A geometric approach to $1$-singular Gelfand-Tsetlin $\mathfrak {gl}_n$-modules

TL;DR

This paper introduces a geometric construction of 1-singular Gelfand-Tsetlin modules for a0n using complex geometry, universal rings, and local distributions, providing a new perspective on these algebraic structures.

Contribution

It presents a novel geometric approach to constructing 1-singular Gelfand-Tsetlin modules via universal rings and distributions, extending previous algebraic methods.

Findings

Constructed a universal ring a0o and associated module a0s with basis from local distributions.

Showed that a0s is a natural module over a0o and a Lie algebra a0h.

Derived a universal 1-singular Gelfand-Tsetlin a0gla0n(a0C) module from previous work.

Abstract

This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with basis $\mathcal B_o = \mathcal B(\mathcal D_o)$ formed from some local distributions such that $\mathcal S$ is a natural $\mathcal D_o$-module. For any homomorphism of rings $\mathcal U(\mathfrak{h}) \to \mathcal D_o$, where $\mathfrak{h}$ is a Lie algebra, it follows that $\mathcal S$ is also an $\mathfrak{h}$-module. We observe that the homomorphism of rings constructed in [FO] is a homomorphism of type $\mathcal U(\mathfrak{gl}_n(\mathbb C)) \to \mathcal D_o$. Using this observation we obtain a construction of the universal $1$-singular Gelfand-Tsetlin $\mathfrak{gl}_n(\mathbb C)$-module from [FRG].

Equations

$$\mathcal{U}({\mathfrak{g}}_1)\subset \mathcal{U}({\mathfrak{g}}_2)\subset \cdots \subset \mathcal{U}({\mathfrak{g}}_{n-1})\subset \mathcal{U}({\mathfrak{g}}_n).$$

$$\sigma (f)=f\circ {\sigma }^{-1},s(f)=f\circ s^{-1},s(\sigma )=s\circ \sigma \circ s^{-1}.$$

$$\sum _i{f}_i{\sigma }_i\circ \sum _j{f}_j^{\prime }{\sigma }_j^{\prime }:=\sum _{ij}{f}_i{\sigma }_i(f_j^{\prime }){\sigma }_i\circ {\sigma }_j^{\prime }.$$

$$\begin{split}E_{k,k+1}(v)&=-\sum_{i=1}^{k}\frac{\prod_{j=1}^{k+1}(x_{ki}-x_{k+1,j})}{\prod_{j\neq i}^{k}(x_{ki}-x_{kj})}(v+\delta_{ki});\\ E_{k+1,k}(v)&=\sum_{i=1}^{k}\frac{\prod_{j=1}^{k-1}(x_{ki}-x_{k-1,j})}{\prod_{j\neq i}^{k}(x_{ki}-x_{kj})}(v-\delta_{ki});\\ E_{k,k}(v)&=\Big{(}\sum_{i=1}^{k}(x_{ki}+i-1)-\sum_{i=1}^{k-1}(x_{k-1,i}+i-1)\Big{)}(v),\end{split}$$

$$E_{k,k+1}=-\sum _{i=1}^k\frac{{\prod }_{j=1}^{k+1}(x_{ki}-x_{k+1,j})}{{\prod }_{j\ne i}^k(x_{ki}-x_{kj})}{\sigma }_{ki}^{-1}.$$

$$A_1\circ \cdots \circ A_m=\sum _{i,j}\frac{G_i{\sigma }_i(H_j^m)}{z_1{\sigma }_i(z_1)}{\sigma }_i\circ {\sigma }_j.$$

$$\begin{array}{r}{D}_{\sigma }^1:=\frac{1}{2}e{v}_v\circ (\sigma +\tau (\sigma )),D_{{\sigma }^{\prime }}^2:=e{v}_v\circ \frac{({\sigma }^{\prime }-\tau ({\sigma }^{\prime }))}{2z_1},\sigma ,{\sigma }^{\prime }\in \gimel ,\tau ({\sigma }^{\prime })\ne {\sigma }^{\prime }.\end{array}$$

$$D_{\tau (\sigma )}^1=D_{\sigma }^1\text{and}{D}_{\tau ({\sigma }^{\prime })}^2=-D_{{\sigma }^{\prime }}^2.$$

$$\begin{array}{r}e{v}_v\circ (\sum _i{h}_i{\sigma }_i)=\sum _i{g}_i(v)\cdot D_{{\sigma }_i}^2+\sum _i\frac{\partial g_i}{\partial z_1}(v)D_{{\sigma }_i}^1,\end{array}$$

$$e{v}_v\circ \sum _i{h}_i{\sigma }_i=\sum _i{g}_i(v)e{v}_v\circ \frac{{\sigma }_i}{z_1}+\sum _i\frac{\partial g_i}{\partial z_1}(v)e{v}_v\circ {\sigma }_i.$$

$$E_{rs}(T(\sigma (v)))$$

$$E_{rs}(\mathcal{D}T({\sigma }^{\prime }(v))$$

$$2D_{\sigma }^2(T(v))=\mathcal{D}T({\sigma }^{\prime }(v))-\mathcal{D}T(\tau ({\sigma }^{\prime }(v))),2D_{\sigma }^1(T(v))=T(\sigma (v))+T(\tau (\sigma (v))).$$

Full text

1 Introduction

This paper is devoted to a new elementary geometric construction of the universal $1$-singular Gelfand-Tsetlin module. Denote $\mathfrak{g}_{k}:=\mathfrak{gl}_{k}(\mathbb{C})$, where $k=1,\ldots n$, and consider the flag $\mathfrak{g}_{1}\subset\mathfrak{g}_{2}\subset\cdots\subset\mathfrak{g}_{n-1}\subset\mathfrak{g}_{n}$ of Lie algebras, where $\mathfrak{g}_{k-1}\subset\mathfrak{g}_{k}$ is the inclusion with respect to the left top corner. This flag gives rise to the following flag of universal enveloping algebras

[TABLE]

Denote by $\mathcal{Z}_{k}$ the center of $\mathcal{U}(\mathfrak{g}_{k})$. Then the subalgebra $\Gamma\subset\mathcal{U}(\mathfrak{g}_{n})$, generated by $\mathcal{Z}_{k}$, where $k=1,\ldots,n$, is a maximal commutative subalgebra [Ov1]. It is called the Gelfand-Tsetlin subalgebra. A $\mathcal{U}(\mathfrak{g}_{n})$-module $M$ is called a Gelfand-Tsetlin module if the action of $\Gamma$ on $M$ is locally finite.

In the classical Gelfand-Tsetlin theory [GT] an explicit construction of an action of $\mathfrak{g}_{n}$ with respect to a basis consisting of Gelfand-Tsetlin tableaux is given providing explicit formulas for $\mathfrak{g}_{n}$-action. These formulas for $\mathfrak{g}_{n}$-action are called classical Gelfand-Tsetlin formulas. It was noticed in [DOF1, DOF2, DOF3, DOF4] that the classical Gelfand-Tsetlin formulas may be used to obtain a family of infinite dimensional Gelfand-Tsetlin modules: so-called generic regular Gelfand-Tsetlin modules. An essential progress in the theory of Gelfand-Tsetlin modules was done in [Ov1, Ov2] and later in [FO]. In particular the following important construction was obtained there. Let $V\simeq\mathbb{C}^{n(n+1)/2}$ be the vector space of all Gelfand-Tsetlin tableaux of fixed order $n$, see the main text for details. Denote by $\gimel$ a certain abelian group acting freely on $V$ and by $\mathcal{M}\star\gimel$ the sheaf of meromorphic functions on $V$ with values in $\gimel$. Then there exists a ring structure on $\mathcal{R}:=H^{0}(V,\mathcal{M}\star\gimel)$ such that the classical Gelfand-Tsetlin formulas define a ring homomorphism $\Phi:\mathcal{U}(\mathfrak{g}_{n})\to\mathcal{R}$. In the case when $\operatorname{Im}\Phi$ is holomorphic at a neighborhood of the orbit $\gimel(v)$ of a point $v\in V$, we can define a $\mathfrak{g}_{n}$-module structure on the vector space with the basis $\{ev_{v},\,\,v\in\gimel(v)\}$, where $ev_{v}$ is the evaluation map at the point $v$. These $\mathfrak{g}_{n}$-modules are exactly generic regular Gelfand-Tsetlin modules.

This construction does not work if $\operatorname{Im}\Phi$ is not holomorphic in any neighborhood of $\gimel(v)$. The study of the case when $\operatorname{Im}\Phi$ is not holomorphic in $\gimel(v)$ but has at most one simple pole, or in other words $\operatorname{Im}\Phi$ is $1$-singular, was initiated by V. Futorny, D. Grantcharov and E. Ramirez in [FGR]. The authors [FGR] constructed the universal $1$-singular Gelfand-Tsetlin $\mathfrak{gl}_{n}(\mathbb{C})$-module using additional formal variables that were called derivative tableaux. For another construction of the universal $1$-singular Gelfand-Tsetlin $\mathfrak{gl}_{n}(\mathbb{C})$-module see [Z], which was posted to the arXiv when the present paper was in preparation.

In the present paper we define a subring $\mathcal{D}_{v}$ of $\mathcal{R}$, where $v$ is a certain point of a $1$-singular $\gimel$-orbit. To the ring $\mathcal{D}_{v}$ we associate the vector space $\mathcal{S}=\mathcal{S}(\mathcal{D}_{v})$ with basis $\mathcal{B}=\mathcal{B}(\mathcal{D}_{v})$ consisting of some local distributions supported at $\gimel(v)$ such that $\mathcal{S}$ is a natural $\mathcal{D}_{v}$-module. In particular this implies the following universal property of $\mathcal{D}_{v}$: for any homomorphism of rings $\Psi:\mathcal{U}(\mathfrak{h})\to\mathcal{D}_{v}$ the vector space $\mathcal{S}$ is also an $\mathfrak{h}$-module. Due to this we call the ring $\mathcal{D}_{v}$ the universal ring. Further, we observe that $\Phi(\mathcal{U}(\mathfrak{g}))\subset\mathcal{D}_{v}$. Hence our construction gives rise to a $\mathfrak{g}$-module structure on $\mathcal{S}$ that is isomorphic to the universal $1$-singular Gelfand-Tsetlin module obtained in [FGR]. Our observation leads to a new geometric interpretation of the universal $1$-singular Gelfand-Tsetlin module from [FGR] that allows to simplify proofs from [FGR] and avoid the use of formal variables. Moreover, similar ideas that we present here can be used in the case of other singularities, see [EMV].

Acknowledgements: E. V. was partially partially supported by SFB TR 191 and by the Universidade Federal de Minas Gerais.

2 Preliminaries

A Gelfand-Tsetlin tableau is a tableau $(a_{ki})$ of complex numbers, where $1\leq i\leq k\leq n$ and $n\geq 2$. Further we will consider the set $V$ of all Gelfand-Tsetlin tableaux as a complex manifold that is isomorphic to $\mathbb{C}^{n(n+1)/2}$. Let $\gimel\simeq\mathbb{Z}^{n(n-1)/2}$ be the free abelian group generated by $\sigma_{st}$, where $1\leq t\leq s\leq n-1$. We fix the following action of $\gimel$ on $V$: $\sigma_{st}(x)=(x_{ki}+\delta_{ki}^{st})$, where $x=(x_{ki})\in V$ and $\delta_{ki}^{st}$ is the Kronecker delta. This is $\delta_{ki}^{st}=1$ if $(ki)=(st)$ and $\delta_{ki}^{st}=0$ otherwise. Further we put $G=S_{1}\times S_{2}\times\cdots\times S_{n}$, so $G$ is the product of symmetric groups $S_{i}$. The group $G$ acts on $V$ in the following way $(s(x))_{ki}=x_{ks_{k}(i)},$ where $s=(s_{1},\ldots,s_{n})\in G$. Denote by $\mathcal{M}$ and by $\mathcal{O}$ the sheaves of meromorphic and holomorphic functions on $V$, respectively. Let us take $f\in H^{0}(V,\mathcal{M})$, $s\in G$ and $\sigma\in\gimel$. We set

[TABLE]

These formulas define an action of $\gimel$ on $H^{0}(V,\mathcal{M})$ and actions of $G$ on $H^{0}(V,\mathcal{M})$ and on $\gimel$, respectively.

Denote by $\mathcal{M}\star\gimel$ the sheaf of meromorphic functions on $V$ with values in $\gimel$. An element of $H^{0}(V,\mathcal{M}\star\gimel)$ is a finite sum $\sum\limits_{i}f_{i}\sigma_{i}$, where $f_{i}\in\mathcal{M}$ and $\sigma_{i}\in\gimel$. In other words, $\mathcal{M}\star\gimel$ is the sheaf of meromorphic sections of the trivial bundle $V\times\bigoplus_{\sigma\in\gimel}\mathbb{C}\sigma\to V$. There exists a structure of a skew group ring on $H^{0}(V,\mathcal{M}\star\gimel)$, see [FO]. Indeed,

[TABLE]

Here $f_{i},f^{\prime}_{j}\in H^{0}(V,\mathcal{M})$ and $\sigma_{i},\sigma^{\prime}_{j}\in\gimel$. This skew group ring we denote by $\mathcal{R}$. To simplify notations we use $\circ$ for the multiplication in $\mathcal{R}$ and for the product in $\gimel$. On $H^{0}(V,\mathcal{M}\star\gimel)$ we will consider also the following multiplication $A*B:=B\circ A$.

Recall that a Gelfand-Tsetlin tableau is called generic if $x_{rt}-x_{rs}\notin\mathbb{Z}$ for any $r$ and $s\neq t$. The definition of a standard Gelfand-Tsetlin tableau can be found in [FGR]. The classical Gelfand-Tsetlin formulas have the following form in terms of generators of $\mathfrak{gl}_{n}(\mathbb{C})$, see for instance [FGR], Theorems $3.6$ and $3.8$, for details.

[TABLE]

Here $E_{st}\in\mathfrak{gl}_{n}(\mathbb{C})$ are standard generators and $v\in V$ is either a standard or generic Gelfand-Tsetlin tableau with coordinates $v=(x_{ki})$ and $(v\pm\delta_{ki})=\sigma^{\pm 1}_{ki}(v)$. Assume that $v$ is a generic Gelfand-Tsetlin tableau. Theorem $3.8$ in [FGR] says that Formulas (1) define a $\mathfrak{g}$-module structure on the vector space spanned by the elements of the orbit $\gimel(v)$.

Let us identify the point $v\in V$ with the evaluation map $ev_{v}:H^{0}(V,\mathcal{O})\to\mathbb{C}$. Then we can define the map $E_{st}\mapsto\Phi(E_{st})\in\mathcal{R}$ using the equality $ev_{v}\circ\Phi(E_{st})=E_{st}(v)$ for $v$ generic. Since generic points are dense in $V$, $\Phi(E_{st})$ is a well-defined element of $\mathcal{R}$. For example,

[TABLE]

In [FO] the following theorem was proved.

**Theorem 1. ** The map $\Phi:\mathcal{U}(\mathfrak{g})\to\mathcal{R}$ is a homomorphism of rings. Here $E_{st}\mapsto\Phi(E_{st})$, where $E_{st}\in\mathfrak{gl}_{n}(\mathbb{C})$, is as above.

Remark. Note that $\operatorname{Im}(\Phi)$ is $G$-invariant. This fact can be verified directly.

From Theorem 2 it follows that for any generic $x\in V$ the formula $\Phi(E_{st})(ev_{y})=ev_{y}\circ\Phi(E_{st})$ defines an action of $\mathfrak{gl}_{n}(\mathbb{C})$ on the vector space spanned by local distributions $ev_{y}$, where $y\in\gimel(x)$. Here $ev_{y}\circ(f\sigma)=ev_{y}(f)ev_{y}\circ\sigma$ and $ev_{y}\circ\sigma(g)=ev_{y}(\sigma(g))$ for $g\in\mathcal{O}$. Since elements $\Phi(E_{st})$ are holomorphic in a sufficiently small neighborhood of the orbit $\gimel(x)$, the expression $ev_{y}\circ\Phi(E_{st})$ is well-defined. More generally, any homomorphism of rings $\Psi:\mathcal{U}(\mathfrak{h})\to\mathcal{R}$, where $\mathfrak{h}$ is any Lie algebra, such that the image $\Psi(\mathfrak{h})$ is holomorphic in a neighborhood of $\gimel(x)$ defines an action of $\mathfrak{gl}_{n}(\mathbb{C})$ on the vector space spanned by the local distributions $ev_{y}$, where $y\in\gimel(x)$. The interpretation of a point $y\in V$ as a local distribution $ev_{y}$ suggests the possibility to define a $\mathfrak{gl}_{n}(\mathbb{C})$-module structure on other local distributions, i.e. on linear maps $D_{y}:\mathcal{O}_{y}\to\mathbb{C}$ with $\mathfrak{m}_{y}^{p}\subset\operatorname{Ker}(D_{y})$, where $p>0$ and $\mathfrak{m}_{y}$ is the maximal ideal in the local algebra $\mathcal{O}_{y}$. This idea we develop in the present paper.

The main problem is that the ring $\mathcal{R}$ does not act on the vector space of local distributions, because of singularities. In the next section we will construct the universal ring $\mathcal{D}_{v}\subset\mathcal{R}^{G_{v}}$, where $v$ is a certain $1$-singular point in $V$ and $G_{v}\subset G$ is the stabilizer of $v$. We will show that $\mathcal{D}_{v}$ acts on $G_{v}$-invariant holomorphic functions $H^{0}(V,\mathcal{O}^{G_{v}})$, where the action is given by $(f\circ\sigma)(F)=fF\circ\sigma^{-1}$. This action induces an action $(f\circ\sigma)(D_{y})=D_{y}\circ(f\circ\sigma)$ of $(\mathcal{D}_{v},*)$ on $G_{v}$-invariant holomorphic local distributions $D:H^{0}(V,\mathcal{O}^{G_{v}})\to\mathbb{C}$ supported at $\gimel(v)$. By Theorem 2 we have also a structure of $\mathfrak{gl}_{n}(\mathbb{C})$-module on the vector space of these local distributions. Further we will consider local distributions $ev_{v}\circ A:H^{0}(V,\mathcal{O}^{G_{v}})\to\mathbb{C}$, where $A\in\mathcal{D}_{v}$. Clearly this vector space is $\mathcal{D}_{v}$- and hence $\mathfrak{gl}_{n}(\mathbb{C})$-submodule. The last step is to find a basis $\mathcal{B}_{v}$ for the vector space spanned by $\{ev_{v}\circ A,\,\,|\,\,A\in\mathcal{D}_{v}\}$. This basis we call the universal basis for the universal ring $\mathcal{D}_{v}$.

Our construction implies that for any homomorphism $\Psi:\mathcal{U}(\mathfrak{h})\to\mathcal{D}_{v}$, where $\mathfrak{h}$ is a Lie algebra, $\mathcal{B}_{v}$ is a basis for the corresponding $\mathfrak{h}$-module. We will see that $\operatorname{Im}(\Phi)\subset\mathcal{D}_{v}$ and that $\mathcal{B}_{v}$ coincide with the basis constructed in [FGR]. We develop these ideas in the case of any point $x\in V$ in [EMV].

3 Main result

A point $x=(x_{kj})\in V$ is called $1$-singular if there exist $x_{ki}$ and $x_{kj}$, where $i\neq j$, such that $x_{ki}-x_{kj}\in\mathbb{Z}$ and $x_{rs}-x_{rt}\notin\mathbb{Z}$, where $s\neq t$, for each $(r,s,t)\neq(k,i,j)$. Note that the generators from (1) have one simple pole at the orbit $\gimel(x)$ for any $1$-singular point $x$. Let us fix an $1$-singular point $x^{0}=(x^{0}_{kj})\in V$ such that $x^{0}_{ki}-x^{0}_{kj}\in\mathbb{Z}$. We put $z_{1}=x_{ki}-x_{kj}$, $z_{2}=x_{ki}+x_{kj}$ and we denote by $z_{3},z_{4},\ldots$ other coordinates $(x_{st})$ in $V$. So $(z_{i})$ are new coordinates in $V$. From now on we fix a point $v=(0,z^{0}_{2},\ldots,z^{0}_{n})\in\gimel(x^{0})$ and a sufficiently small neighborhood $W$ of the orbit $\gimel(v)=\gimel(x^{0})$ that is invariant with respect to the group $\gimel$ and with respect to $\tau\in G$, where $\tau$ is defined by $\tau(z_{1})=-z_{1}$ and $\tau(z_{i})=z_{i}$, $i>1$. From now on we will consider restrictions of elements of $\mathcal{R}$ on $W$. We denote by $G_{v}=\{\operatorname{id},\tau\}\subset G$ the stabilizer of $v$.

We say that an element $A\in\mathcal{R}$ is at most $1$-singular at $v$, if $A=\sum_{i}h_{i}\sigma_{i}$, where $h_{i}$ are holomorphic at $v$ or have the form $h_{i}=g_{i}/z_{1}$, where $g_{i}$ are holomorphic at $v$. We need the following proposition.

**Proposition 1. ** Let $A_{j}=\sum\limits_{i}(H^{j}_{i}/z_{1})\sigma_{i}\in\mathcal{R}^{G_{v}}$, where $j=1,\ldots,m$ and $H^{j}_{i}$ are holomorphic in $W$. Then the product $A_{1}\circ\cdots\circ A_{m}$ is at most $1$-singular at $v$.

Proof. Assume by induction that for $k=m-1$ our statement holds. In other words, assume that $A_{1}\circ\cdots\circ A_{m-1}=\sum\limits_{i}(G_{i}/z_{1})\sigma_{i}$, where $G_{i}$ are holomorphic at $v$. We have

[TABLE]

Assume that this product is two singular at $v$. Note that $\sigma_{i}(H^{m}_{j})$ is holomorphic in $W$. Therefore, $\sigma_{i_{0}}(z_{1})=z_{1}$ for a certain $i_{0}$ and, hence, $\tau(\sigma_{i_{0}})=\sigma_{i_{0}}$. Further, $\tau(\sum\limits_{i}(G_{i}/z_{1})\sigma_{i})=\sum\limits_{i}(G_{i}/z_{1})\sigma_{i}$, since $A_{1}\circ\cdots\circ A_{n}$ is $\tau$-invariant. Therefore $\tau(G_{i_{0}}/z_{1}\sigma_{i_{0}})=\tau(G_{i_{0}}/z_{1})\sigma_{i_{0}}=G_{i_{0}}/z_{1}\sigma_{i_{0}}$. Hence $\tau(G_{i_{0}}/z_{1})$ is also $\tau$-invariant and $\tau(G_{i_{0}})=-G_{i_{0}}$. Therefore, $G_{i_{0}}=z_{1}G^{\prime}_{i_{0}}$, where $G^{\prime}_{i_{0}}$ is holomorphic at $v$. Therefore, $G_{i_{0}}\sigma_{i_{0}}(H^{m}_{j})/z_{1}\sigma_{i_{0}}(z_{1})=G^{\prime}_{i_{0}}\sigma_{i_{0}}(H^{m}_{j})/z_{1}$ is $1$-singular. The proof is complete.$\Box$

Remark. Elements $A_{j}=\sum\limits_{i}(H^{j}_{i}/z_{1})\sigma_{i}$ as in Proposition 3 generate a subring $\mathcal{D}_{v}$ in $\mathcal{R}^{G_{v}}$. We call this ring the universal ring of $v$. By Proposition 3 any element in $\mathcal{D}_{v}$ is at most $1$-singular at $v$. If $A=\sum\limits_{i}(H_{i}/z_{1})\sigma_{i}$ is a generator of $\mathcal{D}_{v}$ and $F\in H^{0}(W,\mathcal{O}^{G_{v}})$, then $A(F)=F^{\prime}/z_{1}$ is at most $1$-singular at $v$, holomorphic in $W\setminus\{v\}$ and $G_{v}$-invariant. Therefore $\tau(F^{\prime})=-F^{\prime}$ and hence $A(F)$ is holomorphic. So we defined an action of $\mathcal{D}_{v}$ on $H^{0}(W,\mathcal{O}^{G_{v}})$.

We put $g_{i}:=z_{1}h_{i}$, where $\sum\limits_{i}h_{i}\sigma_{i}\in\mathcal{R}$. Consider the following set of $G_{v}$-invariant local distributions defined on $H^{0}(W,\mathcal{O}^{G_{v}})$:

[TABLE]

Note that $(\sigma^{\prime}-\tau(\sigma^{\prime}))/z_{1}$ and $\sigma+\tau(\sigma)$ are elements of $\mathcal{D}_{v}$, hence Formulas (2) are well-defined. Moreover we have the following equalities

[TABLE]

Denote $\Delta:=\{\sigma\in\gimel\,\,|\,\,\sigma(x_{ki},x_{ki})=(x_{ki}+m_{1},x_{ki}+m_{2}),\,\,m_{1}\leq m_{2}\}$ and consider the set $\mathcal{B}_{v}:=\{D^{1}_{\sigma},D^{2}_{\sigma^{\prime}}\,\,|\,\,\sigma,\sigma^{\prime}\in\Delta,\,\,\tau(\sigma^{\prime})\neq\sigma^{\prime}\}$. The set $\mathcal{B}_{v}$ is a set of linearly independent distributions defined on $H^{0}(W,\mathcal{O}^{G_{v}})$. To see this we should apply $D^{i}_{\sigma}$ to a linear combination $\alpha+\beta z_{1}^{2}$ of $G_{v}$-invariant functions $1$ and $z_{1}^{2}$. Hence $\mathcal{B}_{v}$ is a basis of the vector subspace $S$ in $H^{0}(W,\mathcal{O}^{G_{v}})^{*}$ spanned by elements from $\mathcal{B}_{v}$. In the next proposition we show that $S$ is a $\mathcal{D}_{v}$- and $\mathfrak{gl}_{n}(\mathbb{C})$-module. This $\mathfrak{gl}_{n}(\mathbb{C})$-module is isomorphic to the universal $1$-singular Gelfand-Tsetlin module constructed in [FGR], see Section $4$ for details.

**Proposition 2. ** Let us take $\sum\limits_{i}h_{i}\sigma_{i}\in\mathcal{D}_{v}$. Then we have

[TABLE]

where $g_{i}=z_{1}h_{i}$. Note that in the case if $h_{i}$ is holomorphic, we have $g_{i}(v)=0$ and $\frac{\partial g_{i}}{\partial z_{1}}(v)=h_{i}(v)$. Therefore, $S$ is a $\mathcal{D}_{v}$-module with basis $\mathcal{B}_{v}$.

Proof. Using the series expansion $g_{i}=g_{i}|_{z_{1}=0}+\frac{\partial g_{i}}{\partial z_{1}}|_{z_{1}=0}z_{1}+\cdots$, we get

[TABLE]

Note that $ev_{v}\circ z_{1}^{m}\sigma_{i}=0$ for $m>1$. Using the symmetrization $2ev_{v}\circ\sum\limits_{i}h_{i}\sigma_{i}=ev_{v}\circ\sum\limits_{i}h_{i}\sigma_{i}+\tau(ev_{v}\circ\sum\limits_{i}h_{i}\sigma_{i})$, we obtain the result.$\Box$

Let $X$ be one of generators (1). Clearly $\Phi(X)\in\mathcal{D}_{v}$, see Remark after Theorem 2.

Theorem 2. [Main result 1] The vector space $S$ spanned by elements of $\mathcal{B}$ is a $\mathfrak{gl}_{n}(\mathbb{C})$-module. The action is given by Formulas (4).

Proof. The result follows from Theorem 2 and Proposition 3. Indeed, $\operatorname{Im}(\Phi)\subset\mathcal{D}_{v}$, hence we get a $\mathfrak{gl}_{n}(\mathbb{C})$-module.$\Box$

In fact we proved a more general result than it is formulated in Theorem 3.

Theorem 3. [Main result 2] Let $\mathfrak{h}$ be a Lie algebra and $\Psi:\mathcal{U}(\mathfrak{h})\to\mathcal{D}_{v}$ be a homomorphism of rings. Then the vector space $S$ spanned by elements from $\mathcal{B}$ is an $\mathfrak{h}$-module. In other words the basis $\mathcal{B}$ is universal for any homomorphisms $\Psi:\mathcal{U}(\mathfrak{h})\to\mathcal{D}_{v}$.

4 Appendix

Theorem 3 recovers one of the main results of [FGR], a construction of the universal $\mathfrak{gl}_{n}(\mathbb{C})$-module. Another main result of [FGR] is that in many cases the $\mathfrak{gl}_{n}(\mathbb{C})$-module $S$ is irreducible, see Theorem $4.14$. Let us give an explicit correspondence between the basis constructed in [FGR] and our basis $\mathcal{B}$. We use notations from [FGR]. In [FGR] the authors consider the basis $\{T(\sigma(v)),\,\,\mathcal{D}T(\sigma^{\prime}(v))\}$, where $\sigma,\sigma^{\prime}\in\gimel$, such that $T(\sigma(v))-T(\tau(\sigma(v)))=0$, $\mathcal{D}T(\sigma^{\prime}(v))+\mathcal{D}T(\tau(\sigma^{\prime}(v)))=0$ and $\tau(\sigma^{\prime})\neq\sigma^{\prime}$, see Remark $4.5$ in [FGR]. The element $T(\sigma(v))$ was considered as a point in $V$ and $\mathcal{D}T(\sigma^{\prime}(v))$ was considered as a formal additional variable. (In our notations, $T(\sigma(v))$ is just $\sigma(v)\in V$, where $v$ as above.) Further, in [FGR] the action of $\mathfrak{gl}_{n}(\mathbb{C})$ is given by the following formulas, [FGR, Theorem 4.11]:

[TABLE]

where $x=(x_{ki})$ are coordinates in a neighborhood of $v$. The explicit correspondence between the bases is given by the following formulas

[TABLE]