl}}(n)$ in the theorem above imply that the standard tableaux $T(v)$ form an eigenbasis for the action of the standard Cartan subalgebra $\mathfrak{h}$. The following result shows that such basis is an eigenbasis for the Gelfand-Tsetlin subalgebra.

Theorem 4.6** (Zhelobenko, [Zh74]).**

Let $L(\lambda)$ be the simple finite dimensional module over $\mathfrak{gl}(n)$ of highest weight $\lambda=(\lambda_{1},\ldots,\lambda_{n})$, with basis as described in Theorem 4.4. The action of the generators $c_{rs}$ of $\Gamma$ (see (1)) is given by

[TABLE]

where $\gamma_{rs}(v)$ are the symmetric polynomials defined in (2).

As a direct consequence of Theorem 4.4 and Theorem 4.6, any simple finite dimensional $\mathfrak{gl}(n)$-module is a Gelfand-Tsetlin module with one-dimensional Gelfand-Tsetlin subspaces.

Remark 4.7**.**

Whenever we refer to finite dimensional $\mathfrak{sl}(n)$-modules we will use the same vector space and the Gelfand-Tselin formulas for generators $E_{r,r+1}$ or $E_{r+1,r}$, for the action of the generators of a Cartan subalgebra $\{h_{1},\ldots,h_{n-1}\}$ we define $h_{i}(T(v)):=(E_{ii}-E_{i+1,i+1})(T(v))$. We also fix the action of the central element $E_{11}+\ldots+E_{nn}$ as zero.

Example 4.8**.**

*Let us to denote by $M$ the simple highest weight $\mathfrak{gl}(3)$-module with highest weight $(1,0,-1)$. $M$ is a finite dimensional module of dimension $8$. The tableaux realization guaranteed by Theorem 4.4 consist of a vector space spanned by the set of all standard tableaux of height $3$ with top row $(1,-1,-3)$.

1$$-1$$-3*         *1$$-1$$-3

$T_{1}$*=       1$$-1             $T_{2}$= *1$$-1

$1$*                                 *[math]

1$$-1$$-3*         *1$$-1$$-3

$T_{3}$*=       1$$-2             $T_{4}$= *1$$-2

$1$*                                 *[math]

1$$-1$$-3*         *1$$-1$$-3

$T_{5}$*=       [math]$-2$             $T_{6}$= *[math]$-2$

$-1$*                                 *[math]

1$$-1$$-3*         *1$$-1$$-3

$T_{7}$*=       1$$-2             $T_{8}$= *[math]$-1$

$-1$*                                 *[math]

*By Theorem 4.4, the module $M$ is isomorphic to $\operatorname{Span}_{\mathbb{C}}\{T_{i}\ |\ i=1,\ldots,8\}$ endowed with the action of $\mathfrak{gl}(3)$ given by the Gelfand-Tsetlin formulas.

Since the action of $E_{11}+E_{22}+E_{33}$ is fixed to be trivial and $\mathfrak{h}=\operatorname{Span}_{\mathbb{C}}\{h_{1}=E_{11}-E_{22},\ h_{2}=E_{22}-E_{33}\}$, $M$ becomes an $\mathfrak{sl}(3)$-module with weight support

[TABLE]

*Then as an $\mathfrak{sl}(3)$-module, $M$ is isomorphic to $L(1,1)$ (the simple finite dimensional $\mathfrak{sl}(3)$-module of highest weight $(1,1)$). The following picture shows the weights lattice of the $\mathfrak{sl}(3)$-module $M$. Note that $M_{(0,0)}$ is $2$-dimensional with basis $\{T_{4},\ T_{8}\}$.

(-2,1)$$(0,0)$$(2,-1)$$(-1,-1)$$(1,-2)$$(-1,2)$$(1,1)

In particular, the basis elements of $M_{(0,0)}$ can not be distinguish by the action of the Cartan subalgebra. However, using $\Gamma$ the module decomposes as a direct sum of $1$-dimensional $\Gamma$-submodules.

The following theorem will give us information about the dimension of Gelfand-Tsetlin subspaces for simple Gelfand-Tsetlin modules and the possible number of non-isomorphic Gelfand-Tsetlin modules with a given Gelfand-Tsetlin character in its support.

Theorem 4.9** ([FO14], Theorem 6.1; [Ov02]).**

Let $U=U(\operatorname{\mathfrak{gl}}(n))$, ${\Gamma}\subset U$ the Gelfand-Tsetlin subalgebra, ${\mathsf{m}}\in\operatorname{Specm}{\Gamma}$. Set $Q_{n}=1!2!\ldots(n-1)!$.

• (i)

For a Gelfand-Tsetlin module $M$, such that $M({\mathsf{m}})\neq 0$ and $M$ is generated by some $x\in M({\mathsf{m}})$ (in particular for an simple module), one has

[TABLE]

• (ii)

The number of isomorphism classes of simple Gelfand-Tsetlin modules $N$ such that $N({\mathsf{m}})\neq 0$ is always nonzero and does not exceed $Q_{n}$.

The theorem above shows that elements of $\operatorname{Specm}{\Gamma}$ classify the simple Gelfand-Tsetlin $\operatorname{\mathfrak{gl}}(n)$-modules (and, hence, $\mathfrak{sl}(n)$-modules) up to some finiteness and up to an isomorphism of Gelfand-Tsetlin modules which contains two different Gelfand-Tsetlin characters.

In [Maz98], Gelfand-Tsetlin modules with tableaux realization and action given by the Gelfand-Tsetlin formulas are studied, but such modules $V$ satisfy $\dim(V(\chi))\leq 1$ for all $\chi\in\Gamma^{*}$. In what follows we will consider a more general definition of tableaux realization, which will allow to consider certain classes of modules with $\dim(V(\chi))$ greater than $1$.

For any Gelfand-Tsetlin tableau $T(v)\in T_{n}(\mathbb{C})$ we consider the set

[TABLE]

If an indecomposable Gelfand-Tsetlin module $V$ has a tableaux realization and $T(v)$ is one of the basis tableaux then it has a basis which is a subset of $\mathcal{B}(T(v))$. On the other hand, we might have a module with a basis consisting of a subset of tableau from $\mathcal{B}(T(v))$ but without a tableaux realization. This may happen, for example, when $V$ has a Gelfand-Tsetlin character of multiplicity more than $1$. For this reason we extend the notion of modules with a tableaux realization.

Definition 4.10**.**

We say that a Gelfand-Tsetlin module $M$ admits a generalized tableaux realization with respect to a Gelfand-Tsetlin subalgebra $\Gamma$ if $M$ has a basis $\mathcal{B}_{M}$ labelled by a subset of $\mathcal{B}(T(v))$ for some tableau $T(v)$, such that every $m_{T(w)}\in\mathcal{B}_{M}$, $T(w)\in\mathcal{B}(T(v))$, is a generalized eigenvector of $c_{rs}$ of eigenvalue $\gamma_{rs}(w)$, for all $r,s$. We will denote by $\mathcal{GT}_{T(v)}(n)$ the full subcategory of the category of Gelfand-Tsetlin modules $\mathcal{GT}(n)$ which consists of modules with a generalized tableaux realization with respect to $\Gamma$ whose basis contains $T(v)$.

The subcategory $\mathcal{GT}_{T(v)}(n)$ is closed under the operations of taking submodules and quotients. Moreover, as our main result will imply, simple modules of the categories $\mathcal{GT}_{T(v)}(3)$ for all $T(v)$ exhaust all simple Gelfand-Tsetlin modules for $\mathfrak{sl}(3)$.

Conjecture: Simple modules of the categories $\mathcal{GT}_{T(v)}(n)$ for all $T(v)$ exhaust all simple Gelfand-Tsetlin modules for $\mathfrak{sl}(n)$.

Remark 4.11**.**

There are modules in $\mathcal{GT}(n)$ that do not belong to $\mathcal{GT}_{T(v)}(n)$ for any $T(v)$. For example, consider $n=2$ and a simple weight module $V(\lambda,\gamma)$ with a weight $\lambda\in\mathbb{C}$ in its weight support and on which the Casimir element $c_{22}$ acts as a multiplication by $\gamma\in\mathbb{C}$, where $\gamma\neq(\lambda+k)^{2}+2(\lambda+2k)$ for any integer $k$. Then $V(\lambda,\gamma)$ has a non-split self-extension which remains a weight module but on which $c_{22}$ does not act semisimply. This self-extension is an indecomposable Gelfand-Tsetlin module that does not admit a generalized tableaux realization.

Definition 4.12**.**

We will call the subcategory $\mathcal{GT}_{T(v)}(n)$ * the block of $\mathcal{GT}(n)$ generated by $T(v)$.*

From now on, whenever $n$ is clear from the context, we will write $\mathcal{GT}_{T(v)}$ instead of $\mathcal{GT}_{T(v)}(n)$.

4.3. Generic modules

Observing that the coefficients in the Gelfand-Tsetlin formulas in Theorem 4.4 are rational functions on the entries of the tableaux, Y. Drozd, V. Futorny and S. Ovsienko [DFO94] extended the Gelfand-Tsetlin construction to more general modules. In the case when all denominators are nonzero for all possible integral shifts, one can use the same formulas and define a new class of infinite dimensional $\operatorname{\mathfrak{gl}}(n)$-modules: generic Gelfand-Tsetlin modules (cf. [DFO94], section 2.3.).

Definition 4.13**.**

A Gelfand-Tsetlin tableau $T(v)$ (equivalently, $v\in T_{n}(\mathbb{C})$) is called generic if $v_{ki}-v_{kj}\notin\mathbb{Z}$ for all $1\leq i\neq j\leq k\leq n-1$. A Gelfand-Tsetlin character $\chi_{v}$ associated to a generic tableau $T(v)$ (see Remark 4.2) will be called a generic Gelfand-Tsetlin character.

Recall that ${\mathcal{B}}(T(v))=\{T(v+z)\ |\ z\in T_{n-1}(\mathbb{Z})\}$ for any Gelfand-Tableau $T(v)$.

Theorem 4.14** ([DFO94], Section 2.3).**

Let $T(v)$ be a generic Gelfand-Tsetlin tableau of height $n$.

• (i)

The vector space $V(T(v))=\operatorname{Span}_{\mathbb{C}}{\mathcal{B}}(T(v))$ has a structure of a $\mathfrak{gl}(n)$-module with action of the generators of $\mathfrak{gl}(n)$ given by the Gelfand-Tsetlin formulas.

• (ii)

The action of the generators of $\Gamma$ on the basis elements of ${\mathcal{B}}(T(v))$ is given by (5).

• (iii)

The $\mathfrak{gl}(n)$-module $V(T(v))$ is a Gelfand-Tsetlin module with Gelfand-Tsetlin multiplicities equal to $1$.

The module $V(T(v))$ constructed in Theorem 4.14 will be extensively used in future and will be referred as the generic Gelfand-Tsetlin module associated to $T(v)$. In general $V(T(v))$ need not to be simple. Because $\Gamma$ has simple spectrum on $V(T(v))$ for $T(w)$ in $\mathcal{B}(T(v))$ we may define the simple $U$-module in $V(T(v))$ containing $T(w)$ to be the simple subquotient of $V(T(v))$ containing $T(w)$.

Remark 4.15**.**

By Theorem 4.14(iii), given two different tableaux $T(w)$ and $T(w^{\prime})$ in $\mathcal{B}(T(v))$, there exists an element of $\Gamma$ that has different eigenvalues for $T(w)$ and $T(w^{\prime})$. Whenever we say that $\Gamma$ “separates” tableaux of $V(T(v))$ we will refer to this property.

4.3.1. Gelfand-Tsetlin formulas in terms of permutations

In this subsection we will rewrite the Gelfand-Tsetlin formulas in terms of permutations. These formulas will be very useful when verifying certain identities for the action of $\mathfrak{g}$ on $V(T(v))$.

Let $\widetilde{S}_{m}$ denotes the subset of $S_{m}$ consisting of the transpositions $(1,i)$, $i=1,...,m$. For $\ell<m$, set $\Phi_{\ell m}=\widetilde{S}_{m-1}\times\cdots\times\widetilde{S}_{\ell}$. For $\ell>m$ we set $\Phi_{\ell m}=\Phi_{m\ell}$. Finally we define, $\Phi_{\ell\ell}=\{\mbox{Id}\}$. Every $\sigma$ in $\Phi_{\ell m}$ will be written as an $|\ell-m|$-tuple of transpositions and by $\sigma[t]$ we will denote the $t$-th component of the tuple.

Remark 4.16**.**

Recall that in order to have well defined action of $\sigma\in\Phi_{\ell m}$ on $T_{n-1}(\mathbb{C})$, for $w\in T_{n-1}(\mathbb{C})$ and $\sigma\in\Phi_{\ell m}$ on $w$ we set

[TABLE]

Definition 4.17**.**

Let $1\leq r<s\leq n$. Define

[TABLE]

Furthermore, define $\varepsilon_{rr}=0$ and $\varepsilon_{sr}=-\varepsilon_{rs}$.

Definition 4.18**.**

For each generic vector $w$ and any $1\leq t\leq n-1$ define

[TABLE]

Lemma 4.19**.**

For each $m>k$ the action of $E_{mk}$ is given by the expression:

[TABLE]

Proof.

The case $k=m+1$ follows from the Gelfand-Tsetlin formulas. The general case follows by induction on $m-k$ using the relation

[TABLE]

for any generic vector $v$. ∎

Lemma 4.20**.**

For each $r<s$ the action of $E_{rs}$ is given by the expression:

[TABLE]

Proof.

The case $s=r+1$ follows from the Gelfand-Tsetlin formulas. The general case follows by induction in $s-r$ using the relation

[TABLE]

for any generic vector $v$. ∎

Definition 4.21**.**

For each generic vector $w\in T_{n}(\mathbb{C})$ and any $1\leq r,s\leq n$ we define

[TABLE]

Proposition 4.22**.**

Let $v\in T_{n}(\mathbb{C})$ be any generic vector and $z\in T_{n-1}(\mathbb{Z})$. The Gelfand-Tsetlin formulas for the $U$-module $V(T(v))$ can be written as follows:

[TABLE]

Proof.

Follows from Lemmas 4.19 and 4.20 and the fact that $\sum_{i=\ell}^{m-1}\sigma(\varepsilon_{i+1,i})=\sigma(\varepsilon_{m,\ell})$ when $m>\ell$ and $\sum_{i=m}^{\ell-1}\sigma(\varepsilon_{i,i+1})=\sigma(\varepsilon_{m,\ell})$ when $m<\ell$. ∎

Example 4.23**.**

Let us write explicitly the functions $e_{rs}(w)$ from Definition 4.21 and the Gelfand-Tsetlin formulas in Proposition 4.22 in the case of $\mathfrak{gl}(3)$. Let $v\in T_{3}(\mathbb{C})$ be a generic vector, $z\in T_{2}(\mathbb{Z})$ and $w=v+z$. Set also $\tau$ to be the permutation that interchanges the entries in positions $(2,1)$ and $(2,2)$. Considering

[TABLE]

The action of $\mathfrak{gl}(3)$ on any tableau is given by:

[TABLE]

The explicit description of all simple generic modules for $\mathfrak{gl}(3)$ was obtained first in [R12]. The classification of simple generic modules $\mathfrak{gl}(n)$ was completed in [FGR15]. Let us discuss briefly the main results in the last classification.

Definition 4.24**.**

Let $T(v)$ be a fixed Gelfand-Tsetlin tableau. For any $T(w)\in\mathcal{B}(T(v))$, and for any $1<r\leq n$, $1\leq s\leq r$ and $1\leq u\leq r-1$ we define:

[TABLE]

[TABLE]

A basis for the simple subquotients of $V(T(v))$ is provided in the following theorem.

Theorem 4.25** ([FGR15], Theorems 6.8 and 6.14).**

Let $T(v)$ be a fixed generic Gelfand-Tsetlin tableau and $T(w)$ in $\mathcal{B}(T(v))$.

• (i)

The module $U\cdot T(w)$ has a basis of tableaux

[TABLE]

• (ii)

The simple module containing $T(w)$ has a basis of tableaux

[TABLE]

The action of $\mathfrak{gl}(n)$ on $T(w^{\prime})\in\mathcal{N}(T(w))$ is given by the Gelfand-Tsetlin formulas. The action of $\operatorname{\mathfrak{gl}}(n)$ on $T(w^{\prime})\in\mathcal{I}(T(w))$ is given by the Gelfand-Tsetlin formulas with the convention that all tableau $T(w^{\prime}\pm\delta^{ki})$ for which $\Omega^{+}(T(w^{\prime}\pm\delta^{ki}))\neq\Omega^{+}(T(w))$ are omitted in the sums for $E_{k,k+1}(T(w^{\prime}))$ and $E_{k+1,k}(T(w^{\prime}))$.

Corollary 4.26**.**

Let $T(v)$ be a generic Gelfand-Tsetlin tableau. The module $V(T(v))$ is simple if and only if $\Omega(T(v))=\emptyset$.

Example 4.27**.**

Consider $a,b,c\in\mathbb{C}$ such that $\{a-b,a-c,b-c\}\bigcap\mathbb{Z}=\emptyset$ and $v=(a,b,c|a,b+2|a)$, then

abc

$T(v)$=      ab+2**

a

then $\Omega(T(v))=\{(3,1,1),(3,2,2),(2,1,1)\}$, $\Omega^{+}(T(v))=\{(3,1,1),(2,1,1)\}$. So, by Theorem 4.25, the simple subquotient of $V(T(v))$ containing $T(v)$ has a basis

[TABLE]

Example 4.28**.**

(See also [Maz98], Section 4.3) Let $a_{1},\ldots,a_{n}$ be complex numbers such that $a_{i}-a_{j}\notin\mathbb{Z}$ for any $i\neq j$. Denote by $T(v)$ the Gelfand-Tsetlin tableau of height $n$ with entries $v_{ij}$, such that $v_{rs}=a_{s}$ for $1\leq s\leq r\leq n$. The tableau $T(v)$ is a generic Gelfand-Tsetlin tableau and by Theorem 4.25 a basis for an simple $\mathfrak{gl}(n)$-module containing $T(v)$ has a basis

[TABLE]

Moreover, we can easily check that $\operatorname{Span}_{\mathbb{C}}\mathcal{I}(T(v))$ is a submodule of $V(T(v))$ isomorphic to the simple Verma module $M(a_{1},a_{2}+1,\ldots,a_{n}+n-1)$.

Using the description of simple subquotients of $V(T(v))$, we will also be able to describe the Loewy series decomposition for $V(T(v))$. We will use the convention that the first module in the list is the socle of $V(T(v))$.

Theorem 4.29**.**

Let $T(v)$ be a generic tableau and set $t:=|\Omega(T(v))|$. The Loewy series decomposition of the Gelfand-Tsetlin module $V(T(v))$ is given by

[TABLE]

where, $D_{i}={\rm Span}_{\mathbb{C}}\{T(w)\in\mathcal{B}(T(v))\ |\ |\Omega^{+}(T(w))|=i\}$ and $0\leq i\leq t$. If $D_{i}=\emptyset$ for some $1<i<t$ we omit this term in the Loewy decomposition.

Proof.

Let us show first that $D_{t}$ is a simple submodule of $V(T(v))$. By Theorem 4.25(i), if $|\Omega^{+}(T(w))|=t$, the module generated by $T(w)$ is simple and hence, equal to $Span_{\mathbb{C}}\{T(w^{\prime})\ |\ |\Omega^{+}(T(w^{\prime}))|=t\}$. That is, $V(T(v))$ has a unique simple submodule $M$, namely $M=D_{t}$.

Set $M_{t+1}:=V(T(v))$ and define $M_{i}:=M_{i+1}/D_{i}$. Note that

[TABLE]

So, by Theorem 4.25(ii) any element basis of $D_{i}$ is a basis element of a simple submodule of $M_{i+1}$ and, then $D_{i}$ is the sum of all simple submodules of $M_{i+1}$. ∎

Remark 4.30**.**

We will often apply Theorem 4.29 in the following way. If $V_{j}$, $j\in J$, are all non-isomorphic simple subquotients of $V(T(v))$, and $T(w_{j})\in V_{j}$, then the modules Loewy series components of $V(T(v))$ are precisely:

[TABLE]

where $\{D_{ij}\}_{j=1}^{r_{i}}$ is the set of all $V_{j}$ such that $|\Omega^{+}(w_{j})|=i$.

Although Theorem 4.25 gives a nice relation between the category $\mathcal{GT}_{T(v)}$ (see Definition 4.10) and $\Omega(T(v))$, it is not true that $\mathcal{GT}_{T(v)}$ is completely determined by $\Omega(T(v))$ - see the next example.

Example 4.31**.**

Consider the tableaux $T(v)$ and $T(v^{\prime})$ such that $\Omega(T(v))=\Omega(T(v^{\prime}))$ but $\mathcal{GT}_{T(v)}$ is not equivalent to $\mathcal{GT}_{T(v^{\prime})}$. Take

a$$a$$a*         *a$$a+1$$a+2

$T(v)$*=     a$$y       $T(v^{\prime})$=    *a$$y

$z$*                                 *$z$

Then $\Omega(T(v))=\Omega(T(v^{\prime}))$. The Loewy series of $T(v)$ is $D_{3},D_{0}$, however, the Loewy series of $T(v^{\prime})$ is $D_{3},D_{2},D_{1},D_{0}$.

4.4. Singular Gelfand-Tsetlin modules

The construction of simple finite-dimensional modules and of generic modules presented in Sections 4.2 and 4.3 have one common feature - an explicit basis parameterized by a set of Gelfand-Tsetlin tableaux. In the finite-dimensional case all the entries of the tableaux $T(v)$ in the basis satisfy $v_{ki}-v_{kj}\in\mathbb{Z}$, while in the generic case they satisfy $v_{ki}-v_{kj}\notin\mathbb{Z}$ for any $k\neq n$. We may consider the standard and generic tableaux as the two extreme cases of the singular Gelfand-Tsetlin tableaux where the latter are defined below.

Definition 4.32**.**

A vector $v\in T_{n}(\mathbb{C})$ will be called singular if there exist $1\leq s<t\leq r\leq n-1$ such that $v_{rs}-v_{rt}\in\mathbb{Z}$. The vector $v$ will be called $1$-singular if there exist $k,i,j$ with $1\leq i<j\leq k\leq n-1$ such that $v_{ki}-v_{kj}\in\mathbb{Z}$ and $v_{rs}-v_{rt}\notin\mathbb{Z}$ for all $(r,s,t)\neq(k,i,j)$, $r\neq n$. If $v$ is $1$-singular, the tableau $T(v)$ will be called $1$-singular tableau. A Gelfand-Tsetlin character $\chi_{v}$ associated to a singular (respectively, $1$-singular) tableau $T(v)$ (see Remark 4.2) will be called a singular (respectively, $1$-singular) character.

4.4.1. Construction of $1$-singular Gelfand-Tsetlin modules

In [FGR16] an explicit construction of modules with a generalized tableaux realization (see Definition 4.10) associated with any $1$-singular Gelfand-Tsetlin tableau was provided. In this section we provide the main details of this construction.

Set $v$ a vector of variables with $\frac{n(n+1)}{2}$ entries indexed by $(r,s)$ such that $1\leq s\leq r\leq n$. By ${\mathcal{F}}$ we will denote the space of rational functions on $v_{ij}$, $1\leq j\leq i\leq n$, with poles on the hyperplanes $v_{rs}-v_{rt}=0$. Note that $V(T(v))$ is defined for all generic $v$ and that $V(T(v))=V(T(v^{\prime}))$ whenever $v-\sigma(v^{\prime})\in T_{n-1}({\mathbb{Z}})$ for some $\sigma\in G$. Thus $V(T(v))$ is defined for elements $v$ in the (generic) complex torus $T=T_{n}({\mathbb{C}})/T_{n-1}({\mathbb{Z}})$. Denote by $T_{n}({\mathbb{C}})_{\rm gen}$ the set of all generic vectors $v$ in $T_{n}({\mathbb{C}})$ such that $V(T(v))$ is simple, equivalently, $v_{rs}-v_{r-1,t}\notin\mathbb{Z}$ for any $r,s,t$. Until the end of this section we fix $(i,j,k)$ such that $1\leq i<j\leq k\leq n-1$.

By ${\mathcal{H}}$ we denote the hyperplane $v_{ki}-v_{kj}=0$ in $T_{n}(\mathbb{C})$, also by $\tau\in S_{n-1}\times\cdots\times S_{1}$ we denote the transposition on the $k$th row interchanging the $i$th and $j$th entries. $\overline{\mathcal{H}}$ stands for the subset of all $w$ in $T_{n}(\mathbb{C})$ such that $w_{tr}\neq w_{ts}$ for all triples $(t,r,s)$ except for $(t,r,s)=(k,i,j)$. Finally, by ${\mathcal{F}}_{ij}$ denote the subspace of ${\mathcal{F}}$ consisting of all functions that are smooth on $\overline{\mathcal{H}}$.

Let us fix $\bar{v}$ in ${\mathcal{H}}$ such that $\bar{v}_{ki}=\bar{v}_{kj}$ and all other differences $v_{mr}-v_{ms}$ are noninteger. In other words, $\bar{v}\in{\mathcal{H}}$ and $\bar{v}+T_{n-1}(\mathbb{Z})\subset\overline{\mathcal{H}}$.

Remark 4.33**.**

For any generic vector $w$ we can choose a representative of the class $w+T_{n-1}(\mathbb{Z})$ of $w$ in $T=T_{n}(\mathbb{C})_{\rm gen}/T_{n-1}(\mathbb{Z})$ as “close” as possible to $\bar{v}$ as follows. Let $m_{rs}:=\lfloor{\rm Re}(\bar{v}_{rs}-w_{rs})\rfloor$ (the integer part of the real part of $\bar{v}_{rs}-w_{rs}$), $m$ be the vector in $T_{n-1}(\mathbb{Z})$ with components $m_{rs}$, and $\bar{v}[w]:=w+m$. Then $\mathcal{S}=\{\bar{v}[w]\;|\;w\in T_{n}(\mathbb{C})_{\rm gen}\}$ is a set of representatives of $T$.

Our goal is construct a module $V(T(\bar{v}))$ with Gelfand-Tsetlin support $\{\chi_{\bar{v}+m}\ |\ m\in T_{n-1}(\mathbb{Z})\}$. We will refer to this module as the $1$-singular universal tableaux Gelfand-Tsetlin module associated with $\bar{v}$, or simply as the universal module.

We formally introduce the complex vector space $V(T(\bar{v}))$ as the one spanned by vectors $\{T(\bar{v}+z),{\mathcal{D}}T({\bar{v}}+z)\ |\ z\in T_{n-1}(\mathbb{Z})\}$ subject to the relations $T(\bar{v}+z)-T(\bar{v}+\tau(z))=0$ and ${\mathcal{D}}T({\bar{v}}+z)+{\mathcal{D}}T({\bar{v}}+\tau(z))=0$. We will refer to $T(u)$ as the regular Gelfand-Tsetlin tableau associated with $u$ and to ${\mathcal{D}}T(u)$ as the derivative Gelfand-Tsetlin tableau associated with $u$.

Remark 4.34**.**

Although $\{T(\bar{v}+z),\,\mathcal{D}T(\bar{v}+z)\;|\;z\in T_{n-1}(\mathbb{Z})\}$ is not a basis, we have the following natural basis of $V(T(\bar{v}))$:

[TABLE]

Set ${\mathcal{V}}_{\rm gen}=\bigoplus_{v\in S}V(T(v))$ and ${\mathcal{V}}^{\prime}=V(T(\bar{v}))\oplus{\mathcal{V}}_{\rm gen}$. Then ${\mathcal{F}}\otimes{\mathcal{V}}_{\rm gen}$ is a $\mathfrak{gl}(n)$-module with the trivial action on ${\mathcal{F}}$. We next define a $\mathfrak{gl}(n)$-module structure on $V(T(\bar{v}))$.

The evaluation map $\mbox{ev}(\bar{v}):{\mathcal{F}}_{ij}\otimes{\mathcal{V}}^{\prime}\to{\mathcal{V}}^{\prime}$ is the linear map defined by

$\mbox{ev}(\bar{v})(fT(v+z))=f(\bar{v})T(\bar{v}+z),\ \mbox{ev}(\bar{v})(f{\mathcal{D}}T(\bar{v}+z))=f(\bar{v}){\mathcal{D}}T(\bar{v}+z)$. Furthermore, $\mathcal{D}^{\bar{v}}:{\mathcal{F}}_{ij}\otimes V(T(v))\to V(T(\bar{v}))$ will denote the linear map defined by

$\mathcal{D}^{\bar{v}}(fT(v+z))=\mathcal{D}^{\bar{v}}(f)T(\bar{v}+z)+f(\bar{v})\mathcal{D}T(\bar{v}+z)$, where $\mathcal{D}^{\bar{v}}(f)=\frac{1}{2}\left(\frac{\partial f}{\partial v_{ki}}-\frac{\partial f}{\partial v_{kj}}\right)(\bar{v})$, $z\in T_{n-1}(\mathbb{Z})$, $f\in{\mathcal{F}}_{ij}$ and $v\in{\mathcal{S}}$. We may think of $\mathcal{D}^{\bar{v}}$ as the map

[TABLE]

This map extends to a linear map ${\mathcal{F}}_{ij}\otimes{\mathcal{V}}_{\rm gen}\to V(T(\bar{v}))$ which we will also denote by $\mathcal{D}^{\bar{v}}$.

Theorem 4.35** ([FGR16] Theorems 4.9 and 5.6).**

$V(T(\bar{v}))$* has structure of Gelfand-Tsetlin module over $\mathfrak{gl}(n)$ with action of the generators of $\operatorname{\mathfrak{gl}}(n)$ given by*

[TABLE]

and action of the generators of $\Gamma$ given by

[TABLE]

where $v$ is a generic vector in the set of representatives $\mathcal{S}$, and $z,w\in T_{n-1}(\mathbb{Z})$ with $w\neq\tau(w)$.

Remark 4.36**.**

In the case of $\mathfrak{gl}(3)$ we can give the following interpretation of the basis elements of the module $V(T(\bar{v}))$. Let $T(v)$ be a generic tableau such that $V(T(v))$ is simple, and let $T(\bar{v})$ be such that $\bar{v}_{21}=\bar{v}_{22}$:

v_{31}$$v_{32}$$v_{33}*         *\bar{v}_{31}$$\bar{v}_{32}$$\bar{v}_{33}

$T(v)$*=    v_{21}$$v_{22}        $T(\bar{v})$ =    *\bar{v}_{21}$$\bar{v}_{22}

$v_{11}$*                                   *$\bar{v}_{11}.$

Then $T(\bar{v}+(m,n,k))$ and $\mathcal{D}T(\bar{v}+(m,n,k))$ can be considered as formal limits in the following way:

[TABLE]

[TABLE]

One essential property of generic Gelfand-Tsetlin modules described in Theorem 4.14 is that $\Gamma$ “separates” the basis elements of $\mathcal{B}(T(v))$, that is, for any two different tableaux $T_{1},\ T_{2}$ in $\mathcal{B}(T(v))$ there exists an element $\gamma\in\Gamma$ such that $\gamma\cdot T_{1}=T_{1}$ and $\gamma\cdot T_{2}=0$ (see Remark 4.15). In the case of $1$-singular modules $V(T(\bar{v}))$ this is also true and follows from the fact that no derivative tableau $\mathcal{D}(T(\bar{v}+w))$ is an eigenvector for the action of $c_{k2}\in\Gamma$. A detailed proof can be found in [GoR18], Section $\S 5$.

Theorem 4.37**.**

Let $\bar{v}$ be any $1$-singular vector and $\mathcal{B}(T(\bar{v}))$ be as before. Then $\Gamma$ separates the tableaux in the basis $\mathcal{B}(T(\bar{v}))$.

In the case of $T_{3}(\mathbb{C})$ (equivalently, Gelfand-Tsetlin tableaux of height $3$) every singular vector is a $1$-singular vector. Therefore, in the case of $\mathfrak{gl}(3)$ the $1$-singular modules exhaust all singular Gelfand-Tsetlin modules.

Example 4.38**.**

The simple Verma $\mathfrak{sl}(3)$-module $M=M(-1,-1)$ admits a tableaux realization as a subquotient of the module $V(T(\bar{v}))$, where $\bar{v}$ is the vector $(-1,-1,-1,-1,-1,-1)$. This module contains Gelfand-Tsetlin characters of dimension $2$. For example, if $\chi$ is the Gelfand-Tsetlin character associated with the tableaux $T(\bar{v}+(-1,0,-1))$ and $\mathcal{D}T(\bar{v}+(0,-1,-1))$, then $\dim(M(\chi))=2$ (see §7.5 (C13) for details).

5. Gelfand-Tsetlin modules of $\mathfrak{sl}(3)$ and modules of $C({\mathfrak{h}})$

From now on we focus on the case $n=3$ and $\operatorname{\mathfrak{g}}=\mathfrak{sl}(3)$. We fix the standard Gelfand-Tsetlin subalgebra ${\Gamma}$ of $\operatorname{\mathfrak{g}}$, that is the one corresponding to the chain whose second component is generated by $E_{12}$ and $E_{21}$. The corresponding category of Gelfand-Tsetlin modules $\mathcal{GT}(3)$ will be denoted simply by $\mathcal{GT}$.

Let $C({\mathfrak{h}})$ be the centralizer of the Cartan subalgebra ${\mathfrak{h}}$ in $U(\operatorname{\mathfrak{g}})$, where

[TABLE]

In this section we collect some properties of modules in $\mathcal{GT}$ that are related to the category of modules of $C({\mathfrak{h}})$. The results are based on the works [F86b], [F89], [BFL95], but for reader’s convenience we provide proofs for some statemenrs.

The following result provides an important relation between the simple $C({\mathfrak{h}})$-modules and the simple weight modules (for a proof, see for example [F86a]):

Lemma 5.1**.**

For any simple $C({\mathfrak{h}})$-module $W$ there exists an simple weight $\operatorname{\mathfrak{g}}$-module $M$ such that $M_{\lambda}\simeq W$ for some $\lambda\in{\mathfrak{h}}^{*}$. Conversely, if $M$ is a simple weight $\operatorname{\mathfrak{g}}$-module then $M_{\lambda}$ is a simple $C({\mathfrak{h}})$-module.

Denote $A:=E_{12}E_{21}$, $B:=E_{23}E_{32}$. Recall that the center of $U({\mathfrak{g}})$ is generated by $c_{32}$ and $c_{33}$, see (1). For convenience we will also use the following generators of the center of $U({\mathfrak{g}})$:

[TABLE]

The next two lemmas provide important technical properties of $C({\mathfrak{h}})$ and the simple $C({\mathfrak{h}})$-modules.

Lemma 5.2** ([BFL95], Lemma 1.1).**

The centralizer $C({\mathfrak{h}})$ is an associative algebra generated by $H_{1}$, $H_{2}$, $A$, $B$, $c_{1}$, and $c_{2}$.

Lemma 5.3** ([F86b]).**

Let $W$ be a simple $C({\mathfrak{h}})$-module, and let $H_{i}=h_{i}{\rm Id}$, $c_{i}=\gamma_{i}{\rm Id}$ on $W$, for some constants $h_{i},\gamma_{i}$, $i=1,2$. Then the following identities hold on $W$.

[TABLE]

where

[TABLE]

Let $M$ be a simple module in $\mathcal{GT}$. In particular, it is a weight module. Consider any $\lambda\in{\mathfrak{h}}^{*}$ from the weight support of $M$. The central elements $c_{1}$ and $c_{2}$ act on $M$, and hence on $M_{\lambda}$, as a multiplication by some complex scalars $\gamma_{1}$ and $\gamma_{2}$, respectively. If $\lambda(H_{i})=h_{i}$, then $H_{i}=h_{i}{\rm Id}$, $i=1,2$, on $M_{\lambda}$. Since $M$ is a Gelfand-Tsetlin module, then each component $M(\chi)$ is finite dimensional. Hence, one can choose a basis of $M_{\lambda}$ with respect to which $A|_{M_{\lambda}}$ has a Jordan canonical form $A_{\lambda}$.

Consider the following polynomial in $2$ variables:

[TABLE]

Recall $r=\frac{1}{2}(h_{1}^{2}-2h_{1})$. Note that $g_{\lambda}(x,y)$ depends only on $h_{1}=\lambda(H_{1})$.

Definition 5.4**.**

Let $\lambda\in{\mathfrak{h}}^{*}$.

• (i)

A sequence $(\mu_{i})_{i\in J}$ is $\lambda$-connected (or, simply, connected) if $g_{\lambda}(\mu_{i},\mu_{i+1})=0$ for all $i\in J$, for some connected subset $J$ of $\mathbb{Z}$. A subsequence $(\mu_{j^{\prime}})_{j^{\prime}\in J^{\prime}}$ of a connected sequence $(\mu_{j})_{j\in J}$ will be called a connected subsequence if $J^{\prime}$ is connected.

• (ii)

A $\lambda$-connected sequence $(\mu_{j})_{j\in J}$ with $\mu_{i}\neq\mu_{j}$ for any $i\neq j$, is called $\lambda$-connected chain.

• (iii)

We will say that a set $B$ is $\lambda$-connected if the elements of $B$ can be ordered in a $\lambda$-connected chain $(\mu_{j})_{j\in J}$. We will call the sequence $(\mu_{j})_{j\in J}$ the $\lambda$-connected chain associated to $B$.

We note that if $B$ is $\lambda$-connected, then there are at most two $\lambda$-connected chains associated to $B$.

Lemma 5.5**.**

Let $M$ be a simple Gelfand-Tsetlin module and $\lambda\in{\mathfrak{h}}^{*}$ a weight of $M$. Then the distinct eigenvalues of $A_{\lambda}$ is a $\lambda$-connected set.

Proof.

This lemma follows from Lemmas 2.2 and 2.3 in [BFL95], but for reader’s convenience, a brief proof is written. Let $[A_{\lambda}]={\rm diag}(A_{i})$ be the matrix of $A_{\lambda}$ in a fixed Jordan canonical basis of $M_{\lambda}$, where $A_{i}$ corresponds to the generalized eigenspace with eigenvalue $\mu_{i}$, and let $[B_{\lambda}]=(B_{ij})$ be the block matrix of $B_{\lambda}$ relative to this basis, for which $B_{ii}$ and $A_{i}$ are in the same position. If $i\neq j$, looking at the $(i,j)$-th block of the matrix equation corresponding to the first relation in Lemma 5.3, we obtain:

[TABLE]

Applying the above identity to suitable elements of the basis of $M_{\lambda}$, one can see that $B_{ij}=0$ if $(\mu_{i}+\mu_{j})-\frac{1}{2}(\mu_{i}-\mu_{j})^{2}+r\neq 0$ (for details see the proof of [BFL95, Lemma 2.2]). Since $M$ is simple, then $M_{\lambda}$ is a simple $C(\mathfrak{h})$-module. However, if the set of eigenvalues of $A$ is a union of two “disconnected” sets, then one easily can prove that $M_{\lambda}$ is a direct sum of two modules, which is a contradiction (for details see the proof of [BFL95, Lemma 2.3]). ∎

Until the end of this section we assume that $M$ is a simple Gelfand-Tsetlin module, $\lambda\in{\rm Supp}\ M$.

Lemma 5.6**.**

If $(\mu_{j})_{j\in J}$ is a $\lambda$-connected sequence with $|J|\geq 3$, then the following recurrence relation holds for any $i$:

[TABLE]

Proof.

As $(\mu_{j})_{j\in J}$ is a connected sequence, we have $g_{\lambda}(\mu_{i},\mu_{i-1})=0$ and $g_{\lambda}(\mu_{i},\mu_{i+1})=0$. By solving both quadratic equations in terms of $\mu_{i}$ we have:

[TABLE]

therefore, $\mu_{i+1}+\mu_{i-1}=2\mu_{i}+2$. ∎

Definition 5.7**.**

Let $(\mu_{j})_{j\in J}$ be a $\lambda$-connected sequence. We say that $(\mu_{j})_{j\in J}$ is:

• (i)

degenerate* if $\mu_{i}=-\frac{1}{2}r$ for some $i\in J$.*

• (ii)

critical* if $\mu_{i}=-1/4-\frac{1}{2}r$ for some $i\in J$.*

• (iii)

singular* if $(\mu_{j})_{j\in J}$ is a connected subsequence of a degenerate or a critical connected sequence.*

• (iv)

generic* if it is not singular.*

Example 5.8**.**

if $\lambda=(0,0)$, then the sequence $\{0,2,6,12,\ldots\}=\{n(n+1)\ |\ n\geq 0\}$ is a degenerate connected chain and $C_{i}:=\{n(n+1)\ |\ n\geq i\}$ is a connected chain that is not degenerate for each $i>0$.

Lemma 5.9**.**

Let $(\mu_{j})_{j\in J}$ be a $\lambda$-connected sequence.

• (i)

If $(\mu_{i})_{j\in J}$ is degenerate, then $\mu_{i}\in\{n(n+1)-\frac{r}{2}\ |\ \ n\geq 0\}$ for all $i\in J$.

• (ii)

If $(\mu_{i})_{j\in J}$ is critical, then $\mu_{i}\in\{n^{2}-\frac{1}{4}-\frac{r}{2}\ |\ \ n\geq 0\}$ for all $i\in J$.

• (iii)

If $(\mu_{i})_{j\in J}$ is generic, then $\mu_{i}\in\{n^{2}+n\sqrt{1+4\mu_{0}+2r}+\mu_{0}\;|\;n\in\mathbb{Z}\}$ for all $i\in J$, where $\sqrt{1+4\mu_{0}+2r}$ is a fixed solution of $x^{2}=1+4\mu_{0}+2r$.

Proof.

By Lemma 5.6, in order to determine a connected sequence, it is enough to know two connected values. In the case of a degenerate sequence we have $\mu_{i}=-\frac{1}{2}r$ for some $i$, and hence, $\mu_{i+1}=\mu_{i}$ or $\mu_{i-1}=\mu_{i}$. Assume $\mu_{i-1}=\mu_{i}$ and $i=0$ for simplicity. Then $\mu_{n}=n(n+1)-\frac{r}{2}$, $n\in{\mathbb{Z}}$, give the unique solution of the recursive equation (13). Therefore, all $\mu_{i}$ are in the set $\{n(n+1)-\frac{r}{2}\ |\ \ n\geq 0\}$. This proves part (i). For parts (ii) and (iii) we reason in the same manner. Namely, for a critical chain, we use that $\mu_{i}=-\frac{1}{4}-\frac{1}{2}r$ for some $i$, hence $\mu_{i+1}=\mu_{i}+1$ or $\mu_{i-1}=\mu_{i}+1$, while in the generic case, given $\mu_{i}$ in the connected sequence, we have $\mu_{i+1}\in\{\mu_{i}+1\pm\sqrt{1+4\mu_{i}+2r}\}$. ∎

The properties of $A_{\lambda}$ are described in the following theorem.

Theorem 5.10**.**

Let $M$ be a simple Gelfand-Tsetlin module. Then the following hold.

• (i)

For every $\lambda\in\operatorname{Supp}M$, every eigenvalue of $A_{\lambda}$ has multiplicity at most $2$.

• (ii)

If $(\mu_{i})_{i\in J}$ is a connected chain of the set of distinct eigenvalues of $A_{\lambda}=A\mid_{M_{\lambda}}$, then

• (a)

if the chain $(\mu_{i})_{i\in J}$ is generic then all eigenvalues of $A_{\lambda}$ are distinct;

• (b)

if the chain $(\mu_{i})_{i\in J}$ is degenerate then the chain can be chosen so that $\mu_{1}=-\frac{r}{2}$, and if the multiplicity of $\mu_{i}$ equals $1$ then the multiplicity of $\mu_{i+1}$ is also $1$;

• (c)

if the chain $(\mu_{i})_{i\in J}$ is critical then the chain can be chosen so that $\mu_{1}+1=\mu_{2}$, the multiplicity of $\mu_{1}$ is $1$, and if the multiplicity of $\mu_{i}$ equals $1$ for $i>1$, then the multiplicity of $\mu_{i+1}$ also equals $1$;

• (d)

if the chain $(\mu_{i})_{i\in J}$ is singular but not degenerate or critical, then all eigenvalues of $A_{\lambda}$ are distinct.

Proof.

The proof of all parts can be found in [F86a], [F89]. The strategy is to apply the relations from Lemma 5.3 to a Jordan form of $A_{\lambda}$. Proofs of parts (b) and (d) can be also found in [BFL95], Theorem 2.7. ∎

As a consequence of Theorem 5.10 we have the following statement.

Corollary 5.11**.**

Let $M$ be a simple weight $\mathfrak{g}$-module. Then for any $\lambda\in\mathfrak{h}^{*}$ and any $i\neq j$ we have $\dim(\ker(E_{ij}|_{M_{\lambda}}))\leq 1$.

Proof.

Suppose that $\ker(E_{ij}|_{M{{}_{\lambda}}})\neq 0$. Consider the Lie subalgebra $\mathfrak{a}$ of $\mathfrak{sl}(3)$ isomorphic to $\mathfrak{sl}(2)$ that is generated by $E_{ij}$ and $E_{ji}$. Then $M$ is a Gelfand-Tsetlin $\mathfrak{a}$-module with respect to the Gelfand-Tsetlin subalgebra generated by $\mathfrak{h}$, the center of $U(\mathfrak{sl}(3))$, and the center of $U(\mathfrak{a})$. The statement follows from Theorem 5.10. ∎

Remark 5.12**.**

In what follows, we give an interpretation of the eigenvalues of $A_{\lambda}$ in terms of tableaux and the Gelfand-Tsetlin formulas. Set for $i\in{\mathbb{Z}}$

abc

$T(v_{i})$=      x+iy-i**

z

The set of all tableaux in $\mathcal{B}(T(v_{0}))$ with fixed $\mathfrak{g}$-weight $\lambda\in{\mathfrak{h}}^{*}$ (see Definition 4.5) is $\{T(v_{i})\ |\ i\in\mathbb{Z}\}$. Moreover, the Gelfand-Tsetlin formulas imply that $A=E_{12}E_{21}$ acts on $T(v_{i})$ as multiplication by $\mu_{i}:=-(x+i-z)(y-i-z)$, and $r=\frac{1}{2}((2z-x-y-1)^{2}-(2z-x-y-1))$. We have:

• (i)

$(\mu_{i})_{i\in\mathbb{Z}}$* is degenerate with $\mu_{j}=-\frac{1}{2}r$ if and only if $x-y\in\{2j-1,2j+1\}$.*

• (ii)

$(\mu_{i})_{i\in\mathbb{Z}}$* is critical with $\mu_{j}=-\frac{1}{4}-\frac{1}{2}r$ if and only if $x-y=2j$.*

In particular, $(\mu_{i})_{i\in\mathbb{Z}}$ is singular if and only the tableau $T(v_{0})$ (respectively, $\chi_{v_{0}}$) is singular and $\{\mu_{i}\}$ is generic if and only if $T(v_{0})$ (respectively, $\chi_{v_{0}}$) is generic. Note also that $\chi(A)=-\frac{1}{4}-\frac{1}{2}r$ if and only if $\chi=\chi_{w}$ with $w_{21}=w_{22}$. Finally, $\chi(A)=-\frac{1}{2}r$ if and only if $\chi=\chi_{w}$ with $|w_{21}-w_{22}|=1$.

Definition 5.13**.**

We say that a Gelfand-Tsetlin character $\chi$ is critical (respectively, degenerate) if $\chi=\chi_{v}$ for some $v$ such that $v_{21}=v_{22}$ (respectively, $|v_{21}-v_{22}|=1$).

Since $A\in{\Gamma}$, we can extend the concepts of generic and singular chains to Gelfand-Tsetlin modules.

Definition 5.14**.**

A Gelfand-Tsetlin module $M$ is called generic if every Gelfand-Tsetlin character of $M$ is generic. A Gelfand-Tsetlin module $M$ is called singular if it has a singular Gelfand-Tsetlin character.

Note that any finite-dimensional module is a singular Gelfand-Tsetlin module, moreover, any $1$-singular module as defined in Section §4.4 is a singular Gelfand-Tsetlin module. Also, generic modules as defined in §4.3 are generic Gelfand-Tsetlin modules.

Proposition 5.15**.**

If a simple Gelfand-Tsetlin $\mathfrak{g}$-module $M$ is singular, then each Gelfand-Tsetlin character of $M$ is singular.

Proof.

The statement follows by a direct computation. Let $\chi$ be a singular Gelfand-Tsetlin character of $M$, $v\in M(\chi)$, $v\neq 0$. If $E_{12}v\neq 0$ then we easily check that $E_{12}v\in M(\chi^{\prime})$ for a singular $\chi^{\prime}$. Similar reasoning applies for $E_{21}v$. Suppose now $E_{23}v\neq 0$. Then $E_{23}v\in M(\chi^{\prime})\oplus M(\chi^{\prime\prime})$, where $\chi^{\prime}$ and $\chi^{\prime\prime}$ are both singular (one of the subspaces can be zero). Moreover, if $\chi$ belongs to a critical (respectively, degenerate) connected chain, then $\chi^{\prime}$ and $\chi^{\prime\prime}$ belong to a degenerate (respectively, critical) connected chain. We reason similarly for $E_{32}$. ∎

Definition 5.16**.**

Given a Gelfand-Tsetlin character $\chi$ and $M$ a Gelfand-Tsetlin module, we say that $M$ is a simple extension of the character $\chi$ if $M$ is simple and $\chi\in\operatorname{Supp}_{GT}(M)$ (i.e. $M(\chi)\neq 0$).

Lemma 5.17**.**

Let $M$ be a simple Gelfand-Tsetlin module, $\chi\in\operatorname{Supp}_{GT}(M)$ and $\lambda=\chi|_{\mathfrak{h}}$. If there exists a basis $\{w_{i}\}_{i\in I}$ of $M_{\lambda}$ such that the action of $B_{\lambda}$ on this basis is completely determined by the eigenvalues of $A_{\lambda}$ and $\chi$, then $M$ is the unique simple extension of $\chi$.

Proof.

Under these conditions, the simple $C({\mathfrak{h}})$-module $M_{\lambda}$ is defined uniquely, moreover, as $M^{(1)}_{\lambda}\simeq M^{(2)}_{\lambda}$ implies $M^{(1)}\simeq M^{(2)}$, the uniqueness follows. ∎

The generic Gelfand-Tsetlin modules are completely determined by any of their characters, as the following result shows.

Theorem 5.18**.**

If $M$ is a generic simple Gelfand-Tsetlin module then for any $\chi\in\operatorname{Supp}_{GT}(M)$ the subspace $M(\chi)$ is one dimensional and $M$ is the unique simple extension of $\chi$.

Proof.

The result can be found in [F86a] and [F89], but for the sake of completeness we provide a proof. Let $\chi\in\operatorname{Supp}_{GT}(M)$ and let $\lambda=\chi|_{\mathfrak{h}}$ be the associated to $\chi$ weight. Let $\mu_{0}\in\mathbb{C}$ be an eigenvalue of the operator $A_{\lambda}=A|_{M_{\lambda}}$. Then all eigenvalues of $A_{\lambda}$ form a connected chain, i.e. belong to a sequence $\mu_{i}=i^{2}+i\sqrt{1+4\mu_{0}+2r}+\mu_{0}$, $i\in\mathbb{Z}$ for some choice of the square root (see Lemma 5.9(iii)).

Using relations between $A$ and $B$ we can choose a basis $\{w_{i}\ |\ i\in\mathbb{Z}\}$ (this set can be finite or bounded from one side or unbounded) of $M_{\lambda}$ such that

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $r$, $r_{1}$, $\eta$, $\tau$ and $\tau_{1}$ are defined in Lemma 5.3.

Hence, in this case $B_{\lambda}$ is completely determined by $\chi$ and $A_{\lambda}$. The uniqueness follows from Lemma 5.17. ∎

Note that in singular cases the subspace $M(\chi)$ can be $2$-dimensional (see Example 4.38). Also, in these cases for a given $\chi\in{\Gamma}^{*}$ there can exist two non-isomorphic simple extensions of $\chi$. Such examples were first constructed in [F86b].

6. simple extensions of singular Gelfand-Tsetlin characters

In this section we provide sufficient conditions for a singular Gelfand-Tsetlin character to admit a unique simple extension.

Theorem 6.1**.**

If $\chi$ is a critical Gelfand-Tsetlin character then $\chi$ admits a unique simple extension.

Proof.

By Theorem 4.9 there exist at most $2$ simple modules $M^{(1)}$ and $M^{(2)}$ such that $\chi\in\operatorname{Supp}_{GT}(M^{(i)})$ for $i=1,2$. Assume that we have two such modules and let $\lambda\in\operatorname{Supp}(M^{(1)})\cap\operatorname{Supp}(M^{(2)})$ such that $\lambda=\chi|_{\mathfrak{h}}$. For $i=1,2$, consider the restriction $A^{(i)}_{\lambda}$ of $A=E_{12}E_{21}$ on $M^{(i)}_{\lambda}$. Since $M^{(i)}$ are Gelfand-Tsetlin modules, then we can choose bases ${\mathcal{B}}_{1}=\{w_{0},\ldots,w_{m}\}$, $0\leq m\leq\infty$ and ${\mathcal{B}}_{2}=\{w^{\prime}_{0},\ldots,w^{\prime}_{k}\}$, $0\leq k\leq\infty$ of $M^{(1)}_{\lambda}$ and $M^{(2)}_{\lambda}$ such that the matrix $[A_{\lambda}^{(i)}]$ of $A^{(i)}_{\lambda}$ with respect to ${\mathcal{B}}_{i}$ is in a Jordan normal form and each eigenvalue of $A^{(i)}_{\lambda}$ has algebraic multiplicity at most $2$.

By Theorem 5.10(ii)(c), the eigenvalue $\chi(A)$ of $A^{(1)}_{\lambda}$ and $A^{(2)}_{\lambda}$ has multiplicity $1$. Suppose first that all eigenvalues of both $A^{(1)}_{\lambda}$ and $A^{(2)}_{\lambda}$ have multiplicity $1$. Then they can be ordered in connected chains $\{\mu_{n}\ |\ 0\leq n\leq m\}$ and $\{\mu_{m}\ |\ 0\leq m\leq k\}$ with

[TABLE]

and $\mu_{i}=i^{2}-\frac{1}{4}-\frac{1}{2}r$ for $i\geq 1$ (see Lemma 5.9(ii)).

Applying the relations from Lemma 5.3 we obtain

[TABLE]

where

[TABLE]

and

[TABLE]

If $m<k$, then $c_{m+1}=0$ implying that $M^{(2)}$ is reducible. Similarly, if $m>k$, then $c_{k+1}=0$ and $M^{(1)}$ is reducible. It follows that $k=m$. But then formulas (14) define uniquely a simple $C({\mathfrak{h}})$-module $M^{(1)}_{\lambda}\simeq M^{(2)}_{\lambda}$. Therefore, $M^{(1)}\simeq M^{(2)}$.

Suppose now that the algebraic multiplicity of some $\mu_{i}$ is two in $M^{(1)}_{\lambda}$. For simplicity assume that

[TABLE]

where $[B_{\lambda}^{(1)}]$ stands for the matrix of $B|_{M^{(1)}_{\lambda}}$ relative to $\mathcal{B}_{1}$. Applying the relations from Lemma 5.3 we obtain $b_{32}=0$. Note that due to the irreducibility of $M_{\lambda}^{(1)}$ as an $C({\mathfrak{h}})$-module we have $b_{12}\neq 0$ and $b_{31}\neq 0$. Hence, using row operations, one can change the basis $\mathcal{B}_{1}$ so that $b_{13}$ becomes [math].

Now, applying the relations from Lemma 5.3 we obtain

[TABLE]

By changing the basis if needed, we can assume $b_{12}=1$. Therefore, the matrix $[B_{\lambda}^{(1)}]$ is completely determined by $\chi$ and the matrix $[A_{\lambda}^{(1)}]$. We can show that the latter holds for any Jordan normal form $[A_{\lambda}^{(1)}]$.

Consider now the matrix $[A^{(2)}_{\lambda}]$. If this Jordan normal form is not equivalent to $[A^{(1)}_{\lambda}]$, then one of the modules $M^{(1)}_{\lambda}$ or $M^{(2)}_{\lambda}$ will be reducible. Indeed, this can be immediately seen from the form of matrices $[B^{(1)}_{\lambda}]$ and $[B^{(2)}_{\lambda}]$. On the other hand, if $[A^{(1)}_{\lambda}]=[A^{(2)}_{\lambda}]$ then $M^{(1)}_{\lambda}\simeq M^{(2)}_{\lambda}$ as $C({\mathfrak{h}})$-modules and hence $M^{(1)}\simeq M^{(2)}$. ∎

If $\chi$ is a singular Gelfand-Tsetlin character in a critical connected chain and $\chi$ is not critical then there might exist two simple extensions of $\chi$ (see [F86b] for examples). On the other hand we have

Corollary 6.2**.**

Suppose that $\chi$ is a singular character in a critical connected chain and $\chi$ is not critical. If $\lambda=\chi|_{\mathfrak{h}}$, then there exists a unique simple extension $M$ of $\chi$ with diagonalizable $A_{\lambda}$.

Proof.

Indeed, if $A_{\lambda}$ is diagonalizable then $B_{\lambda}$ is determined uniquely. As it was shown in the proof of Theorem 6.1, it is sufficient to know one eigenvalue of $A_{\lambda}$ to reconstruct the whole $A_{\lambda}$ in a simple module. Hence, the statement follows. ∎

Lemma 6.3**.**

Let $M$ be a simple Gelfand-Tsetlin module, $\chi$ a degenerate character of $M$ associated with the weight $\lambda\in{\mathfrak{h}}^{*}$. Let $\rho_{1}=-\frac{1}{2}r$, and $\rho_{2}=2-\frac{1}{2}r$ be connected eigenvalues of $A_{\lambda}$ with $\chi(A)=\rho_{1}$. Suppose that both $\rho_{1}$ and $\rho_{2}$ have multiplicity $2$. Then the Gelfand-Tsetlin support of $M$ contains a critical character $\chi^{\prime}$.

Proof.

Let $u_{1},u_{2},u_{3},u_{4}$ be non-zero elements of $M$ such that

[TABLE]

[TABLE]

Suppose that $M_{\lambda+\epsilon_{2}-\epsilon_{3}}$ does not contain a critical character. Then the eigenvalues $\{\mu_{k},\mu_{k+1}\ldots\,\mu_{m}\}$, $k>1$, of $A_{\lambda+\epsilon_{2}-\epsilon_{3}}$ form a part of a critical connected chain but without the critical character $\mu_{1}$. By Theorem 5.10 these eigenvalues are of multiplicity $1$. Let $v_{1},v_{2}$ be eigenvectors of $A_{\lambda+\epsilon_{2}-\epsilon_{3}}$. Then we have:

[TABLE]

Since $u_{1},u_{2},u_{3}$ and $u_{4}$ are linearly independent and their images span at least a two dimensional space we have that $\dim(\ker(E_{23}\mid_{M_{\lambda}}))\geq 2$ which is impossible by Corollary 5.11.∎

From Lemma 6.3 we immediately the following.

Corollary 6.4**.**

Let $\chi$ be a degenerate Gelfand-Tsetlin character such that $\mu_{1}=\chi(A)$ and $\lambda=\chi|_{\mathfrak{h}}$. If $\{\mu_{1},\mu_{2}\}$ is a $\lambda$-connected set, then there exist at most one simple extension of $\chi$ such that both $\mu_{1}$ and $\mu_{2}$ have multiplicity $2$.

Proof.

Indeed, any such simple module $M$ will contain a critical character $\chi^{\prime}$ determined by the condition $E_{23}(M(\chi))\subset M(\chi^{\prime})+M(\chi^{\prime\prime})$. But, by Theorem 6.1, $\chi^{\prime}$ defines $M$ uniquely. ∎

Lemma 6.5**.**

Let $M$ be a simple Gelfand-Tsetlin module such that $M$ is singular but has no critical characters. Then $A$ is diagonalizable on $M$.

Proof.

Fix $\lambda\in\operatorname{Supp}M$, then the distinct eigenvalues of $A_{\lambda}$ form a singular $\lambda$-connected chain. If this chain is critical then $A_{\lambda}$ is diagonalizable since there is no critical eigenvalue, see 5.10(ii)(d). Suppose that the chain is degenerate. Then by Theorem 5.10(ii)(b) one can order the distinct eigenvalues of $A_{\lambda}$ in the following way: $\{\mu_{1},\mu_{2},\ldots,\mu_{m}\}$, where $\mu_{1}=-\frac{1}{2}r$, and if the multiplicity of $\mu_{i}$ equals $1$ then the multiplicity of $\mu_{i+1}$ is also $1$. Suppose that $M$ has a character $\widetilde{\chi}$ such that $\widetilde{\chi}(A)=\mu_{1}$ and $\mu_{1}$ has multiplicity $2$. If $\mu_{2}$ has multiplicity $2$, then by Lemma 6.3 there exists a critical character $\chi^{\prime}$ in the Gelfand-Tsetlin support of every simple extension of $\widetilde{\chi}$, and we obtain a contradiction.

Assume now that $\mu_{1}$ has multiplicity $2$ but $\mu_{2}$ has multiplicity $1$. Consider the weight subspace $M^{\prime}=M_{\lambda+\epsilon_{2}-\epsilon_{3}}$ and $A^{\prime}=A|_{M^{\prime}}$. Observe that $M^{\prime}\neq 0$, since otherwise $\dim(\ker E_{23}|_{M_{\lambda}})\geq 2$ which is a contradiction by Corollary 5.11. If $A^{\prime}$ has no critical eigenvalue then neither does $A_{\lambda+\epsilon_{2}-\epsilon_{3}+k(\epsilon_{1}-\epsilon_{2})}$, for all integer $k$. In this case all these subspaces $M_{\lambda+\epsilon_{2}-\epsilon_{3}+k(\epsilon_{1}-\epsilon_{2})}$, $k\in\mathbb{Z}$ can generate only one eigenvector of $A^{\prime}$ with eigenvalue $\mu_{1}$ and, hence, produce only multiplicity $1$ eigenvalue $\mu_{1}$ of $A^{\prime}$. But this contradicts to the irreducibility of $M$. Therefore, $A^{\prime}$ must contain a critical eigenvalue giving a contradiction again. Therefore, $A_{\lambda}$ is diagonalizable, which completes the proof. ∎

The proof of Lemma 6.5 implies also the following statement.

Corollary 6.6**.**

Let $M$ be a simple Gelfand-Tsetlin module and $\chi$ a Gelfand-Tsetlin character of $M$ associated with $\lambda\in{\mathfrak{h}}^{*}$ and such that $\dim(M(\chi))=2$. Then $M$ has a critical character $\chi^{\prime}$ associated with the weight $\lambda+\epsilon_{2}-\epsilon_{3}$.

Theorem 6.7**.**

Let $M$ be a simple Gelfand-Tsetlin module and $\chi$ be a Gelfand-Tsetlin character such that $\dim(M(\chi))=2$. Then $M$ is the unique simple extension of $\chi$.

Proof.

Let $\lambda=\chi|_{\mathfrak{h}}$. Since $\dim(M(\chi))=2$, the distinct eigenvalues of $A_{\lambda}$ form a singular $\lambda$-connected chain $\{\mu_{1},\ldots,\mu_{m}\}$, $m\leq\infty$. Moreover, there exists $\mu_{i}$ of multiplicity $2$. We proceed with two cases.

Case 1. The chain $\{\mu_{1},\ldots,\mu_{m}\}$ is critical. By Theorem 5.10 all distinct eigenvalues can be ordered in the following way: $\{\mu_{1},\mu_{2},\ldots\,\mu_{m}\},$ where $\mu_{1}+1=\mu_{2}$, multiplicity of $\mu_{1}$ is $1$, and if the multiplicity of $\mu_{i}$ equals $1$ for $i>1$ then the multiplicity of $\mu_{i+1}$ is also $1$. Therefore, the module $M$ has a critical character $\chi^{\prime}$ such that $\chi^{\prime}(A)=\mu_{1}$. Thus, every simple extension of $\chi$ contains $\chi^{\prime}$ in its Gelfand-Tsetlin support. Applying Theorem 6.1 we conclude that $M$ is unique.

Case 2. The chain $\{\mu_{1},\ldots,\mu_{m}\}$ is degenerate. By Theorem 5.10 one can order the distinct eigenvalues of $A_{\lambda}$ in the following way: $\{\mu_{1},\mu_{2},\ldots,\mu_{m}\}$, where $\mu_{1}=-\frac{r}{2}$, and if the multiplicity of $\mu_{i}$ equals $1$ then the multiplicity of $\mu_{i+1}$ is also $1$. Therefore $M$ has a character $\widetilde{\chi}$ such that $\widetilde{\chi}(A)=\mu_{1}$ and $\mu_{1}$ has multiplicity $2$. By Corollary 6.6, $A|_{M_{\lambda+\epsilon_{2}-\epsilon_{3}}}$ must contain a critical eigenvalue. Thus, $M$ is unique by Theorem 6.1. This completes the proof. ∎

Theorem 6.8**.**

Let $M$ be a simple Gelfand-Tsetlin module such that $M$ is singular but has no critical characters. Then for each character $\chi\in\operatorname{Supp}_{GT}(M)$, $M$ is the unique simple extension of $\chi$ with the property that it has no critical characters.

Proof.

Let as usual $\lambda=\chi|_{\mathfrak{h}}$ and $A_{\lambda}=A|_{M_{\lambda}}$. It follows from Lemma 6.5 that $A_{\lambda}$ is diagonalizable. We proceed in two steps.

Step I. Suppose that $\chi$ belongs to a critical connected chain. As $M$ does not have critical characters, $\chi(A)$ belongs to a critical connected chain $\mu_{i}=i^{2}-\frac{1}{4}-\frac{1}{2}r$, $1\leq n\leq i\leq m\leq\infty$ for some integers $n$ and $m$. Then as in Theorem 6.1, there exists a basis $\{w_{i},n\leq i\leq m\}$ such that

[TABLE]

Hence, $B_{\lambda}=B$ is determined uniquely by $\chi$ and $A_{\lambda}$. The uniqueness follows from Lemma 5.17.

Step II. Suppose that $\chi(A)$ belongs to a degenerate chain $\{\mu_{1},\mu_{2},\ldots\}$ where $\mu_{n}=n(n-1)-\frac{1}{2}r$, $n\geq 1$ (see Lemma 5.9(i)). We proceed considering two cases depending on the connected chain $\{\mu_{1},\mu_{2},\ldots\}$.

Case 1. The chain $\{\mu_{1},\mu_{2},\ldots\}$ does not contain a degenerate character, that is the eigenvalues of $A_{\lambda}$ are $\{\mu_{k},\mu_{k+1},\ldots,\mu_{s}\}$ for some $k>1$ and some $s\leq\infty$. Applying relations from Lemma 5.3 one can choose a basis $\{w_{k},...,w_{s}\}$ of $M_{\lambda}$ such that the matrix of $A_{\lambda}$ is diagonal and the matrix of $B_{\lambda}$ has a tridiagonal form as in the generic case. Suppose there exists another simple extension $W$ of $\chi$ satisfying the conditions of the theorem such that the eigenvalues of $A|_{W_{\lambda}}$ are $\{\mu_{d},\mu_{d+1},\ldots,\mu_{t}\}$ for some $d>1$ and some $t\leq\infty$. If $d=k$ and $s=t$, then the diagonal matrices $[A|_{M_{\lambda}}]=[A|_{W_{\lambda}}]$ will give the same matrix of $B_{\lambda}$, hence $M\simeq W$ by Lemma 5.17.

Suppose $d<k$ (note that in this case $k\leq t$). Then applying relations from Lemma 5.3 we obtain that $W_{\lambda}$ has a $C({\mathfrak{h}})$-submodule $U$ such that the eigenvalues of $A|_{U}$ are $\{\mu_{k},\mu_{k+1},\ldots,\mu_{t}\}$. Hence $M$ has a nontrivial proper submodule $\widetilde{M}$ such that the eigenvalues of $A|_{\widetilde{M}_{\lambda}}$ are $\{\mu_{k},\mu_{k+1},\ldots,\mu_{t}\}$. This contradicts the irreducibility of $M$. The case $d>k$ is treated analogously.

Case 2. The chain $\{\mu_{1},\mu_{2},\ldots\}$ does contain a degenerate character, that is the eigenvalues of $A_{\lambda}$ are $\{\mu_{1},\mu_{2},\ldots,\mu_{s}\}$ for some some $s\leq\infty$. Let $\chi^{\prime}$ be the character associated with $\mu_{1}$. Using relations from Lemma 5.3 we see that $r^{2}+2ar+4\tau=0$ and there exists a basis $\{w_{i},1\leq i\leq s\}$ of $M_{\lambda}$ such that

[TABLE]

where

[TABLE]

and $T$ is a root of the equation

[TABLE]

Let $W$ be another simple extension of $\chi$. Then $\mu_{1}$ must be an eigenvalue of $A|_{W_{\lambda}}$, otherwise $M$ is not simple. In fact, the quadratic equation on $T$ shows that there might exist two non-isomorphic simple modules with the same degenerate chain $\{\mu_{1},\ldots,\mu_{s}\}$. We will show that only one such module will satisfy the conditions of the theorem.

The hypothesis that there is no critical characters in all connected chains implies that $E_{23}(M(\chi^{\prime}))\subset M(\tilde{\chi})$, where $\tilde{\chi}$ is a character such that $\tilde{\chi}(A)$ belongs to a critical connected chain (without critical characters by hypothesis). Also $E_{23}(W(\chi^{\prime}))\subset W(\tilde{\chi})$. If both $E_{23}(M(\chi^{\prime}))$ and $E_{23}(W(\chi^{\prime}))$ are non-zero then $M(\tilde{\chi})\neq 0$ and $W(\tilde{\chi})\neq 0$. We immediately conclude that $M\simeq W$ by Theorem 6.1. Suppose $E_{23}(M(\chi^{\prime}))=E_{23}(W(\chi^{\prime}))=0$. Apply the same arguments for $E_{13}$. If both $E_{13}(M(\chi^{\prime}))$ and $E_{13}(W(\chi^{\prime}))$ are non-zero, then $M\simeq W$ as above. On the other hand, if $E_{13}(M(\chi^{\prime}))=E_{13}(W(\chi^{\prime}))=0$, then $M$ and $W$ are simple quotients of the same generalized Verma module generated by a weight vector $v$ such that $E_{23}v=E_{13}v=0$. But such generalized Verma module has a unique simple quotient implying $M\simeq W$. Hence, it remains to consider mixed cases. We finish the proof considering three subcases. Recall that a weight module is pointed if all its nonzero weight spaces are $1$-dimensional. All other cases are considered analogously.

Case 2(a). Suppose, $E_{23}(M(\chi^{\prime})),E_{13}(M(\chi^{\prime}))\neq 0$, and $E_{23}(W(\chi^{\prime})),E_{13}(W(\chi^{\prime}))=0$. Then $W$ is a quotient of the generalized Verma module $M_{1}$ generated by an element $v\in M_{1}$ such that $E_{23}v=E_{13}v=0$. Suppose $E_{32}(W(\chi^{\prime}))\neq 0$ and $E_{31}(W(\chi^{\prime}))\neq 0$. If one of $E_{32}(M(\chi^{\prime}))$ or $E_{31}(M(\chi^{\prime}))$ is non-zero then we are done. Suppose $E_{32}(M(\chi^{\prime}))=E_{31}(M(\chi^{\prime}))=0$ and thus $M$ is a quotient of generalized Verma module $M_{2}$ generated by $v^{\prime}$ such that $E_{32}v^{\prime}=E_{31}v^{\prime}=0$. In order not to have critical characters both $M$ and $W$ must be pointed modules, that is all weight spaces have dimension $1$. Let $\chi^{\prime}(H_{1})=h_{1}$, $\chi^{\prime}(H_{2})=h_{2}$. Comparing the values of Casimir elements on $M$ and $W$ we obtain $h_{1}=-2h_{2}$. This condition guarantees that $M$ and $W$ have common degenerate character $\chi^{\prime}$. Let us find the condition when $W$ is a pointed module. It is sufficient to check when the following system has a non-trivial solution:

[TABLE]

Assume that $E_{12}v\neq 0$. Then we have $h_{2}\alpha-\beta h_{1}-\frac{1}{2}\beta r=0$, and $\alpha+\beta(h_{1}+h_{2}+1)=0$. If $h_{1}=0$, then $M\simeq W\simeq\mathbb{C}$. If $h_{1}\neq 0$ then $h_{1}=2$, $h_{2}=-1$ and $\chi^{\prime}(A)=0$. It follows that $E_{23}v=E_{13}v=E_{21}v=0$ with $h_{1}=2$, $h_{2}=-1$ or $E_{23}v=E_{13}v=E_{12}v=0$ with $h_{1}=-2$, $h_{2}=1$.

Consider first the case $h_{1}=-2$, and let $\mu\in{\mathfrak{h}}^{*}$ be such that $\mu(H_{1})=\mu(H_{2})=-1$. Then $W_{\mu}$ is $1$-dimensional and $E_{12}W_{\mu}=0$. Hence, $W_{\mu}$ is a Gelfand-Tsetlin subspace and $A|_{W_{\mu}}=-{\rm Id}$. But, this is a critical value and, thus, $W$ contains critical characters, which is a contradiction.

Suppose now that $h_{1}=2$ and consider $\mu\in{\mathfrak{h}}^{*}$ such that $\mu(H_{1})=1$, $\mu(H_{2})=-2$. Then $W_{\mu}$ is $1$-dimensional and $E_{21}W_{\mu}=0$. Hence, $W_{\mu}$ is a Gelfand-Tsetlin subspace and $A|_{W_{\mu}}=0$. Again, this is a critical value which is a contradiction.

Suppose now that $E_{32}(W(\chi^{\prime}))=0$. Then $E_{12}(W(\chi^{\prime}))=0$ and $W$ is a highest weight module of highest weight $\chi^{\prime}|_{H}$ and $\chi^{\prime}(H_{2})=h_{2}=0$. Since $\chi^{\prime}$ is degenerate we have $h_{1}=0$ or $h_{1}=-2$. In the case $h_{1}=0$ we obtain $M\simeq W\simeq\mathbb{C}$. Now suppose $h_{1}=-2$ and $A|_{W(\chi^{\prime})}=-2{\rm Id}$. This highest weight module has no critical characters. Since $E_{31}W(\chi^{\prime})\neq 0$ we have $E_{31}M(\chi^{\prime})=0$, otherwise $M\simeq W$ as before. Since $A|_{M(\chi^{\prime})}=-2{\rm Id}$ and all characters have multiplicity $1$, we have $E_{12}M(\chi^{\prime})=0$. Thus, in addition we have $E_{32}M(\chi^{\prime})=0$. Consider a weight $\mu\in{\mathfrak{h}}^{*}$ such that $\mu(H_{1})=-1$, $\mu(H_{2})=1$. The subspace $M_{\mu}$ is $1$-dimensional. In fact, this is a critical Gelfand-Tsetlin subspace, since $A|_{M_{\mu}}=-{\rm Id}$. Hence, $M$ does not satisfy the conditions of the theorem and $W$ again is a unique required module.

Case 2(b). Suppose $E_{23}(M(\chi^{\prime})),E_{13}(W(\chi^{\prime}))\neq 0$, and $E_{23}(W(\chi^{\prime})),E_{13}(M(\chi^{\prime}))=0$. Now we act by $E_{32}$ and $E_{31}$. Suppose first that $E_{32}(M(\chi^{\prime}))=0$ and $E_{31}(W(\chi^{\prime}))=0$. Hence, $M$ contains a non-zero vector $v$ such that $E_{13}v=E_{32}v=E_{12}v=0$. On the other hand, $W$ contains a non-zero vector $v^{\prime}$ such that $E_{31}v^{\prime}=E_{23}v^{\prime}=E_{21}v^{\prime}=0$. Moreover, $H_{1}v=H_{1}v^{\prime}=0$. But, since $A$ is diagonalizable on $M$ and on $W$, we have $E_{21}v=E_{12}v^{\prime}=0$. Thus $M\simeq W\simeq\mathbb{C}$.

Suppose now $E_{31}(M(\chi^{\prime}))=0$ and $E_{32}(W(\chi^{\prime}))=0$. Hence, $M$ contains a non-zero vector $v$ such that $E_{13}v=E_{31}v=0$, and $W$ contains a non-zero vector $v^{\prime}$ such that $E_{32}v^{\prime}=E_{23}v^{\prime}=0$. We obtain that $H_{1}v=H_{1}v^{\prime}=0$ and $H_{2}v=H_{2}v^{\prime}=0$. Moreover, $Av=Av^{\prime}=0$. Since $A$ is diagonalizable on $M$ and $W$ we have $E_{12}v=E_{12}v^{\prime}=0$ and $E_{21}v=E_{21}v^{\prime}=0$. Thus $M\simeq W\simeq\mathbb{C}$.

Finally, let $E_{31}(M(\chi^{\prime}))=0$ and $E_{32}(M(\chi^{\prime}))=0$. Hence, $M$ contains a non-zero vector $v$ such that $E_{13}v=E_{31}v=E_{32}v=0$, implying $E_{12}v=0$. So, either $H_{1}v=H_{2}v=0$ and $M\simeq W\simeq\mathbb{C}$, or $H_{1}v=-2$ and $H_{2}v=2$. In the latter case $W$ contains a non-zero vector $v^{\prime}$ such that $E_{23}v^{\prime}=0$, $H_{1}v^{\prime}=-2v^{\prime}$ and $Av^{\prime}=-2v^{\prime}$. Hence, $E_{12}v^{\prime}=0$ and $E_{13}v^{\prime}=0$ since $A$ is diagonalizable. We have $c_{1}v=c_{1}v^{\prime}$ implying $H_{2}v=H_{2}v^{\prime}=0$ which is a contradiction.

Case 2(c). Suppose $E_{23}(M(\chi^{\prime}))\neq 0$, and $E_{23}(W(\chi^{\prime})),E_{13}(M(\chi^{\prime})),E_{13}(W(\chi^{\prime}))=0$. Now we act by $E_{32}$ and $E_{31}$ on $M(\chi^{\prime})$ and $W(\chi^{\prime})$. Without loss of generality we may assume that $E_{32}(M(\chi^{\prime}))=0$ and $E_{31}(W(\chi^{\prime}))=0$. Therefore $E_{21}W(\chi^{\prime})=[E_{23},E_{31}]W(\chi^{\prime})=0$ and $AW(\chi^{\prime})=0$. Hence, we have either $H_{1}w=H_{2}w=0$ or $H_{1}w=2w$, $H_{2}w=-2w$ for any $w\in W(\chi^{\prime})$. In the first case we obtain $M\simeq W\simeq\mathbb{C}$. Consider the second case. Since $AM(\chi^{\prime})=0$, we must have $E_{21}(M(\chi^{\prime}))=0$ (otherwise $A$ will not be diagonalizable on $M_{\mu}$), where $E_{21}(M(\chi^{\prime}))\subset M_{\mu}$. Thus $E_{23}(M(\chi^{\prime}))=[E_{21},E_{13}]M(\chi)=0$. Which is a contradiction. ∎

We next state the main theorem in this section.

Theorem 6.9**.**

Let $M$ be a simple Gelfand-Tsetlin $\mathfrak{g}$-module and $\chi\in{\rm Supp}_{GT}(M)$. Consider the following conditions:

• (i)

$\chi$* is non-critical and $\dim M(\chi^{\prime})\leq 1$ for any $\chi^{\prime}\in{\Gamma}^{*}$;*

• (ii)

$\chi$* is non critical and $\dim(M(\chi))=2$;*

• (iii)

$\chi$* is critical;*

• (iv)

$M$* has no critical characters.*

If any of (ii)–(iv) holds, then $M$ is the unique simple extension of $\chi$. If (i) holds, then $M$ is the unique simple extension of $\chi$ with property (i), but $\chi$ may have another simple extension with two-dimensional Gelfand-Tsetlin multiplicities.

Proof.

Let $\chi$ be a non critical character of $M$. Suppose first that $M$ is generic. Then $M$ is the only simple extension of $\chi$ by Theorem 5.18. Suppose now that $M$ is singular and satisfies the conditions in (i). Assume first that $\chi(A)$ belongs to a critical chain but $\chi(A)$ itself is non-critical. Then $M$ is the unique simple extension of $\chi$ by Corollary 6.2. Assume now that $\chi(A)$ belongs to a degenerate chain $\{\mu_{1},\mu_{2},\ldots\}$ where $\mu_{n}=n(n-1)-\frac{1}{2}r$, $n\geq 1$ (see Lemma 5.9(i)). Suppose that $M^{(1)}$ and $M^{(2)}$ are two simple extensions of $\chi$ satisfying the conditions of (i). If $\mu_{1}$ is not an eigenvalue of $A^{(i)}_{\lambda}:=A|_{M^{(i)}_{\lambda}}$, where $\lambda=\chi|_{\mathfrak{h}}$, then $M^{(1)}\simeq M^{(2)}$ since $A^{(i)}_{\lambda}$ (and thus $M^{(i)}$) is uniquely determined by $\chi(A)$ in this case. Assume that $\mu_{1}$ is an eigenvalue of both $A^{(i)}_{\lambda}$, $i=1,2$. Consider the weight $\tilde{\lambda}:=\lambda+\epsilon_{2}-\epsilon_{3}$ and the eigenvalues of $A^{(i)}_{\tilde{\lambda}}$. They belong to the same critical chain. If both $A^{(i)}_{\tilde{\lambda}}$ have a common non-critical eigenvalue then $M^{(1)}\simeq M^{(2)}$ by Corollary 6.2. Hence, may assume that $A^{(i)}_{\tilde{\lambda}}$ have distinct eigenvalues. This is only possible if $\dim M^{(i)}_{\lambda}=1$ and $\dim M^{(i)}_{\tilde{\lambda}}\leq 1$, $i=1,2$. Let $B|_{M^{(i)}_{\lambda}}=b_{i}{\rm Id}$. Recall that a weight module is torsion free if all root vectors act injectively on the module. Now consider the following cases.

Case 1. $\chi(A)-k\chi(H_{1})\neq 0$, $b_{i}-m\chi(H_{2})\neq 0$ for $i=1,2$ and $k,m\in{\mathbb{Z}}$. In this case both $M^{(1)}$ and $M^{(2)}$ are pointed torsion free modules. Set $\tilde{\lambda}(H_{i})=\tilde{h}_{i}$, $i=1,2$. Then we have $A^{(1)}_{\tilde{\lambda}}=\tilde{\mu}_{1}{\rm Id}$, $A^{(2)}_{\tilde{\lambda}}=\tilde{\mu}_{2}{\rm Id}$, $B^{(1)}_{\rho}=\tilde{b}_{1}{\rm Id}$ and $B^{(2)}_{\tilde{\lambda}}=\tilde{b}_{2}{\rm Id}$, where we can assume that $\tilde{\mu}_{1}=-\frac{1}{4}-\frac{1}{4}\tilde{h}^{2}_{1}+\tilde{h}_{1}$ and $\tilde{\mu}_{2}=\tilde{\mu}_{1}+1$. Using the second identity in Lemma 5.3 we also have

[TABLE]

Keeping in mind that $\tau_{1}$ and $a$ depend of $\tilde{h}_{1}$ and $\tilde{h}_{2}$, $r$ depends of $\tilde{h}_{1}$ and $b_{i}$, $r_{1}$ depend of $\tilde{h}_{2}$, $(a-2\tilde{\mu}_{1})\tilde{b}_{1}-\tilde{b}_{1}^{2}-r_{1}\tilde{\mu}_{1}-\tau_{1}$ can be express as a polynomial in $\tilde{h}_{1}$, $\tilde{h}_{2}$. Let us consider the two-variable polynomial

[TABLE]

Then by (17), $f_{i}(\tilde{h}_{1},\tilde{h}_{2})=0$. An easy calculation shows that

[TABLE]

Then $f_{i}(\tilde{h}_{1}-2,\tilde{h}_{2}+4)=0$, $i=1,2$. Similarly, using the operator $E_{12}^{2}$ we obtain $f_{i}(\tilde{h}_{1}+4,\tilde{h}_{2}-2)=0$. Hence, $f_{i}(\tilde{h}_{1}+6,\tilde{h}_{2})=0$. If we repeat this argument again we will obtain $f_{i}(\tilde{h}_{1}+12,\tilde{h}_{2})=0$ and so on. Hence if $g(x)=f_{1}(x,\tilde{h}_{2})$, we have shown that $g(t)=0$ implies $g(t+6)=0$, so $g$ has infinitely many roots, thus $g=0$, which is impossible.

Case 2. $\chi(A)-k\chi(H_{1})=0$, $b_{i}-m\chi(H_{2})\neq 0$ for some $k\in{\mathbb{Z}}$ and all $i=1,2$,   $m\in{\mathbb{Z}}$. Without loss of generality we may assume that $\chi(A)=0$. Since $A$ is diagonalizable on both modules, then this immediately implies that both $M^{(1)}$ and $M^{(2)}$ are quotients of generalized Verma modules (induced from an infinite-dimensional simple $\mathfrak{sl}(2)$-module $W$) that have the same central character and the same weight support. Since the generalized Harish-Chandra homomorphism defines $W$ uniquely we conclude that $M^{(1)}\simeq M^{(2)}$.

Case 3. $\chi(A)-k\chi(H_{1})=0$, $b_{1}-m_{1}\chi(H_{2})=0$, $b_{2}-m_{2}\chi(H_{2})\neq 0$, for some $k,m_{1}\in{\mathbb{Z}}$ and all $i=1,2$,   $m_{2}\in{\mathbb{Z}}$. This case is handled in a similar way as Case 2.

Case 4. $\chi(A)-k\chi(H_{1})=0$ and $b_{i}=m_{i}\chi(H_{2})$ for some integer $k,m_{i}$, $i=1,2$. Therefore, as in Case 2, both $M^{(1)}$ and $M^{(2)}$ are quotients of the same generalized Verma module. Hence, $m_{1}=m_{2}$ and $M^{(1)}\simeq M^{(2)}$. This completes the proof of (i).

If $\chi$ is a non critical character such that $\dim(M(\chi))=2$, then $M$ is a unique simple extension of $\chi$ by Theorem 6.7 implying the result for (ii). The uniqueness of the extension if (iii) holds follows immediately from Theorem 6.1. It remains to consider (iv). Suppose $M$ is generic. Then the uniqueness of simple extension for any character of $M$ again follows from Theorem 5.18. If $M$ is singular but without critical characters, then we apply Theorem 6.8. ∎

7. Realizations of all simple Gelfand-Tsetlin modules for $\mathfrak{sl}(3)$

In this section we give an explicit realization of all simple Gelfand-Tsetlin modules of $\mathfrak{sl}(3)$. For this purpose we consider any Gelfand-Tsetlin character $\chi\in\Gamma^{*}$ and construct a Gelfand-Tsetlin module $M$ such that any simple extension of $\chi$ is isomorphic to some subquotient of $M$ (recall that, by Theorem 4.9, the number of non-isomorphic simple extensions is at least one and at most two).

Remark 4.2 provides a natural correspondence between the Gelfand-Tsetlin characters and the Gelfand-Tsetlin tableaux. Hence, given a character $\chi$ we can associate a tableau $T(v)$ and the problem of constructing simple extensions of $\chi$ is reduced to the problem of finding simple modules with tableaux realization containing $T(v)$ as a basis element. Recall that any Gelfand-Tsetlin tableau $T(v)$ of height $3$ is either generic ($v_{21}-v_{22}\notin\mathbb{Z}$) or $1$-singular ($v_{21}-v_{22}\in\mathbb{Z}$), and the constructions in §4 allow us to describe an explicit Gelfand-Tsetlin module $V(T(v))$ for any $T(v)$. This, combined with Theorem 6.9, implies that for the desired classification, it is sufficient to describe all simple subquotients of the modules $V(T(v))$.

7.1. Structure of generic $\mathfrak{sl}(3)$-modules $V(T(v))$

In this subsection we consider all possible generic Gelfand-Tsetlin tableaux $T(v)$ and describe all simple subquotients of the $\mathfrak{sl}(3)$-module $V(T(v))$. The description includes an explicit basis for each simple subquotient, its weight support and its Loewy decomposition. Since $\mathfrak{g}=\mathfrak{sl}(3)$, the action of $E_{11}+E_{22}+E_{33}$ is zero, thus $w_{31}+w_{32}+w_{33}+3=0$ for any Gelfand-Tsetlin tableaux $T(w)$. We first rewrite Theorem 4.14 in the case of $\mathfrak{sl}(3)$.

Theorem 7.1**.**

If $T(v)$ is a generic Gelfand-Tsetlin tableau of height $3$, then the vector space $V(T(v))$ spanned by the set of tableaux $\mathcal{B}(T(v))$ has a structure of a Gelfand-Tsetlin $\mathfrak{sl}(3)$-module with the action of $\mathfrak{sl}(3)$ on $V(T(v))$ given by the Gelfand-Tsetlin formulas:

[TABLE]

[TABLE]

[TABLE]

By Theorem 5.18, a generic character admits a unique simple extension. In order to describe such simple extension, given a tableau $T(w)\in\mathcal{B}(T(v))$ we will describe explicit basis of tableaux for the simple subquotient of $V(T(v))$ that contains $T(w)$.

By (18) and (19) it is clear that $\mathcal{B}(T(v))$ is an eigenbasis for the action of the generators of the Cartan subalgebra $\mathfrak{h}$. In particular, any subquotient of $V(T(v))$ is a weight module. The following proposition describes explicit bases for the weight subspaces of the subquotiens of $V(T(v))$.

Proposition 7.2**.**

Let $M$ be a Gelfand-Tsetlin module with basis of tableaux $\mathcal{B}_{M}\subset\mathcal{B}(T(v))$ for some generic tableau $T(v)$. If $T(w)\in\mathcal{B}_{M}$ is a tableau of weight $\lambda$, then the weight space $M_{\lambda}$ is spanned by the set of tableaux $\{T(w+(i,-i,0))\ |\ i\in\mathbb{Z}\}\cap\mathcal{B}_{M}$.

Proof.

As $\mathcal{B}(T(v))=\mathcal{B}(T(w))$, we just need to characterize tableaux of the form $T(w+(m,n,k))$ in $\mathcal{B}_{M}$ with the same weight $\lambda$ of $T(w)$. By the Gelfand-Tsetlin formulas we have

[TABLE]

In particular, the weight of $T(w)$ is

[TABLE]

Furthermore, a tableau $T(w+(m,n,k))$ in $\mathcal{B}_{M}$ has weight $\lambda$ if $m,n,k$ satisfy $m+n=0$ and $k=0$. ∎

In this section we will use Theorem 4.25 to describe all simple subquotients of the generic $\mathfrak{sl}(3)$-modules $V(T(v))$. Let us recall this result.

Let $T(v)$ be a generic Gelfand-Tsetlin tableau of height $3$ and $T(w)\in\mathcal{B}(T(v))$ and $\Omega^{+}(T(u))=\{(r,s,t)\ |\ u_{rs}-u_{r-1,t}\in\mathbb{Z}_{\geq 0}\}$. The complex vector space with basis $\mathcal{N}(T(w))=\{T(w^{\prime})\in\mathcal{B}(T(v))\ |\ \Omega^{+}(T(w))\subseteq\Omega^{+}(T(w^{\prime}))\}$ is a submodule of $V(T(v))$ containing $T(w)$. Moreover, $\mathcal{I}(T(w)):=\{T(w^{\prime})\in\mathcal{B}(T(v))\ |\ \Omega^{+}(T(w))=\Omega^{+}(T(w^{\prime}))\}$ is a basis of the unique simple extension of $\chi_{w}$, where $\chi_{w}\in\Gamma^{*}$ is given by $\chi_{w}(c_{rs})=\gamma_{rs}(w)$.

By Theorem 4.25, bases of the subquotients of $V(T(v))$ can be described by subsets of $\mathbb{Z}^{3}$. In order to describe these bases, we introduce new notation.

Definition 7.3**.**

Let $T(w)$ be a tableau and $\mathcal{B}$ be a subset of ${\mathbb{Z}}^{3}$. Assume that $M$ is a Gelfand-Tsetlin module with basis $\{T(w+(m,n,k))\;|\;(m,n,k)\in\mathcal{B}\}$. Then we will denote $M$ by $M(\mathcal{B},T(w))$, or simply by $M(\mathcal{B})$ if $T(w)$ is fixed. If $M(\mathcal{B})$ is simple, then we will write $L(\mathcal{B})$ for $M(\mathcal{B})$.

Example 7.4**.**

With the notation of Definition 7.3, the simple module from Example 4.27 can be written as follows:

[TABLE]

where $v=(a,b,c,a,b+2,a)$.

7.2. Realizations of all simple generic Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules

In this section we describe all simple objects in every generic block $\mathcal{GT}_{T(v)}$ (see Definition 4.12 and Remark 4.2). Such description will include an explicit tableaux basis of each simple subquotient $M$ in $\mathcal{GT}_{T(v)}$ and the weight support of $M$. For the weight support we will use Proposition 7.2 and the explicit basis. If the weight multiplicities are finite, a picture of the weight support along with the multiplicities is provided. We also present the components of the Loewy series of the universal module $V(T(v))$. A rigorous proof based on Theorem 4.25, Proposition 7.2, and Theorem 4.29 is given for Case $(G6)$ only, however, for all other cases the reasoning is the same.

Until the end of this section we use the following convention. The entries of the Gelfand-Tsetlin tableaux will be integer shifts of some of the complex numbers $a,b,c,x,y,z$. We also assume that if any two of $a,b,c,x,y,z$ appear in the same row or in consecutive rows of a given tableau, then their difference is not integer.

Remark 7.5**.**

By Theorem 5.18, any generic character has a unique simple extension. In particular, if $T(v)$ is generic, the number of simple subquotients of $V(T(v))$ is equal to the number of simple modules in $\mathcal{GT}_{T(v)}$. This number depends only on $\Omega(T(v))$ (see Theorem 7.6 in [FGR15]).

• $(G1)$

Consider the following Gelfand-Tsetlin tableau:

a$$b$$c

$T(v)$=    x$$y

$z$

The module $V(T(v))$ is simple and, then $\mathcal{GT}_{T(v)}$ has unique (up to isomorphism) simple module, and this module has infinite-dimensional weight multiplicities.

[TABLE]

• $(G2)$

Let $T(v)$ be the tableau

a$$b$$c

x$$y

$x$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $2$ simple subquotients and they have infinite-dimensional weight multiplicities:

[TABLE]

• II.

**Loewy series.

**

[TABLE]

• $(G3)$

Consider the tableau:

a$$b$$c

$T(v)$=    a$$y

$z$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $2$ simple subquotients and they have infinite-dimensional weight multiplicities:

[TABLE]

• II.

Loewy series.

[TABLE]

• $(G4)$

Consider the tableau:

a$$b$$c

$T(v)$=    a$$y

$y$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $4$ simple subquotients. The bases and corresponding weight lattices are given by:

• (i)

Modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Modules with unbounded finite weight multiplicities.

[TABLE]

4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$5$$4$$3$$2$$1$$3$$2$$1$$5$$3$$4$$2$$1$$2$$1$$6$$3$$4$$5$$2$$1$$2$$1$$1$$2$$3$$4$$5$$6$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$5$$4$$3$$2$$1$$3$$2$$1$$5$$4$$3$$2$$1$$2$$1$$6$$5$$4$$3$$2$$1$$2$$1$$1$$6$$5$$4$$3$$2$$1$$\leftarrow L_{2}$$L_{3}\rightarrow

The origin of the picture above corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v)$.

• II.

**Loewy series.

**

[TABLE]

• $(G5)$

Let $T(v)$ be the generic tableau

a$$b$$c

a$$y

$a$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $4$ simple subquotients. The bases and corresponding weight lattices are given by:

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two modules with unbounded finite weight multiplicities. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v+\delta^{11})$.

[TABLE]

2$$1$$2$$1$$1$$3$$2$$1$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$5$$4$$3$$2$$1$$5$$3$$4$$2$$1$$6$$3$$4$$5$$2$$1$$6$$4$$5$$3$$1$$2$$1$$2$$3$$4$$1$$2$$3$$4$$1$$2$$3$$1$$2$$3$$4$$5$$1$$2$$3$$1$$2$$3$$4$$5$$1$$2$$1$$2$$3$$4$$5$$6$$1$$2$$1$$2$$3$$4$$5$$6$$1$$\leftarrow L_{1}$$L_{4}\rightarrow

• II.

**Loewy series.

**

[TABLE]

• $(G6)$

Consider the tableau:

a$$b$$c

$T(v)$=    a$$b

$a$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $8$ simple subquotients. The bases and corresponding weight lattices are given by:

• (i)

Two modules with infinite-dimensional weight spaces

[TABLE]

• (ii)

Six modules with unbounded finite weight multiplicities. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v+\delta^{11}+\delta^{21})$.

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$\leftarrow L_{1}$$L_{2}\rightarrow

4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$5$$4$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$5$$4$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$\leftarrow L_{3}$$L_{6}\rightarrow

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$\leftarrow L_{7}$$L_{8}\rightarrow

• II.

**Loewy series.

**

[TABLE]

Proof.

If $M$ denotes the universal module $V(T(v))$, we proceed as follows: first prove that $L_{1}$ is a simple submodule of $M$, then that $M_{1}:=M/L_{1}$ has simple submodules isomorphic to $L_{2},\ L_{3}$ and $L_{4}$, then prove that $M_{2}:=M_{1}/(L_{2}\oplus L_{3}\oplus L_{4})$ has simple submodules isomorphic to $L_{5},\ L_{6}$ and $L_{7}$, and finally show that $L_{8}=M_{2}/(L_{5}\oplus L_{6}\oplus L_{7})$ is a simple module.

By Theorem 4.25 we see that, $L_{1}$ (respectively $L_{2}$, $L_{3}$, $L_{4}$ , $L_{5}$, $L_{6}$, $L_{7}$ and $L_{8}$) is a simple subquotient of $V(T(v))$ containing the tableau $T(v)$ (respectively $T(v+(0,0,1))$, $T(v+(0,1,0))$, $T(v+(1,0,1))$ , $T(v+(0,1,1))$, $T(v+(1,0,2))$, $T(v+(1,1,1))$, and $T(v+(1,1,2))$). To find the the layers of the Loewy series decomposition for $V(T(v))$ we apply Theorem 4.29 and Remark 4.30. We describe these layers in four steps.

Step 1. $L_{1}$ is a simple submodule of $M$. By Theorem 4.25(i) the module with basis $\mathcal{N}(T(v))=\{T(w)\ |\ \Omega^{+}(T(v))\subseteq\Omega^{+}(T(w))\}$ is a submodule of $M$, but $\Omega^{+}(T(v))=\{(3,1,1),(3,2,2),(2,1,1)\}=\Omega(T(v))$, thus $\mathcal{N}(T(v))=\mathcal{I}(T(v))$. Hence, $L_{1}$ is a simple submodule of $M$.

Step $2$. $M_{1}:=M/L_{1}$ has simple submodules isomorphic to $L_{2},\ L_{3}$ and $L_{4}$. For these modules we have that

[TABLE]

Hence, by Theorem 4.25(i) the modules with bases $\mathcal{I}(T(v+(1,0,1)))\cup\mathcal{I}(T(v))$, $\mathcal{I}(T(v+(0,0,1)))\cup\mathcal{I}(T(v))$ and $\mathcal{I}(T(v+(0,1,0)))\cup\mathcal{I}(T(v))$ are submodules of $M$. Therefore, the modules with bases $\mathcal{I}(T(v+(1,0,1)))$; $\mathcal{I}(T(v+(0,0,1)))$ and $\mathcal{I}(T(v+(0,1,0)))$ are submodules of $M_{1}:=M/L_{1}$ since $L_{1}$ is has basis $\mathcal{I}(T(v))$.

Step $3$. $M_{2}:=M_{1}/(L_{2}\oplus L_{3}\oplus L_{4})$ has simple submodules isomorphic to $L_{5},\ L_{6}$ and $L_{7}$. For these modules we have

[TABLE]

Hence, by Theorem 4.25(i) the modules with bases $\mathcal{I}(T(v+(1,0,1)))\cup\mathcal{I}(T(v))$; $\mathcal{I}(T(v+(0,0,1)))\cup\mathcal{I}(T(v))$ and $\mathcal{I}(T(v+(0,1,0)))\cup\mathcal{I}(T(v))$ are submodules of $M$. Therefore, the modules with bases $\mathcal{I}(T(v+(1,0,1)))$; $\mathcal{I}(T(v+(0,0,1)))$ and $\mathcal{I}(T(v+(0,1,0)))$ are submodules of $M_{2}:=M_{1}/(L_{2}\oplus L_{3}\oplus L_{4})$ because $L_{1}$ has a basis $\mathcal{I}(T(v))$.

Step $4$. $M_{3}/(L_{5}\oplus L_{6}\oplus L_{7})\simeq L_{8}$. In fact, $\Omega^{+}(T(v+(1,1,2)))=\emptyset$ so the submodule of $V(T(v))$ generated by $T(v+(1,1,2))$ is $V(T(v))$ and the simple subquotient containing $T(v+(1,1,2))$ has the same basis as $M_{3}/(L_{5}\oplus L_{6}\oplus L_{7})$ so, we have $M_{3}/(L_{5}\oplus L_{6}\oplus L_{7})\simeq L_{8}$.

∎

• $(G7)$

Consider the tableau:

a$$b$$c

$T(v)$=    a$$b

$z$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $4$ simple subquotients.

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two modules with unbounded finite weight multiplicities.

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$\leftarrow L_{4}$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$L_{1}\rightarrow

The origin of the picture above corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v+\delta^{11}+\delta^{21})$.

• II.

**Loewy series.

**

[TABLE]

• $(G8)$

Set $t\in\mathbb{Z}_{>0}$ and consider the following Gelfand-Tsetlin tableau:

a$$a-t$$c

$T(v)$=    a$$y

$z$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $3$ simple subquotients.

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

A cuspidal module with $t$-dimensional weight spaces:

[TABLE]

• II.

**Loewy series.

**

[TABLE]

• $(G9)$

For each $t\in\mathbb{Z}_{>0}$, let consider the following generic tableau:

a$$a-t$$c

$T(v)$=    a$$y

$a$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $6$ simple subquotients. The bases and corresponding weight lattices are given by:

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two modules with unbounded finite weight multiplicities.

[TABLE]

1$$1$$2$$1$$2$$1$$3$$2$$1$$3$$2$$1$$4$$3$$2$$1$$4$$2$$3$$1$$5$$2$$3$$4$$1$$5$$3$$4$$2$$1$$1$$2$$3$$1$$2$$3$$1$$2$$1$$2$$3$$4$$1$$2$$1$$2$$3$$4$$1$$1$$2$$3$$4$$5$$1$$1$$2$$3$$4$$5$$\leftarrow L_{1}$$L_{6}\rightarrow

• (iii)

Two modules with weight multiplicities bounded by $t$.

[TABLE]

2$$1$$2$$2$$2$$1$$2$$1$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$\leftarrow L_{3}$$L_{4}\rightarrow

The pictures above correspond to the case $t=2$, and the origin is the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v-2\delta^{21})$.

• II.

**Loewy series.

**

[TABLE]

• $(G10)$

For each $t\in\mathbb{Z}_{>0}$, let $T(v)$ be the following Gelfand-Tsetlin tableau:

a$$a-t$$c

a$$y

$y$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $6$ simple subquotients. The bases and corresponding weight lattices are given by:

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two modules with unbounded finite weight multiplicities.

[TABLE]

3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$2$$1$$4$$2$$3$$1$$1$$5$$2$$3$$4$$1$$1$$1$$2$$3$$4$$5$$3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$2$$1$$4$$3$$2$$1$$1$$5$$4$$3$$2$$1$$1$$5$$4$$3$$2$$1$$\leftarrow L_{4}$$L_{3}\rightarrow

• (iii)

Two modules with finite weight multiplicities bounded by $t$.

[TABLE]

2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$1$$2$$2$$2$$2$$2$$2$$2$$1$$2$$1$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$2$$1$$2$$1$$2$$2$$2$$2$$2$$2$$1$$\leftarrow L_{2}$$L_{5}\rightarrow

The pictures above correspond to the case $t=2$, and the origin is the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v-\delta^{22})$.

• II.

**Loewy series.

**

[TABLE]

• $(G11)$

For any $t\in\mathbb{Z}_{>0}$, let $T(v)$ be the tableau:

a$$a-t$$c

a$$c

$a$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $12$ simple subquotients. We provide pictures of the weight lattice corresponding to the case $t=2$, and with origin at the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v)$.

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Six modules with unbounded finite weight multiplicities.

[TABLE]

1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$4$$\leftarrow L_{1}$$L_{2}\rightarrow

4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$1$$2$$1$$5$$4$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$5$$4$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$1$$\leftarrow L_{3}$$L_{10}\rightarrow

1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$4$$3$$2$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$1$$2$$3$$4$$4$$4$$\leftarrow L_{11}$$L_{12}\rightarrow

• (iii)

Four modules with finite weight multiplicities bounded by $t$.

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$1$$2$$2$$1$$2$$1$$1$$1$$1$$2$$2$$2$$1$$2$$2$$1$$2$$2$$1$$2$$2$$2$$1$$1$$2$$\leftarrow L_{6}$$L_{7}\rightarrow

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$L_{9}\rightarrow$$\leftarrow L_{4}

• II.

**Loewy series.

**

[TABLE]

• $(G12)$

Consider $t\in\mathbb{Z}_{>0}$, and $T(v)$ to be the Gelfand-Tsetlin tableau:

a$$b$$b-t

$T(v)$=    a$$b

$a$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $12$ simple subquotients. We provide pictures of the weight lattice corresponding to the case $t=2$, and with origin at the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v+\delta^{11})$.

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Six modules with unbounded weight multiplicities.

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$\leftarrow L_{1}$$L_{2}\rightarrow

4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$5$$4$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$5$$4$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$\leftarrow L_{5}$$L_{6}\rightarrow

1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$1$$2$$3$$4$$4$$4$$\leftarrow L_{11}$$L_{12}\rightarrow

• (iii)

Four modules with finite weight multiplicities bounded by $t$.

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$1$$L_{10}\rightarrow$$\leftarrow L_{8}

1$$1$$1$$1$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$1$$2$$2$$1$$2$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$2$$1$$2$$2$$2$$2$$2$$2$$L_{7}\rightarrow$$\leftarrow L_{3}

• II.

**Loewy series.

**

[TABLE]

• $(G13)$

Consider $t\in\mathbb{Z}_{>0}$. Set $T(v)$ to be the following Gelfand-Tsetlin tableau:

a$$a-t$$c

a$$c

$z$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $6$ simple subquotients. We provide pictures of the weight lattice corresponding to the case $t=2$, and with origin at the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v+\delta^{11})$.

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two modules with unbounded finite weight multiplicities:

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$\leftarrow L_{6}$$1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$L_{1}\rightarrow

• (iii)

Two modules with finite weight multiplicities bounded by $t$.

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$\leftarrow L_{5}$$1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$2$$L_{2}\rightarrow

• II.

**Loewy series.

**

[TABLE]

• $(G14)$

Set $t,s\in\mathbb{Z}_{>0}$ with $t<s$ and let $T(v)$ be the following Gelfand-Tsetlin tableau:

a$$a-t$$a-s

$T(v)$=    a$$y

$a$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $8$ simple subquotients. We provide pictures of the weight lattice corresponding to the case $t=1$, $s=2$, and with origin at the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v)$.

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two modules with unbounded finite weight multiplicities:

[TABLE]

1$$1$$2$$1$$2$$1$$3$$2$$1$$3$$2$$1$$4$$3$$2$$1$$4$$2$$3$$1$$5$$2$$3$$4$$1$$5$$3$$4$$2$$1$$1$$2$$3$$1$$2$$3$$1$$2$$1$$2$$3$$4$$1$$2$$1$$2$$3$$4$$1$$1$$2$$3$$4$$5$$1$$1$$2$$3$$4$$5$$\leftarrow L_{1}$$L_{8}\rightarrow

• (iii)

Four modules with finite weight multiplicities. The modules $L_{4}$ and $L_{7}$ have multiplicities bounded by $t$, and $L_{2}$, $L_{5}$ have multiplicities bounded by $s-t$.

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$\leftarrow L_{4}$$L_{7}\rightarrow

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$\leftarrow L_{2}$$L_{5}\rightarrow

• II.

**Loewy series.

**

[TABLE]

• $(G15)$

Set $t,s\in\mathbb{Z}_{>0}$ with $t<s$ and let $T(v)$ be the following tableau:

a$$a-t$$a-s

$T(v)$=    a$$y

$y$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $8$ simple subquotients. We provide pictures of the weight lattice corresponding to the case $t=1$, $s=2$, and with origin at the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v-\delta^{22})$.

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two modules with unbounded finite weight multiplicities:

[TABLE]

3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$2$$1$$4$$2$$3$$1$$1$$5$$2$$3$$4$$1$$1$$1$$2$$3$$4$$5$$3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$2$$1$$4$$3$$2$$1$$1$$5$$4$$3$$2$$1$$1$$5$$4$$3$$2$$1$$\leftarrow L_{6}$$L_{3}\rightarrow

• (iii)

Four modules with bounded weight multiplicities. The modules $L_{4}$ and $L_{7}$ have multiplicities bounded by $t$, and $L_{2}$, $L_{5}$ have multiplicities bounded by $s-t$.

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$\leftarrow L_{4}$$L_{7}\rightarrow

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$\leftarrow L_{2}$$L_{5}\rightarrow

• II.

**Loewy series.

**

[TABLE]

• $(G16)$

For any $t,s\in\mathbb{Z}_{>0}$ with $t<s$ let $T(v)$ be the following tableau:

a$$a-t$$a-s

$T(v)$=    a$$y

$z$

• I.

simple subquotients.

In this case the module $V(T(v))$ has $4$ simple subquotients. The bases and corresponding weight multiplicities are given by:

• (i)

Two modules with infinite-dimensional weight multiplicities:

[TABLE]

• (ii)

Two cuspidal modules; $L_{2}$ with weight multiplicities $s-t$ and $L_{3}$ with weight multiplicities $t$:

[TABLE]

• II.

**Loewy series.

**

[TABLE]

7.3. Structure of singular $\mathfrak{sl}(3)$-modules $V(T(\bar{v}))$

In this subsection we describe all simple singular Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules. Like in the generic case it is enough to explicitly present all simple subquotients of $V(T(\bar{v}))$ for every $1$-singular vector $\bar{v}$.

7.3.1. Singular Gelfand-Tsetlin formulas

Recall the construction of the $1$-singular $\mathfrak{sl}(3)$-modules $V(T(\bar{v}))$ in §4.4. We adapt this construction to $\operatorname{\mathfrak{g}}=\mathfrak{sl}(3)$. Since the singularity appears in row $2$, we fix $\bar{v}\in T_{3}(\mathbb{C})$ such that $\bar{v}_{21}=\bar{v}_{22}$. Also, we denote by $\tau$ the permutation in $S_{3}\times S_{2}\times S_{1}$ that interchanges the $(2,1)$th and $(2,2)$th entries and is identity on row $2$.

Recall that $V(T(\bar{v}))$, as a vector space, is generated by the set of tableaux $\{T(\bar{v}+z),\mathcal{D}T(\bar{v}+z^{\prime})\ |\ z,z^{\prime}\in\mathbb{Z}^{3}\}$ satisfying the relations $T(\bar{v}+z)-T(\bar{v}+\tau(z))=0$ and ${\mathcal{D}}T(\bar{v}+z)+{\mathcal{D}}T(\bar{v}+\tau(z))=0$. As explained in Remark 4.34,

[TABLE]

is a basis of $V(T(\bar{v}))$.

Definition 7.6**.**

Given $w\in\mathbb{Z}^{3}$, the tableau associated to $w$ with respect to $\mathcal{B}(T(\bar{v}))$ is defined by

[TABLE]

In particular, $\mathcal{B}(T(\bar{v}))=\{\operatorname{Tab}(w)\ |\ w\in\mathbb{Z}^{3}\}$.

We next write explicitly the formulas for the action of $\mathfrak{sl}(3)$ on $V(T(\bar{v}))$. We again use the conventions $v=(v_{31},v_{32},v_{33},v_{21},v_{22},v_{11})$, $\bar{v}=(a,b,c,x,x,z)$, and $w=(0,0,0,m,n,k)$. For any rational function $f$ in $\{v_{ij}\}_{1\leq j\leq i\leq 3}$, $\partial_{v_{ij}}^{\bar{v}}(f)$ will stand for $\frac{\partial f}{\partial v_{ij}}(\bar{v})$.

Recall that by Proposition 4.22, the Gelfand-Tsetlin formulas for generic modules can be written as follows:

[TABLE]

[TABLE]

where $h_{1}(w)=2w_{11}-(w_{21}+w_{22}+1)$, $h_{2}(w)=2(w_{21}+w_{22}+1)-w_{11}$, and $e_{rs}(w)$, $\varepsilon_{rs}$ are defined in Example 4.23.

Using the relations $T(\bar{v}+z)-T(\bar{v}+\tau(z))=0$ and $\mathcal{D}T(\bar{v}+z)+\mathcal{D}T(\bar{v}+\tau(z))=0$ (see Section 4.4) we can write the above formulas in a simpler form. Set for convenience $\bar{w}=\bar{v}+w$. Then:

[TABLE]

If $E=E_{ij}\in\{E_{32},E_{23},E_{31},E_{13}\}$, $\varepsilon$ denotes the corresponding $\varepsilon_{ij}$, and $\tilde{e}(w):=(v_{21}-v_{22})e(w)$, then the action of $E$ on $\operatorname{Tab}(w)$ is given by:

[TABLE]

Remark 7.7**.**

Note that the Gelfand-Tsetlin formulas for singular tableaux have the same coefficients as in the classical formulas for tableaux of the same type (see formulas (20), (21), (22), (23), (24), (25)). More precisely, the action of the generators $E_{ij}$ on a regular tableau is a linear combination of regular tableaux with the same coefficients as those that appear in the classical Gelfand-Tsetlin formulas. On the other hand, the corresponding action on a derivative tableau is a linear combination of both regular and derivative tableaux, and the coefficients of the derivative tableaux are the same as those that appear in the classical formulas.

7.3.2. Explicit formulas and computations

Some explicit computations are included in the following example.

Example 7.8**.**

For any complex numbers $a$ and $v_{ij}$, $1\leq j\leq i\leq 3$, consider the following Gelfand-Tsetlin tableaux.

v_{31}$$v_{32}$$v_{33}*         *a$$a$$a

$T(v)$*=    v_{21}$$v_{22}        $T(\bar{v})$=    *a$$a

$v_{11}$*                                   *$a$

Consider $w=(0,1,0)=\delta^{22}$ and $w^{\prime}=(1,0,0)=\delta^{21}$ in $\mathbb{Z}^{3}$. Then the following hold.

• (i)

Using (24), $E_{32}(T(\bar{v}+w))=\mathcal{D}^{\bar{v}}((v_{21}-v_{22})E_{32}(T(v+w)))$ which equals to:

[TABLE]

• (ii)

Using (25), $E_{32}(\mathcal{D}T(\bar{v}+w^{\prime}))=\mathcal{D}^{\bar{v}}(E_{32}(T(v+w^{\prime})))$ which equals to:

[TABLE]

The computations made in the last example can easily be applied to the formulas (20), (21), (22), (23), (24), (25). As a result we have the following set of formulas.

*Action of the generators on regular tableaux:

\begin{array}[]{lll}E_{21}(T(\bar{w}))&=&T(\bar{w}-\delta^{11}).\end{array}*

\begin{array}[]{lll}E_{12}(T(\bar{w}))&=&-(x+m-z-k)(x+n-z-k)T(\bar{w}+\delta^{11}).\end{array}

\begin{array}[]{lll}E_{32}(T(\bar{w}))&=&\small{\begin{dcases}T(\bar{w}-\delta^{21})+2(x+m-z-k)\mathcal{D}T(\bar{w}-\delta^{21}),&\text{if m=n}\\ \frac{x+m-z-k}{m-n}T(\bar{w}-\delta^{21})-\frac{x+n-z-k}{m-n}T(\bar{w}-\delta^{22}),&\text{if m\neq n}.\end{dcases}}\end{array}

\begin{array}[]{lll}E_{23}(T(\bar{w}))&=&\begin{dcases}\partial_{v_{21}}^{\bar{v}}\left(\prod_{i=1}^{3}(v_{3i}-v_{21}-m)\right)T(\bar{w}+\delta^{22})&\\ \small{-2(a-x-m)(b-x-m)(c-x-m)}\mathcal{D}T(\bar{w}+\delta^{22}),&\text{if m=n}\\ &\\ \frac{(a-x-m)(b-x-m)(c-x-m)}{m-n}T(\bar{w}+\delta^{21})-&\\ \frac{(a-x-n)(b-x-n)(c-x-n)}{m-n}T(\bar{w}+\delta^{22}),&\text{if m\neq n}.\end{dcases}\end{array}

Action of the generators on derivative tableaux (recall that we assume $m>n$):

\begin{array}[]{lll}E_{21}(\mathcal{D}T(\bar{w}))=&\mathcal{D}T(\bar{w}-\delta^{11}).\end{array}

\begin{array}[]{lll}E_{12}(\mathcal{D}T(\bar{w}))=&-(x+m-z-k)(x+n-z-k)\mathcal{D}T(\bar{w}+\delta^{11})+\\ &\frac{m-n}{2}T(\bar{w}+\delta^{11}).\end{array}

\begin{array}[]{lll}E_{32}(\mathcal{D}T(\bar{w}))=&\displaystyle\frac{x+m-z-k}{m-n}\mathcal{D}T(\bar{w}-\delta^{21})-\frac{x+n-z-k}{m-n}\mathcal{D}T(\bar{w}-\delta^{22})+\\ &\frac{1}{2}\left(\frac{1}{m-n}-\frac{2(x+m-z-k)}{(m-n)^{2}}\right)T(\bar{w}-\delta^{21})-\\ &\frac{1}{2}\left(\frac{1}{m-n}-\frac{2(x+n-z-k)}{(n-m)^{2}}\right)T(\bar{w}-\delta^{22}).\end{array}

\begin{array}[]{lll}E_{23}(\mathcal{D}T(\bar{w}))=&-\displaystyle\frac{(a-x-m)(b-x-m)(c-x-m)}{n-m}\mathcal{D}T(\bar{w}+\delta^{21})-\\ &\displaystyle\frac{(a-x-n)(b-x-n)(c-x-n)}{m-n}\mathcal{D}T(\bar{w}+\delta^{22})-\\ &\partial_{v_{22}}^{\bar{v}}\left(\displaystyle\frac{\prod\limits_{i=1}^{3}(v_{3i}-v_{22}-n)}{m-n}\right)T(\bar{w}+\delta^{22})-\\ &2\displaystyle\frac{(a-x-n)(b-x-n)(c-x-n)}{(m-n)^{2}}T(\bar{w}+\delta^{22})+\\ &-\partial_{v_{21}}^{\bar{v}}\left(\displaystyle\frac{\prod\limits_{i=1}^{3}(v_{3i}-v_{21}-m)}{n-m}\right)T(\bar{w}+\delta^{21})-\\ &2\displaystyle\frac{(a-x-m)(b-x-m)(c-x-m)}{(m-n)^{2}}T(\bar{w}+\delta^{21}).\end{array}

Lemma 7.9**.**

The action of $\Gamma$ on $V(T(\bar{v}))$ is given by the formulas:

[TABLE]

Proof.

The identities follow from Theorem 4.35. Indeed,

[TABLE]

∎

7.3.3. Submodules generated by singular tableaux

In this subsection we obtain an analogous to Theorem 4.25(i) for $1$-singular tableaux.

Recall that $\mathcal{B}(T(\bar{v}))=\{\operatorname{Tab}(z)\ |\ z\in\mathbb{Z}^{3}\}$ is a basis of $V(T(\bar{v}))$.

Definition 7.10**.**

Let $\bar{v}$ be a fixed critical vector, $\operatorname{Tab}(w)\in\mathcal{B}(T(\bar{v}))$, and $\bar{w}=\bar{v}+w$. Define:

[TABLE]

Lemma 7.11**.**

Assume that $w_{21}\neq w_{22}$. Then $T(\bar{v}+w)$ belongs to $U\cdot\mathcal{D}T(\bar{v}+w)$.

Proof.

The action of $c_{22}-\gamma_{22}(\bar{v}+w)$ on $\mathcal{D}T(\bar{v}+w)$ is given by the formula (27) and can be easily check that is a nonzero multiple of $(w_{21}-w_{22})T(\bar{v}+w)$. ∎

Lemma 7.12**.**

Suppose $\operatorname{Tab}(w)$ is a critical tableau. If $\operatorname{Tab}(w^{\prime})$ is a derivative tableau such that $\Lambda^{+}(\operatorname{Tab}(w))\subseteq\Lambda^{+}(\operatorname{Tab}(w^{\prime}))$, then $\operatorname{Tab}(w^{\prime})\in U\cdot\operatorname{Tab}(w)$. In particular, the simple subquotient $M$ of $V(T(\bar{v}))$ containing $\operatorname{Tab}(w)$ satisfies $\dim(M_{\chi_{w}})=2$.

Proof.

The statement follows from Remark 7.7 and formulas (21, 22, 23). In fact, the numerators of the coefficients of the derivative tableaux appearing in the decomposition of $gT(\bar{v}+w)$ as linear combination of basis elements, are either zero (if $g$ is a product of generators of the form $E_{21}$, $E_{12}$ or $E_{ii}$) or the same as the numerators of the coefficients that appear in the classical Gelfand-Tsetlin formulas. In the latter case $\operatorname{Tab}(w^{\prime})$ is a derivative tableau, hence we can not have zero coefficients. Therefore, we can use the same arguments as in the proof of Lemma 4.25(i). ∎

Definition 7.13**.**

For any tableau $\operatorname{Tab}(w)\in\mathcal{B}(T(\bar{v}))$ define

[TABLE]

By $\mathcal{C}(w)$ we will denote the set of all critical tableaux in $\mathcal{A}(\operatorname{Tab}(w))$ and by $\mathcal{R}(w)$ we will denote the set of all regular tableaux in $\mathcal{A}(\operatorname{Tab}(w))$. Also, set:

[TABLE]

Lemma 7.14**.**

For any tableau $\operatorname{Tab}(w)$ we have $\mathcal{N}(\operatorname{Tab}(w))\subseteq U\cdot\operatorname{Tab}(w)$.

Proof.

The statement follows from Remark 7.7, Lemma 7.11, and Lemma 7.12. More precisely, as the Gelfand-Tsetlin formulas for singular tableaux have the same coefficients as in the classical formulas for tableaux of the same type, we can use the reasoning in the proof of Theorem 6.8 in [FGR15] and adapt it to the singular case. ∎

The following lemma together with Lemma 7.11 gives a sufficient condition in order to have modules with Gelfand-Tsetlin multiplicity $2$.

Lemma 7.15**.**

Suppose that $\operatorname{Tab}(w)$ is a regular tableau such that $\Omega^{+}(\operatorname{Tab}(w))=\Omega^{+}(\operatorname{Tab}(w^{\prime}))$ for some critical tableau $\operatorname{Tab}(w^{\prime})$, then $\operatorname{Tab}(\tau(w))\in U\cdot\operatorname{Tab}(w)$.

Proof.

The statement follows directly from Lemma 7.12. ∎

Corollary 7.16**.**

Let $\operatorname{Tab}(w)$ be a regular tableau associated to a Gelfand-Tsetlin character $\chi$. If $\{Tab(w^{\prime})\ |\ \Omega^{+}(\operatorname{Tab}(w^{\prime}))=\Omega^{+}(\operatorname{Tab}(w))\}$ does not contain critical tableaux, then any simple subquotient $N$ of $V(T(\bar{v}))$ satisfies $\dim(N_{\chi})\leq 1$.

Proof.

Since $\operatorname{Tab}(w)$ is regular, $w_{21}<w_{22}$. Then $\operatorname{Tab}(\tau(w))$ is a derivative tableau such that $\operatorname{Tab}(w)\in U\cdot\operatorname{Tab}(\tau(w))$ (see Lemma 7.11). Therefore, it is enough to prove that $\operatorname{Tab}(\tau(w))\notin U\cdot\operatorname{Tab}(w)$. However, this follows from Theorem 4.35 and the fact that we can not obtain critical tableaux from $\operatorname{Tab}(w)$ with the same $\Omega^{+}(\operatorname{Tab}(w))$, in particular we can not obtain derivative tableaux $\operatorname{Tab}(w^{\prime})$ such that $\Omega^{+}(\operatorname{Tab}(w^{\prime}))=\Omega^{+}(\operatorname{Tab}(w))$. ∎

Remark 7.17**.**

By definition of $\mathcal{N}(\operatorname{Tab}(w))$, any $\operatorname{Tab}(w^{\prime})$ in $\mathcal{N}(\operatorname{Tab}(w))$ satisfies the relation $|\Omega^{+}(\operatorname{Tab}(w))|\leq|\Omega^{+}(\operatorname{Tab}(w^{\prime}))|$. However it is possible to have $\operatorname{Tab}(w^{\prime})\in U\cdot\operatorname{Tab}(w)$ with $|\Omega^{+}(\operatorname{Tab}(w^{\prime}))|=|\Omega^{+}(\operatorname{Tab}(w))|-1$. For instance, consider $\bar{v}=(a,b,c,x,x,x)$ such that $\{a-x,b-x,c-x\}\cap\mathbb{Z}=\emptyset$ and $w=(0,0,0)$, then $|\Omega^{+}(\operatorname{Tab}(w))|=2$ while $E_{32}\operatorname{Tab}(w)=\operatorname{Tab}(w-\delta^{21})$ and $|\Omega^{+}(\operatorname{Tab}(w-\delta^{21}))|=1$.

Let us write $\operatorname{Tab}(w^{\prime})\prec_{g}\operatorname{Tab}(w)$ if $\operatorname{Tab}(w^{\prime})$ appears with no zero coefficient in the decomposition of $g\cdot\operatorname{Tab}(w)$ for some generator $g\in\mathfrak{gl}(n)$.

Lemma 7.18** ([GoR18] Lemma 7.4).**

Suppose that $\operatorname{Tab}(w^{\prime})\prec_{g}\operatorname{Tab}(w)$ with $g\in\mathfrak{gl}(n)$ of the form $E_{k,k+1}$ or $E_{k+1,k}$, then $|\Omega^{+}(\operatorname{Tab}(w^{\prime}))|\geq|\Omega^{+}(\operatorname{Tab}(w))|-1$. Moreover, the complete list of Gelfand-Tsetlin tableaux $\operatorname{Tab}(w)$ and $\operatorname{Tab}(w^{\prime})$ such that $\operatorname{Tab}(w^{\prime})\prec_{g}\operatorname{Tab}(w)$ and $|\Omega^{+}(\operatorname{Tab}(w^{\prime}))|=|\Omega^{+}(\operatorname{Tab}(w))|-1$ is as follows.

• (i)

a$$b$$c*         *a$$b$$c

$\operatorname{Tab}(w)$*= x$$x-t      $\operatorname{Tab}(w^{\prime})$= *x-t$$x

$x$*                                  *$x+1$

for $t\in\mathbb{Z}_{>0}$.

• (ii)

a$$b$$c*         *a$$b$$c

$\operatorname{Tab}(w)$*= x$$x-t      $\operatorname{Tab}(w^{\prime})$= *x-t$$x-1

$x$*                                  *$x$

for $t\in\mathbb{Z}_{>0}$.

• (iii)

a$$b$$c*         *a$$b$$c

$\operatorname{Tab}(w)$*= x$$x      $\operatorname{Tab}(w^{\prime})$= *x-1$$x

$x$*                                  $x$.*

• (iv)

x$$b$$c*         *x$$b$$c

$\operatorname{Tab}(w)$*= x$$x-t      $\operatorname{Tab}(w^{\prime})$= *x-t$$x+1

$z$*                                  *$z$

for $t\in\mathbb{Z}_{>0}$, $b\neq x$, and $c\neq x$.

• (v)

x$$b$$c*         *x$$b$$c

$\operatorname{Tab}(w)$*= x$$x      $\operatorname{Tab}(w^{\prime})$= *x$$x+1

$z$*                                  *$z$

for $b\neq x$ and $c\neq x$.

Remark 7.19**.**

For the tableaux in Lemma 7.18(iv)(v), one may consider $x$ at positions $3i$, $i=1,2,3$, obtaining the same property for $\Omega^{+}$.

Definition 7.20**.**

We will say that a tableau $\operatorname{Tab}(w)$ is of type (I) if it can be written in the form of one of the tableaux $\operatorname{Tab}(w)$ of parts (i), (ii) or (iii) of Lemma 7.18 for some $x,a,b,c,t,z$. We also say that the tableau is of type (II)i if can be written in the form of one of the tableaux $\operatorname{Tab}(w)$ of parts (iv) or (v) of Lemma 7.18 for some $x,b,c,t,z$ and $x$ appear in the top row in position $3i$.

Remark 7.21**.**

With the notation of Lemma 7.18 for tableaux of type (I) we have $\Omega^{+}(\operatorname{Tab}(w^{\prime}))=\Omega^{+}(\operatorname{Tab}(w))\setminus\{(2,1,1)\}$ and for tableaux of type (II)i we have $\Omega^{+}(\operatorname{Tab}(w^{\prime}))=\Omega^{+}(\operatorname{Tab}(w))\setminus\{(3,i,2)\}$.

Definition 7.22**.**

Let $\operatorname{Tab}(w)\in\mathcal{B}(T(\bar{v}))$ and $(r,s,t)\in\Omega^{+}(\operatorname{Tab}(w))$. Set

[TABLE]

where $\operatorname{Tab}(w^{\prime})$ is any tableau such that $\Omega^{+}(\operatorname{Tab}(w))\setminus\{(r,s,p)\}=\Omega^{+}(\operatorname{Tab}(w^{\prime}))$. Also, define:

[TABLE]

[TABLE]

[TABLE]

Lemma 7.23**.**

Let $\operatorname{Tab}(w)$ be any Gelfand-Tsetlin tableau and $\operatorname{Tab}(w^{\prime})\in\mathcal{N}(\operatorname{Tab}(w))$. We have $\widehat{\mathcal{A}}(\operatorname{Tab}(w^{\prime}))\subseteq U\cdot\operatorname{Tab}(w)$.

Proof.

As $\operatorname{Tab}(w^{\prime})\in\mathcal{N}(\operatorname{Tab}(w))$, by Lemma 7.14, Lemma 7.18, and Remark 7.21 we have $\mathcal{N}^{(i)}(\operatorname{Tab}(w^{\prime}))\subseteq U\cdot\operatorname{Tab}(w)$ for $i=1,2$. ∎

The following theorem summarize the results of this section.

Theorem 7.24**.**

Let $\operatorname{Tab}(w)\in\mathcal{B}(T(\bar{v}))$. The submodule $U\cdot\operatorname{Tab}(w)$ has the following basis of tableaux:

[TABLE]

Proof.

The statement follows from Lemmas 7.18 and 7.23. ∎

Definition 7.25**.**

Let $M$ be a Gelfand-Tsetlin module with basis $\mathcal{B}_{M}\subseteq\mathcal{B}(T(\bar{v}))$ for some $1$-singular vector $\bar{v}$. We say that $\operatorname{Tab}(w)\in\mathcal{B}_{M}$ is $\Omega^{+}$-maximal in $M$ if $|\Omega^{+}(\operatorname{Tab}(w))|$ is maximal for all $\operatorname{Tab}(w)$ in $\mathcal{B}_{M}$. Also, denote by $U\cdot_{M}\operatorname{Tab}(w)$ the submodule of $M$ generated by $\operatorname{Tab}(w)$.

The next two corollaries follow from Theorem 7.24 and will be useful when describing the simple subquotients of $V(T(\bar{v}))$.

Corollary 7.26**.**

Let $M$ be a Gelfand-Tsetlin module with basis $\mathcal{B}_{M}\subseteq\mathcal{B}(T(\bar{v}))$ for some $1$-singular vector $\bar{v}$. If $\operatorname{Tab}(w)\in\mathcal{B}_{M}$ is a regular tableau that is $\Omega^{+}$-maximal in $M$, then $U\cdot_{M}\operatorname{Tab}(w)$ is a simple submodule of $M$.

Proof.

It is enough to proof that $\operatorname{Tab}(w)$ belongs to $U\cdot_{M}\operatorname{Tab}(w^{\prime})$ for any $\operatorname{Tab}(w^{\prime})$ in $U\cdot_{M}\operatorname{Tab}(w)$. As $\operatorname{Tab}(w^{\prime})$ in $U\cdot_{M}\operatorname{Tab}(w)$ and $\operatorname{Tab}(w)$ is a regular tableau, we have $\Omega^{+}(\operatorname{Tab}(w))\subseteq\Omega^{+}(\operatorname{Tab}(w^{\prime}))\cup\{(r,s,t)\}$ for some $(r,s,t)$. As $\operatorname{Tab}(w)$ is $\Omega^{+}$-maximal, we should have $\Omega^{+}(\operatorname{Tab}(w))=\Omega^{+}(\operatorname{Tab}(w^{\prime}))\cup\{(r,s,t)\}$ for some $(r,s,t)$. Therefore, $U\cdot_{M}\operatorname{Tab}(w)\subseteq U\cdot_{M}\operatorname{Tab}(w^{\prime})$ and, then we have $\operatorname{Tab}(w)\in U\cdot_{M}\operatorname{Tab}(w^{\prime})$. ∎

Corollary 7.27**.**

Let $M$ be a Gelfand-Tsetlin module with basis $\mathcal{B}_{M}\subseteq\mathcal{B}(T(\bar{v}))$ for some $1$-singular vector $\bar{v}$. If $\{\operatorname{Tab}(w)\in\mathcal{B}_{M}\ |\ \operatorname{Tab}(w)\text{ is }\Omega^{+}\text{-maximal}\}$ does not contain regular tableaux, then for any $\Omega^{+}$-maximal tableau $\operatorname{Tab}(w)$ the submodule $U\cdot_{M}\operatorname{Tab}(w)$ is a simple submodule of $M$.

Proof.

The proof is analogous to the proof of Corollary 7.26. ∎

In order to describe the basis of the simple subquotients of $V(T(\bar{v}))$ we modify Definition 7.3 to singular vectors.

Definition 7.28**.**

Let $\bar{v}$ be any $1$-singular vector and $\mathcal{B}$ be a subset of ${\mathbb{Z}}^{3}$. By $\operatorname{Tab}(\mathcal{B})$ we will denote the set of tableaux $\{\operatorname{Tab}(m,n,k)\;|\;(m,n,k)\in\mathcal{B}\}\subseteq\mathcal{B}(T(\bar{v}))$. Assume that $M$ is a Gelfand-Tsetlin module with basis $\operatorname{Tab}(\mathcal{B})$. Then we will denote $M$ by $M(\mathcal{B},\bar{v})$, or simply by $M(\mathcal{B})$ if $\bar{v}$ is fixed. If $M(\mathcal{B})$ is simple, we will write $L(\mathcal{B})$ for $M(\mathcal{B})$.

Example 7.29**.**

Let $\bar{v}=(a,b,c|a,a|z)$. Below we give a basis for the submodule of $V(T(\bar{v}))$ generated by $\operatorname{Tab}(0,0,0)$. In this case $\mathcal{B}(T(\bar{v}))$ does not contain tableaux of type $(I)$, $(II)_{2}$ or $(II)_{3}$. However, the set of all tableaux of type $(II)_{1}$ is $\{\operatorname{Tab}(0,n,k)\ |\ n\in\mathbb{Z}_{\leq 0}\}$. By definition we have

[TABLE]

[TABLE]

Therefore, by Theorem 7.24, the submodule of $V(T(\bar{v}))$ generated by $\operatorname{Tab}(0,0,0)$ has basis:

[TABLE]

7.4. The singular block containing $L(-\rho)$

In this subsection, we describe all simple subquotients of the module $V(T(\bar{v}))$.

Next we give an algorithm, which based on Theorem 7.24 and Corollaries 7.26 and 7.27, provides an explicit basis of all simple subquotients of a module $M$ with basis $\mathcal{B}_{M}\subseteq\mathcal{B}(T(\bar{v}))$.

1. Step $1$.

If there is an $\Omega^{+}$-maximal regular tableau in $\mathcal{B}_{M}$, choose any such tableau $\operatorname{Tab}(w)$. By Corollary 7.26, $U\cdot_{M}\operatorname{Tab}(w)$ is a simple submodule of $M$. 2. Step $2$.

If there are no $\Omega^{+}$-maximal regular tableaux in $\mathcal{B}_{M}$, consider any $\Omega^{+}$-maximal (derivative) tableau $\operatorname{Tab}(w)$. By Corollary 7.27 the module $U\cdot_{M}\operatorname{Tab}(w)$ will be a simple submodule of $M$. 3. Step $3$.

Using the bases of $M$ and $U\cdot\operatorname{Tab}(w)$ (see Theorem 7.24), we find a basis of $M/(U\cdot_{M}\operatorname{Tab}(w))$. 4. Step $4$.

Start over the procedure with the module $M^{\prime}:=M/(U\cdot_{M}\operatorname{Tab}(w))$.

Example 7.30**.**

Let $\bar{v}=(a,a,a|a,a|a)$. Below we define explicit bases of all simple subquotient of $V(T(\bar{v}))$.

Note that none of the tableaux in $\mathcal{B}(T(\bar{v}))$ can be of type $(II)_{i}$, $i=1,2,3$. Therefore $\mathcal{N}^{(2)}(\operatorname{Tab}(w))=\mathcal{N}(\operatorname{Tab}(w))$ for any $\operatorname{Tab}(w)\in\mathcal{B}(T(\bar{v}))$. Moreover, the set of all tableaux of type $(I)$ is $\{\operatorname{Tab}(m,n,k)\ |\ n\leq m=k\}$. Now we apply Steps 1–4 described above to the module $V(T(\bar{V}))$.

• (1)

The tableau $\operatorname{Tab}(0,0,0)$ is $\Omega^{+}$-maximal on $V(T(\bar{v}))$. By Corollary 7.26, $U\cdot\operatorname{Tab}(0,0,0)$ is a simple submodule of $V(T(\bar{v}))$ and by Theorem 7.24, the submodule $U\cdot\operatorname{Tab}(0,0,0)$ has a basis:

[TABLE]

Denote this module by $L_{1}$, and $M_{1}=V(T(\bar{v}))/L_{1}$.

• (2)

Now, the derivative tableau $\operatorname{Tab}(0,-2,-1)$ is $\Omega^{+}$-maximal in $M_{1}$. By Theorem 7.24 , $U\cdot\operatorname{Tab}(0,-2,-1)$ has a basis:

[TABLE]

Moreover, by Corollary 7.27, $U\cdot_{M_{1}}\operatorname{Tab}(0,-2,-1)$ is a simple submodule of $M_{1}$ and has a basis

[TABLE]

Denote by $L_{3}$ this module and $M_{2}=M_{1}/L_{3}$.

• (3)

The tableau $\operatorname{Tab}(0,1,0)$ is $\Omega^{+}$-maximal in $M_{2}$ and $U\cdot\operatorname{Tab}(0,1,0)$ has a basis $\widehat{\mathcal{A}}(\operatorname{Tab}(0,1,0))\cup\widehat{\mathcal{A}}(\operatorname{Tab}(0,0,0))$ which is equal to

[TABLE]

Therefore, $U\cdot_{M_{2}}\operatorname{Tab}(0,1,0)$ has basis

[TABLE]

call this module $L_{2}$ and $M_{3}=M_{2}/L_{2}$.

• (4)

There are not $\Omega^{+}$-maximal regular tableaux in $M_{3}$, so we choose the derivative tableau $\operatorname{Tab}(1,0,0)$ which is $\Omega^{+}$-maximal in $M_{3}$. By Corollary 7.27 the module $U\cdot_{M_{3}}\operatorname{Tab}(1,0,0)$ is a simple submodule of $M_{3}$ with basis

[TABLE]

call this module $L_{5}$ and $M_{4}=M_{3}/L_{5}$.

• (5)

The tableau $\operatorname{Tab}(0,1,1)$ is $\Omega^{+}$-maximal in $M_{4}$ and $U\cdot_{M_{4}}\operatorname{Tab}(0,1,1)$ is a simple submodule of $M_{4}$ with basis

[TABLE]

call this module $L_{4}$ and $M_{5}=M_{4}/L_{4}$.

• (6)

The derivative tableau $\operatorname{Tab}(1,0,1)$ is $\Omega^{+}$-maximal in $M_{5}$. Therefore, by Corollary 7.27 $U\cdot_{M_{5}}\operatorname{Tab}(1,0,1)$ a simple submodule of $M_{5}$ with basis

[TABLE]

call this module $L_{7}$ and $M_{6}=M_{5}/L_{7}$.

• (7)

The tableau $\operatorname{Tab}(0,1,2)$ is $\Omega^{+}$-maximal in $M_{6}$ so, $U\cdot_{M_{6}}\operatorname{Tab}(0,1,2)$ is a simple submodule of $M_{6}$ and has a basis

[TABLE]

call this module $L_{6}$ and $M_{7}=M_{6}/L_{6}$.

• (8)

The tableau $\operatorname{Tab}(1,0,2)$ is $\Omega^{+}$-maximal in $M_{7}$ and $U\cdot_{M_{7}}\operatorname{Tab}(1,0,2)$ is a simple submodule of $M_{7}$ with a basis

[TABLE]

Call this module $L_{9}$ and $M_{8}=M_{7}/L_{9}$.

• (9)

The tableau $\operatorname{Tab}(1,1,0)$ is $\Omega^{+}$-maximal in $M_{8}$. The module $U\cdot_{M_{8}}\operatorname{Tab}(1,2,2)$ is a simple submodule of $M_{8}$ with a basis

[TABLE]

call this module $L_{8}$ and $M_{9}=M_{8}/L_{8}$.

• (10)

The tableau $\operatorname{Tab}(1,1,2)$ is $\Omega^{+}$-maximal in $M_{9}$ and $U\cdot\operatorname{Tab}(1,1,2)$ has a basis $\operatorname{Tab}(\mathbb{Z}^{3})$. Therefore, $U\cdot_{M_{9}}\operatorname{Tab}(1,1,2)$ is has a basis

[TABLE]

call this module $L_{10}$.

Remark 7.31**.**

The reasoning in Example 7.30 can be applied also when finding the Loewy series decomposition of $V(T(\bar{v}))$. More precisely, the Loewy series decomposition of $V(T(\bar{v}))$ for $\bar{v}=(a,a,a|a,a|a)$ is:

[TABLE]

7.5. Realizations of all simple singular Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules

In this section we will describe all simple objects in every block $\mathcal{GT}_{T(v)}$ defined by a singular Gelfand-Tsetlin character $\chi_{v}$ (see Definition 4.12 and Remark 4.2). Such description will include an explicit tableaux basis of each simple subquotient $M$ in $\mathcal{GT}_{T(v)}$ and the weight support of $M$. For the weight support we will use Proposition 7.32 and the explicit basis to give a description of the weight multiplicities, when the multiplicities are finite, a picture of the weight lattice is provided. We also present the components of the Loewy series of the universal module $V(T(v))$. A rigorous proof based on Theorem 7.24, and Corollaries 7.26 and 7.27 was given for Case $(C13)$, see §7.4, Example 7.30. For all other cases the reasoning is the same.

The simple subquotients will be defined by their corresponding sets in $\mathbb{Z}^{3}$, equivalently, by their bases in $\mathcal{B}(T(\bar{v}))$. We should note that all subsets of $\mathbb{Z}^{3}$ that define a simple subquotient are defined by a set of inequalities of the form $a\leq b$ or $a<b$ where $a,b$ are elements in the set $\{m,n,k,0,-t,-s\}$.

As we did for the description of generic blocks we will characterize the weight spaces for subquotients of the singular module $V(T(\bar{v}))$.

Proposition 7.32**.**

Let $M$ be a singular Gelfand-Tsetlin module with basis of tableaux $\mathcal{B}_{M}\subseteq\mathcal{B}(T(\bar{v}))$. If $\operatorname{Tab}(z)\in\mathcal{B}_{M}$ is a tableau of weight $\lambda$, then the weight space $M_{\lambda}$ is spanned by the set of tableaux $\{\operatorname{Tab}(z+(i,-i,0))\ |\ i\in\mathbb{Z}\}\cap\mathcal{B}_{M}$.

Proof.

The action of the generators of $\mathfrak{h}$ in $M$ is given by the same expressions as in the case of generic modules, therefore, we can use the same argument of the proof of Proposition 7.2. ∎

We now describe the sets $\mathcal{B}\subseteq\mathbb{Z}^{3}$ that define all simple subquotients of $V(T(\bar{v}))$. For convenience, the modules listed in one row are isomorphic. Recall that $\tau$ denote the transposition $(\operatorname{id},(1,2),\operatorname{id})\in S_{3}\times S_{2}\times S_{1}$. It is worth noting that all isomorphisms between simple subquotients of $V(T(\bar{v}))$ are $\tau$-induced, that is all isomorphisms between simple subquotients are given by $L(\mathcal{B})\mapsto L(\tau(\mathcal{B}))$.

Remark 7.33**.**

In general it is not true that if $\mathcal{B}\subseteq\mathbb{Z}^{3}$ defines a subquotient of $V(T(\bar{v}))$ then $\tau(\mathcal{B})$ defines also a subquotient of $V(T(\bar{v}))$.

Remark 7.34**.**

We should note that for singular $\mathfrak{sl}(3)$-modules we may have characters with unique simple extension or with two non-isomorphic simple extensions. In particular, the number of simple subquotients of the singular blocks in general will not coincide with the number of non-isomorphic modules in the block.

Until the end of this section we use the following convention. The entries of the Gelfand-Tsetlin tableaux we will use will be integer shifts of some of the complex numbers $a,b,c,x,z$. We also assume that if any two of $a,b,c,x,z$ appear in the same row or in consecutive rows of a given tableau, then their difference is not integer. For convenience, in the description of basis of simple subquotients, isomorphic modules are listed in the same row.

• $(C1)$

Consider the Gelfand-Tsetlin tableau:

a$$b$$c

$T(\bar{v})$=    x$$x

$z$

In this case the module $V(T(\bar{v}))$ is simple and all its weight spaces are infinite dimensional.

[TABLE]

• $(C2)$

Consider the following Gelfand-Tsetlin tableau:

a$$b$$c

$T(\bar{v})$=    x$$x

$x$

• I.

Number of simples in the block: $2$

• II.

simple subquotients.

In this case the module $V(T(\bar{v}))$ has $2$ simple subquotients and they have infinite-dimensional weight multiplicities:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C3)$

Consider the following Gelfand-Tsetlin tableau:

a$$b$$c

$T(\bar{v})$=    a$$a

$z$

• I.

Number of simples in the block: $2$

• II.

simple subquotients.

We have $2$ simple subquotients and they have infinite-dimensional weight spaces.

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C4)$

Consider the Gelfand-Tsetlin tableau:

a$$b$$c

$T(\bar{v})$=    a$$a

$a$

• I.

Number of simples in the block: $5$

• II.

simple subquotients.

In this case we have $6$ simple subquotients. The origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}+\delta^{11}+\delta^{21})$.

• (i)

Modules with unbounded finite weight multiplicities:

[TABLE]

2$$1$$2$$1$$1$$3$$2$$1$$3$$2$$1$$4$$3$$2$$1$$4$$3$$2$$1$$5$$4$$3$$2$$1$$5$$3$$4$$2$$1$$6$$3$$4$$5$$2$$1$$6$$4$$5$$3$$1$$2$$1$$2$$3$$4$$1$$2$$3$$4$$1$$2$$3$$1$$2$$3$$4$$5$$1$$2$$3$$1$$2$$3$$4$$5$$1$$2$$1$$2$$3$$4$$5$$6$$1$$2$$1$$2$$3$$4$$5$$6$$1$$\leftarrow L_{1}$$L_{6}\rightarrow

4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$5$$4$$3$$2$$1$$3$$2$$1$$5$$3$$4$$2$$1$$2$$1$$6$$3$$4$$5$$2$$1$$2$$1$$1$$2$$3$$4$$5$$6$$1$$4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$5$$4$$3$$2$$1$$3$$2$$1$$5$$4$$3$$2$$1$$2$$1$$6$$5$$4$$3$$2$$1$$2$$1$$1$$6$$5$$4$$3$$2$$1$$\leftarrow L_{4}$$L_{3}\rightarrow

• (ii)

Two isomorphic modules with infinite-dimensional weight spaces:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C5)$

For any $t\in\mathbb{Z}_{>0}$, consider the following Gelfand-Tsetlin tableau:

a$$a-t$$c

$T(\bar{v})$=    a$$a

$a$

• I.

Number of simples in the block: $11$

• II.

simple subquotients.

We have $16$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}-\delta^{21})$. The pictures correspond to the case $t=2$.

• (i)

Six modules with weight multiplicities bounded by $t$, two pairs of isomorphic modules and two more modules:

[TABLE]

[TABLE]

1$$1$$1$$2$$1$$1$$2$$2$$1$$1$$2$$2$$2$$1$$1$$2$$2$$2$$2$$1$$1$$1$$2$$1$$1$$2$$1$$1$$2$$2$$2$$2$$1$$1$$2$$1$$1$$2$$2$$2$$L_{7}\rightarrow$$L_{2}\rightarrow

2$$2$$2$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$1$$2$$2$$2$$2$$1$$1$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$1$$2$$2$$1$$2$$1$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$1$$2$$2$$1$$2$$1$$2$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$2$$1$$2$$2$$1$$2$$2$$1$$2$$1$$\leftarrow L_{4}$$L_{13}\rightarrow

• (ii)

Modules with unbounded weight multiplicities:

[TABLE]

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$L_{1}\rightarrow$$L_{6}\rightarrow

3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$L_{3}\rightarrow$$L_{9}\rightarrow

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$L_{11}\rightarrow$$L_{16}\rightarrow

• (iii)

Two isomorphic modules with infinite-dimensional weight spaces:

[TABLE]

• III.

**Loewy series.

**

$L_{1},\ L_{2}\oplus L_{3},\ L_{4}\oplus L_{5}\oplus L_{6},\ L_{7}\oplus L_{8}\oplus L_{9}\oplus L_{10},\ L_{11}\oplus L_{12}\oplus L_{13},\ L_{14}\oplus L_{15},\ L_{16}.$

• $(C6)$

Set $t\in\mathbb{Z}_{>0}$ and consider the following Gelfand-Tsetlin tableau:

a$$a-t$$c

$T(\bar{v})$=    a$$a

$z$

• I.

Number of simples in the block: $4$

• II.

simple subquotients.

In this case we have $5$ simple subquotients. The origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}-\delta^{21})$. We provide the pictures corresponding to the case $t=2$.

• (i)

Two modules with unbounded weight multiplicities:

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$\leftarrow L_{5}$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$L_{1}\rightarrow

• (ii)

A cuspidal module with $t$-dimensional weight multiplicities:

[TABLE]

• (iii)

Two isomorphic modules with infinite-dimensional weight spaces:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C7)$

Consider the following Gelfand-Tsetlin tableau:

a$$a$$c

$T(\bar{v})$=    a$$a

$a$

• I.

Number of simples in the block: $7$

• II.

simple subquotients.

The module $V(T(\bar{v}))$ has $10$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}+\delta^{21}+\delta^{11})$.

• (i)

Eight modules with unbounded weight multiplicities:

[TABLE]

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$L_{1}\rightarrow$$L_{2}\rightarrow

3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$L_{3}\rightarrow$$L_{6}\rightarrow

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$L_{9}\rightarrow$$L_{10}\rightarrow

• (ii)

Two isomorphic modules with infinite-dimensional weight spaces:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C8)$

Consider the following Gelfand-Tsetlin tableau:

a$$a$$c

$T(\bar{v})$=    a$$a

$z$

• I.

Number of simples in the block: $3$

• II.

simple subquotients.

The module $V(T(\bar{v}))$ has $4$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}+\delta^{21}+\delta^{11})$.

• (i)

Modules with unbounded weight multiplicities:

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$\leftarrow L_{4}$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$L_{1}\rightarrow

• (ii)

Two isomorphic modules with infinite-dimensional weight multiplicities:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C9)$

Let $s,t\in\mathbb{Z}_{>0}$ be such that $t<s$. Consider the following Gelfand-Tsetlin tableau:

a$$a-t$$a-s

$T(\bar{v})$=    a$$a

$z$

• I.

Number of simples in the block: $7$

• II.

simple subquotients.

The module $V(T(\bar{v}))$ has $10$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}-\delta^{22})$. The pictures correspond to the case $t=1$, $s=2$.

• (i)

Two modules with unbounded weight multiplicities:

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$\leftarrow L_{10}$$1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$L_{1}\rightarrow

• (ii)

Three modules with weight multiplicities bounded by $t$:

[TABLE]

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$\leftarrow L_{8}$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$L_{2}\rightarrow

• (iii)

Three modules with weight multiplicities bounded by $s-t$:

[TABLE]

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$\leftarrow L_{5}$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$L_{3}\rightarrow

• (iv)

Two isomorphic modules with infinite-dimensional weight spaces

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C10)$

Let $t,s\in\mathbb{Z}_{>0}$ be such that $t<s$. Consider the following Gelfand-Tsetlin tableau:

a$$a-t$$a-s

$T(\bar{v})$=    a$$a

$a$

• I.

Number of simples in the block: $20$

• II.

simple subquotients.

The module $V(T(\bar{v}))$ has $32$ simple subquotients, two of them are isomorphic to the simple finite dimensional module with highest weight $\lambda=(t-1,s-t-1)$. Also, there are two isomorphic modules with infinite-dimensional weight spaces. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}-\delta^{22})$. We provide the pictures corresponding to the case $t=1$, $s=2$ (i.e. the principal block).

• (i)

Two isomorphic finite dimensional modules with weight multiplicities of degree $\min\{t,s-t\}$ and highest weight $\lambda=(t-1,s-t-1)$.

[TABLE]

• (ii)

Twenty weight modules with weight multiplicities bounded by $t$ or $s-t$. There are eight pairs of isomorphic modules and four more modules in the list:

[TABLE]

[TABLE]

The following picture describes the weight support of the above listed modules. Recall that for the modules listed in the left picture, the weight multiplicities are bounded by $t$, while for the modules in the right picture, the multiplicities are bounded by $s-t$.

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$L_{21}\downarrow$$L_{14}\rightarrow$$L_{2}\rightarrow$$\leftarrow L_{10}$$\leftarrow L_{26}$$L_{6}\uparrow$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$L_{11}\rightarrow$$L_{7}\rightarrow$$\leftarrow L_{17}$$\leftarrow L_{24}$$L_{16}\downarrow$$L_{3}\uparrow

In the pictures above, the point in the middle is the $\mathfrak{sl}(3)$-weight associated to the tableau $T(v-\delta^{22})$ and corresponds to the trivial (i.e. the finite-dimensional) module.

• (iii)

Eight simple Verma modules (with unbounded set of weight multiplicities). There are two pairs of isomorphic modules and four more modules in the list.

[TABLE]

[TABLE]

The weight supports are listed below.

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$\leftarrow L_{1}$$L_{4}\rightarrow

4$$3$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$\leftarrow L_{5}$$L_{19}\rightarrow

1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$1$$2$$3$$4$$4$$4$$\leftarrow L_{28}$$L_{32}\rightarrow

• (iv)

Two weight modules with infinite weight multiplicities. In this case we have one pair of isomorphic modules:

[TABLE]

• III.

**Loewy series.

**

$L_{1},\ L_{2}\oplus L_{3},\ L_{4}\oplus L_{5}\oplus L_{6}\oplus L_{7},\ L_{8}\oplus L_{9}\oplus L_{10}\oplus L_{11}\oplus L_{12}\oplus L_{13},\\ \ L_{14}\oplus L_{15}\oplus L_{16}\oplus L_{17}\oplus L_{18}\oplus L_{19},\ L_{20}\oplus L_{21}\oplus L_{22}\oplus L_{23}\oplus L_{24}\oplus L_{25},\\ L_{26}\oplus L_{27}\oplus L_{28}\oplus L_{29},\ L_{30}\oplus L_{31},\ L_{32}.$

• $(C11)$

Set $t\in\mathbb{Z}_{>0}$ and consider the following Gelfand-Tsetlin tableau:

a$$a$$a-t

$T(\bar{v})$=    a$$a

$a$

• I.

Number of simples in the block: $13$

• II.

simple subquotients.

The module $V(T(\bar{v}))$ has $20$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v})$. The pictures correspond to the case $t=1$.

• (i)

Six modules with finite dimensional weight spaces of dimension at most $t$, given by:

[TABLE]

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$1$$L_{9}\rightarrow$$L_{4}\rightarrow$$\leftarrow L_{8}$$\leftarrow L_{14}$$L_{10}\downarrow$$L_{2}\uparrow

In the pictures above, the highest weight of $L_{4}$ corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v})$.

• (ii)

Six modules with unbounded finite weight multiplicities:

[TABLE]

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$2$$2$$2$$1$$2$$3$$3$$3$$1$$2$$3$$4$$4$$1$$2$$3$$4$$5$$5$$\leftarrow L_{1}$$L_{5}\rightarrow

3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$2$$1$$2$$1$$1$$1$$3$$2$$1$$2$$1$$2$$1$$1$$1$$\leftarrow L_{3}$$L_{11}\rightarrow

1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$4$$3$$2$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$1$$2$$3$$4$$4$$4$$\leftarrow L_{16}$$L_{20}\rightarrow

• (iii)

Two isomorphic modules with infinite-dimensional weight multiplicities:

[TABLE]

• III.

**Loewy series.

$L_{1},\ L_{2}\oplus L_{3}\oplus L_{4},\ L_{5}\oplus L_{6}\oplus L_{7},\ L_{8}\oplus L_{9}\oplus L_{10}\oplus L_{11}\oplus L_{12}\oplus L_{13},\\ L_{14}\oplus L_{15}\oplus L_{16},\ L_{17}\oplus L_{18}\oplus L_{19},\ L_{20}.$**

• (C12)

For any $t\in\mathbb{Z}_{>0}$ consider the tableau:

a$$a$$a-t

$T(\bar{v})$=    a$$a

$z$

• I.

Number of simples in the block: $4$

• II.

simple subquotients.

We have $5$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v})$. We provide pictures of the weight lattice corresponding to the case $t=1$.

• (i)

Two modules with unbounded weight multiplicities:

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$\leftarrow L_{5}$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$L_{1}\rightarrow

• (ii)

A cuspidal module with $t$-dimensional weight multiplicities:

[TABLE]

• (iii)

Two isomorphic modules with infinite-dimensional weight spaces:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C13)$

Consider the following Gelfand-Tsetlin tableau:

a$$a$$a

$T(\bar{v})$=    a$$a

$a$

• I.

Number of simples in the block: $7$

• II.

simple subquotients.

The module $V(T(\bar{v}))$ has $10$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}+\delta^{21}+\delta^{11})$.

• (i)

Modules with unbounded weight multiplicities:

[TABLE]

[TABLE]

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$\leftarrow L_{1}$$L_{3}\rightarrow

3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$4$$3$$2$$1$$3$$2$$1$$3$$2$$1$$2$$1$$2$$1$$1$$\leftarrow L_{2}$$L_{6}\rightarrow

1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$3$$3$$3$$3$$2$$1$$4$$4$$4$$3$$2$$1$$5$$5$$5$$4$$3$$2$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$1$$2$$3$$3$$3$$3$$1$$2$$3$$4$$4$$4$$1$$2$$3$$4$$5$$5$$5$$\leftarrow L_{8}$$L_{10}\rightarrow

• (ii)

Two isomorphic modules with infinite-dimensional weight spaces:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

• $(C14)$

Consider the following Gelfand-Tsetlin tableau:

a$$a$$a

$T(\bar{v})$=    a$$a

$z$

• I.

Number of simples in the block: $3$

• II.

simple subquotients.

The module $V(T(\bar{v}))$ has $4$ simple subquotients. In this case, the origin of the weight lattice corresponds to the $\mathfrak{sl}(3)$-weight associated to the tableau $T(\bar{v}+\delta^{21}+\delta^{11})$.

• (i)

Modules with unbounded weight multiplicities:

[TABLE]

1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$\leftarrow L_{4}$$1$$1$$1$$1$$1$$1$$1$$1$$2$$2$$2$$2$$2$$2$$2$$2$$2$$3$$3$$3$$3$$3$$3$$3$$3$$4$$4$$4$$4$$4$$4$$4$$4$$4$$5$$5$$5$$5$$5$$5$$5$$5$$L_{1}\rightarrow

• (ii)

Two isomorphic modules with infinite-dimensional weight multiplicities:

[TABLE]

• III.

**Loewy series.

**

[TABLE]

8. Localization on Gelfand-Tsetlin modules

8.1. Localization and twisted localization functors

We first recall the definition of the localization functor of $U$-modules. For details we refer the reader to [De80] and [M00].

For every root $\alpha\in\Delta$ the multiplicative set ${\bf F}_{\alpha}:=\{f_{\alpha}^{n}\,|\,n\in{\mathbb{Z}}_{\geq 0}\}\subset U$ satisfies Ore’s localizability conditions because $\mbox{ad}\,f_{\alpha}$ acts locally nilpotent on $U$. By $\mathcal{D}_{\alpha}U$ we denote the localization of $U$ relative to ${\bf F}_{\alpha}$. For every weight module $M$, $\mathcal{D}_{\alpha}M=\mathcal{D}_{\alpha}U\otimes_{U}M$ is the $\alpha$–localization of $M$. If $f_{\alpha}$ is injective on $M$, then $M$ can be naturally viewed as a submodule of $\mathcal{D}_{\alpha}M$. Furthermore, if $f_{\alpha}$ is injective on $M$, then it is bijective on $M$ if and only if $\mathcal{D}_{\alpha}M=M$.

For $x\in\operatorname{\mathbb{C}}$ and $u\in\mathcal{D}_{\alpha}U$ we set

[TABLE]

where $\binom{x}{i}=\frac{x(x-1)...(x-i+1)}{i!}$. Since $\mbox{ad}\,f_{\alpha}$ is locally nilpotent on $U_{\alpha}$, the sum above is actually finite. Note that for $x\in{\mathbb{Z}}$ we have $\Theta_{x}(u)=f_{\alpha}^{x}uf_{\alpha}^{-x}$. For a $\mathcal{D}_{\alpha}U$-module $M$ by $\Phi_{\alpha}^{x}M$ we denote the $\mathcal{D}_{\alpha}U$-module $M$ twisted by the action

[TABLE]

where $u\in\mathcal{D}_{\alpha}U$, $v\in M$, and $v^{x}$ stands for the element $v$ considered as an element of $\Phi_{\alpha}^{x}M$. Since $v^{n}=f_{\alpha}^{-n}\cdot v$ whenever $n\in{\mathbb{Z}}$ it is convenient to set $f_{\alpha}^{x}\cdot v:=v^{-x}$ in $\Phi_{\alpha}^{-x}M$ for $x\in\operatorname{\mathbb{C}}$.

In what follows we set $\mathcal{D}_{\alpha}^{x}M:=\Phi_{\alpha}^{x}(\mathcal{D}_{\alpha}M)$ and refer to it as a twisted localization of $M$. One easily check that if $M$ is a weight $\mathfrak{g}$-module, then $\mathcal{D}_{\alpha}^{x}M$ iis a weight module as well, in particular, $v^{x}\in M_{\lambda+x\alpha}$ whenever $v\in M_{\lambda}$. Furthermore, one easily verifies the following proposition.

Proposition 8.1**.**

Let $\alpha$ be a root and $x\in\mathbb{C}$.

• (i)

${\mathcal{D}}_{\alpha}^{x}$* is an exact functor from the category of $U$-modules to the category of ${\mathcal{D}}_{\alpha}U$-modules*

• (ii)

If $M\subset N$ are $U$-modules such that $M$ is $f_{\alpha}$-injective and $N$ is $f_{\alpha}$-bijective, then $N=\mathcal{D}_{\alpha}M$.

In the case when $f_{\alpha}$ acts injectively on $M$, set $\mathcal{QD}_{\alpha}M:={\mathcal{D}}_{\alpha}M/M$. Also, if $\alpha=\varepsilon_{i}-\varepsilon_{j}$ we will write $\mathcal{D}_{ij}$, $\mathcal{D}^{x}_{ij}$, and $\mathcal{QD}_{ij}$ for $\mathcal{D}_{\alpha}$, $\mathcal{D}^{x}_{\alpha}$, and $\mathcal{QD}_{\alpha}$, respectively.

8.2. Localization functors in the case of $\mathfrak{sl}(3)$

From now on we consider $U=U(\mathfrak{sl}(3))$. In this section we study the relation between the tableaux bases of a module and its localized module.

Our goal is to apply localization functors to Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules and realize all simple Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules as subquotients of twisted localized modules.

With this in mind, our first step is to obtain conditions on the bases of the modules that guarantee injectivity or surjectivity of the operator $f_{\alpha}$. For simplicity, we will work with $\alpha=\varepsilon_{1}-\varepsilon_{2}$, hence $f_{\alpha}=E_{21}$.

8.2.1. Injectivity and surjectivity of the operator $E_{21}$

In this subsection, we assume that $V$ is the generic module $V(T(v))$, or the singular module $V(T(\bar{v}))$. By $\mathcal{B}$ we denote the lattice of tableaux $\mathcal{B}(T(v))$ (or $\mathcal{B}(T(\bar{v}))$). Also, the tableaux basis of a Gelfand-Tsetlin module $M$ that is a subquotient of $V$ will be denoted by $\mathcal{B}_{M}\subseteq\mathcal{B}$.

Remark 8.2**.**

Since $V$ is a weight module, every subquotient $M$ of $V$ is a weight module. Hence, in order to check injectivity or surjectivity of $E_{21}$ on $M$, it is enough to check those properties on weight spaces of $M$. Also, recall that for a weight $\lambda=(\lambda_{1},\lambda_{2})$ in the weight support of $M$, $E_{21}(M_{\lambda})\subseteq M_{(\lambda_{1}-2,\lambda_{2}+1)}$.

To unify the notation, in the case of a generic tableau $T(v)$ it will be convenient to write $\operatorname{Tab}(w):=T(v+w)$. Then, the action of $E_{21}$ on $\operatorname{Tab}(w)$ (generic or singular) is given by the formula:

[TABLE]

From Propositions 7.2 and 7.32, if $\operatorname{Tab}(w)\in\mathcal{B}_{M}$ is a weight vector of weight $\lambda$, then the weight space $M_{\lambda}$ is spanned by $\{\operatorname{Tab}(w+(i,-i,0))\ |\ i\in\mathbb{Z}\}\bigcap\mathcal{B}_{M}$. If $w_{i}$ denotes the vector $w+(i,-i,0)\in\mathbb{Z}^{3}$, any element of $M_{\lambda}$ will be of the form $u=\sum_{i\in I}a_{i}T(w_{i})$ for some finite subset $I$ of $\mathbb{Z}$.

Lemma 8.3**.**

The operator $E_{21}$ acts injectively on $M$ if and only if $\operatorname{Tab}(w)\in\mathcal{B}_{M}$ implies $\operatorname{Tab}(w-\delta^{11})\in\mathcal{B}_{M}$.

Proof.

Suppose first that there exists $\operatorname{Tab}(w)\in\mathcal{B}_{M}$ such that $\operatorname{Tab}(w-\delta^{11})\notin\mathcal{B}_{M}$, then $E_{21}(\operatorname{Tab}(w))=\operatorname{Tab}(w-\delta^{11})=0$ (on $M$), which implies that $E_{21}$ is not injective. On the other hand, suppose $u:=\sum_{i\in I}a_{i}\operatorname{Tab}(w_{i})\in M_{\lambda}$ with $\operatorname{Tab}(w_{i})\in\mathcal{B}_{M}$ for any $i$. If $u$ is such that $0=E_{21}(u)=\sum_{i\in I}a_{i}\operatorname{Tab}(w_{i}-\delta^{11})$, then $a_{i}T(w_{i}-\delta^{11})=0$ for any $i\in I$. Since,by hypothesis $T(w_{i}-\delta^{11})\in\mathcal{B}_{M}$, we have $a_{i}=0$ for any $i\in I$. ∎

Lemma 8.4**.**

The operator $E_{21}$ acts surjectively on $M$ if and only if $T(w)\in\mathcal{B}_{M}$ implies $T(w+\delta^{11})\in\mathcal{B}_{M}$.

Proof.

Any element of $M_{(\lambda_{1}-2,\lambda_{2}+1)}$ is of the form $u^{\prime}=\sum\limits_{i\in I}a_{i}\operatorname{Tab}(w_{i}-\delta^{11})$ and a direct computation using (32) shows that $E_{21}\left(\sum\limits_{i\in I}a_{i}\operatorname{Tab}(w_{i})\right)=u^{\prime}$. ∎

8.2.2. twisted localization with respect to $E_{21}$

Recall that for $x\in\operatorname{\mathbb{C}}$ and $u\in\mathcal{D}_{12}U$ we have

[TABLE]

Lemma 8.5**.**

Let $\{c_{ij}\}_{1\leq j\leq i\leq 3}$ be the generators of $\Gamma$ defined in (1). Then

[TABLE]

Proof.

We first note that if $u$ commutes with $E_{21}$, then $\Theta_{x}(u)=u$. Since the generators $\{c_{ij}\}_{2\leq j\leq i\leq 3}$ commute with $E_{21}$, the first part of the lemma is proven. For the second part we use that $c_{11}=E_{11}$ and $(\mbox{ad}\,E_{21})^{2}(E_{11})=0$. ∎

As an immediate consequence of Lemma 8.5 we have the following corollary that will be frequently applied.

Corollary 8.6**.**

Let $M$ be any Gelfand-Tsetlin module on which $E_{21}$ acts injectively.

• (i)

The twisted localized module $\mathcal{D}_{12}^{x}M$ is also a Gelfand-Tsetlin module.

• (ii)

If $v\in M$ has Gelfand-Tsetlin character $\chi=(a_{11},a_{21},a_{22},a_{31},a_{32},a_{33})$, then $v^{x}\in\mathcal{D}_{12}^{x}M$ has Gelfand-Tsetlin character $\tilde{\chi}=(a_{11}+x,a_{21},a_{22},a_{31},a_{32},a_{33})$.

Next, for a Gelfand-Tsetlin module $M$ with tableaux basis $\mathcal{B}_{M}$ and injective action of $E_{21}$, we explicitly describe the tableaux basis of $\mathcal{D}_{12}^{x}M$. For this, we introduce some notation.

For $\mathcal{B}\subseteq\mathbb{Z}^{3}$, denote by $\mathcal{B}+\delta^{11}$ the region $\{(m,n,k)\;|\;(m,n,k-1)\in\mathcal{B}\}$. Set $\mathcal{B}+t\delta^{11}=(\mathcal{B}+(t-1)\delta^{11})+\delta^{11}$ for $t\in{\mathbb{N}}$ and $\mathcal{B}+{\mathbb{N}}\delta^{11}=\bigcup_{t=0}^{\infty}(\mathcal{B}+t\delta^{11})$.

Recall the Definition 7.3 for $L(\mathcal{B};\ v)$.

Proposition 8.7**.**

Let ${\mathcal{B}}\subset{\mathbb{Z}}^{3}$ and $L(\mathcal{B})=L(\mathcal{B};\ v)$ be a simple Gelfand-Tsetlin module. Assume that $E_{21}$ acts injectively on $L(\mathcal{B};\ v)$. Then $\mathcal{D}_{12}^{x}L(\mathcal{B};\ v)\simeq M(\mathcal{B}+{\mathbb{N}}\delta^{11};\ v+x\delta^{11})$ and $\mathcal{QD}^{x}_{12}L(\mathcal{B};\ v)\simeq M((\mathcal{B}+{\mathbb{N}}\delta^{11})\setminus\mathcal{B};\ v+x\delta^{11})$. In particular, if $x$ is an integer we have $\mathcal{D}_{12}L(\mathcal{B})\simeq M(\mathcal{B}+{\mathbb{N}}\delta^{11})$ and $\mathcal{QD}_{12}L(\mathcal{B})\simeq M((\mathcal{B}+{\mathbb{N}}\delta^{11})\setminus\mathcal{B})$.

Corollary 8.8**.**

Let $M$ be a simple module in $\mathcal{GT}$, generated by a tableau $\operatorname{Tab}(w)$ and such that $E_{21}$ acts injectively on $M$. Then for any $x\in\mathbb{C}$, $\mathcal{D}_{12}^{x}M$ has a subquotient isomorphic to a simple $\mathfrak{sl}(3)$-module in $\mathcal{GT}$ generated by the tableau $\operatorname{Tab}(w+x\delta^{11})$.

8.3. Simple Gelfand-Tsetlin modules and localization functors

In this section we will describe the simple Gelfand-Tsetlin $\mathfrak{sl}(3)$-modules via localization functors and subquotients starting with some simple $E_{21}$-injective Gelfand-Tsetlin module. In order to give such description we rely on Lemmas 8.3, 8.4, and Proposition 8.7. In fact, in order to use Proposition 8.7 we have to check if the corresponding module defined in such region is $E_{21}$-bijective.

For convenience, we denote by $L_{i}^{(Gj)}$ the simple module $L_{i}$ in the $j$th generic block from the list in §7.2, and by $L_{i}^{(Cj)}$ the simple module $L_{i}$ in the $j$th singular block in the list in §7.5. For example, $L_{1}^{(G2)}$ stands for the module

$L\left(\left\{k\leq m\right\};T(a,b,c|x,y|x)\right)$.

Below we list of all simple modules which are $E_{21}$-injective.

• (i)

Simple $E_{21}$-injective generic modules:

[TABLE]

• (ii)

Simple $E_{21}$-injective singular modules:

[TABLE]

Finally, we apply (twisted) localization functors on the modules above and obtain all simple modules in the block as shown in the following tables.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Corollary 8.9**.**

Every simple Gelfand-Tsetlin module can be obtained via a composition of a twisted localization functor and taking a subquotient from a simple $E_{21}$-injective Gelfand-Tsetlin module.