Decomposition of the polynomials over the spherical subalgebra

TL;DR

This paper explores the structure of modules over the spherical subalgebra of the rational Cherednik algebra, establishing a lowest weight category description and drawing analogies with invariant differential operator decompositions.

Contribution

It introduces a lowest weight category framework for modules over the spherical Cherednik algebra, generalizing known decompositions for invariant differential operators across all finite reflection groups.

Findings

Established a lowest weight description for the category of D_c-modules over the polynomial ring.

Identified the algebra Rc_c as an analogue of the Cartan algebra in this setting.

Extended the analogy between Cherednik algebra modules and invariant differential operator modules.

Abstract

Given a finite subgroup $W \subset \GL(\fh)$ of the linear group of a finite-dimensional complex vector field $\fh$, it is a well-studied problem to describe the structure of the symmetric algebra $B= \sym(\fh^*)$ as a representation of $G$, and also as a module over the ring of invariant differential operators under $W$ in the ring $\D(\fh)$ of differential operators on $\fh$. Since the rational Cherednik algebra $H_c(W,\fh)$ and the spherical algebra $eH_ce$ are respectively universal deformations of the ring $\D(\fh)$ and the ring $\D(\fh)^W$of $W$-invariant differential operators, we would like to build an analogy between the decomposition of modules over the invariant differential operators in \cite{ Nonk1, Nonk2, Nonk3} and the decomposition of modules over the sperical subalgebra of the rational Cherednik algebra. The ring $\D_c= eH_ce$ inherits the natural grading of $B$, and we let $\D_c^0 \subset \D_c$ and $\D_c^{-} \subset \D_c$ be subset of elements of degree 0 end strictly negative degree, respectively. Our main result is that there is for all finite reflection groups a lowest weight description of the category of $\D_c$-modules of $B$ where the ring $\Rc_c= \D_c^0 / \Dc_c^0 \cap \D_c \D_c^{-}$ plays the very important role of Cartan algebra.

Equations

$$w\cdot x\cdot w^{-1}=w(x),w\cdot x\cdot w^{-1}=w(y),\forall y\in \mathfrak{h},x\in {\mathfrak{h}}^{\ast },w\in W,$$

$$x_1\cdot x_2=x_2\cdot x_1,y_1{y}_2=y_2{y}_1,\forall y_1,y_2\in \mathfrak{h},x_1,x_2\in {\mathfrak{h}}^{\ast }$$

$$y\cdot x-x\cdot y=⟨y,x⟩-\sum _{\alpha \in R/\{\pm 1\}}{c}_{\alpha }\cdot ⟨y,\alpha ⟩⟨{\alpha }^{\vee },x⟩\cdot s_{\alpha },\forall y\in \mathfrak{h},x\in {\mathfrak{h}}^{\ast }.$$

$$\mathbb{C}[\mathfrak{h}]{\otimes }_{\mathbb{C}}\mathbb{C}[W]{\otimes }_{\mathbb{C}}\mathbb{C}[{\mathfrak{h}}^{\ast }]\cong H_c.$$

$$[x_i,x_j]=[y_i,y_j]=0,[y_i,x_j]=c{s}_{ij},[y_i,x_i]=1-c\sum _{j\ne i}{s}_{ij},$$

$$\mathbf{h}=\sum _i{x}_i{y}_i+\frac{1}{2}\dim \mathfrak{h}-\sum _{s\in S}\frac{2c_s}{1-{\lambda }_s}s.$$

$$[\mathbf{h},x]=x,x\in {\mathfrak{h}}^{\ast },[\mathbf{h},y]=-y,y\in \mathfrak{h}.$$

$$\mathbb{C}[\mathfrak{h}]v_0=\{f{v}_0|f\in \mathbb{C}[\mathfrak{h}]\}$$

$$D_y={\partial }_y-\sum _{s\in S}\frac{2c_s⟨y,{\alpha }_s⟩}{(1-{\lambda }_s){\alpha }_s}(1-s)$$

$$x(f{v}_0)$$

$$y(f{v}_0)$$

$$w(f{v}_0)$$

$$L_c:=\Delta -\sum _{s\in S}\frac{c_s({\alpha }_s,{\alpha }_s)}{{\alpha }_s}{\partial }_{{{\alpha }_s}^{\vee }}\in \mathcal{D}({\mathfrak{h}}_{reg}{)}_{-}$$

$$(\text{mbox}PWB)\mathbb{C}[\mathfrak{h}]\otimes \mathbb{C}[{\mathfrak{h}}^{\ast }]\otimes \mathbb{C}[W]\to {\mathbb{D}}_c,f(x)\otimes g(D)\otimes w\mapsto f(x)\cdot g(D)\cdot w,$$

$$\mathbf{h}=\nabla +C,$$

$$\mathbf{h}$$

$$\mathbb{C}[V{]}^W\to \mathbf{e}(\mathbb{C}[V]⋊W)\mathbf{e}$$

$$M(\tau )=H_c{\otimes }_{\mathbb{C}[W]⋉\mathbb{C}[{\mathfrak{h}}^{\ast }]}\tau .$$

$$M(\tau )\cong \mathbb{C}[\mathfrak{h}]{\otimes }_{\mathbb{C}}\tau$$

$$\sum _{w^{\prime }\in W}{P}_{w^{\prime }}{w}^{\prime }\cdot \sum _{w"\in W}{Q}_{w"}w"=\sum _{w\in W}\sum _{w^{\prime }w"=w}(P_{w^{\prime }}{Q}_{w"})w.$$

$$B\cong B^G[G]=B^G\otimes \mathbb{C}[G]=\underset{\chi \in \hat{G}}{⨁}{m}_{\chi }{B}^G\otimes V_{\chi },$$

$$B=\underset{\chi \in \hat{W}}{⨁}{B}_{\chi },$$

$$B_{\chi }=V_{\chi }{\otimes }_{\mathbb{C}}{N}_{\chi }.$$

$$N_{\chi }\cong {\mathrm{Hom}}_W(V_{\chi },B),$$

$$V_{\chi }\cong {\mathrm{Hom}}_{{\mathcal{D}}_c}(N_{\chi },B),$$

$${\mathrm{Hom}}_W(V_{\lambda },B)=⨁m_{\chi }{\mathrm{Hom}}_G(V_{\lambda },B^G\otimes V_{\chi })\ne \{0\}$$

$$B=\underset{\chi \in \hat{W}}{⨁}{N}_{\chi }\otimes V_{\chi }.$$

$$\text{mbox}(T){\mathcal{D}}_c={\mathcal{D}}_c^{-}\oplus {\mathcal{D}}_c^0\oplus {\mathcal{D}}_c^{+}$$

$$\mathcal{L}:{\mathrm{Mod}}_{{\mathcal{D}}_c}(M)$$

$$N$$

$$\delta :{\mathrm{Mod}}_{{\mathcal{R}}_c}(\mathcal{L}(M))$$

$$V$$

Full text

Decomposition of the polynomials over the spherical subalgebra

Ibrahim Nonkané and Jean Kaboré

Departement d’économie et de mathématiques appliquées, IUFIC, Université Thomas Sankara, Burkina faso

nonkane_ibrahim@yahoo.fr, ibrahim.nonkane@uts.bf, kaborejean7750@gmail.com

1. Introduction

Given a finite subgroup $W\subset\operatorname{GL}(\mathfrak{h})$ of the linear group of a finite-dimensional complex vector field $\mathfrak{h}$, it is a well-studied problem to describe the structure of the symmetric algebra $B=\operatorname{sym}(\mathfrak{h}^{*})$ as a representation of $G$, and also as a module over the ring of invariant differential operators under $W$ in the ring $\mathcal{D}(\mathfrak{h})$ of differential operators on $\mathfrak{h}$. Since the rational Cherednik algebra $H_{c}(W,\mathfrak{h})$ and the spherical algebra $eH_{c}e$ are respectively universal deformations of the ring $\mathcal{D}(\mathfrak{h})$ and the ring $\mathcal{D}(\mathfrak{h})^{W}$of $W$-invariant differential operators, we would like to build an analogy between the decomposition of modules over the invariant differential operators in [10, 11, 12] and the decomposition of modules over the sperical subalgebra of the rational Cherednik algebra. The ring $\mathcal{D}_{c}=eH_{c}e$ inherits the natural grading of $B$, and we let $\mathcal{D}_{c}^{0}\subset\mathcal{D}_{c}$ and $\mathcal{D}_{c}^{-}\subset\mathcal{D}_{c}$ be subset of elements of degree 0 end strictly negative degree, respectively. Our main result is that there is for all finite reflection groups a lowest weight description of the category of $\mathcal{D}_{c}$-modules of $B$ where the ring ${{\cal R}}_{c}=\mathcal{D}_{c}^{0}/\mathcal{D}_{c}^{0}\cap\mathcal{D}_{c}\mathcal{D}_{c}^{-}$ plays the very important role of Cartan algebra.

Theorem 1.1**.**

The decomposition of $\mathbb{C}[{\mathfrak{h}}]$ as a $\mathcal{D}_{c}$-module is the same as the decomposition of ${\operatorname{Hom}}(\displaystyle\frac{\mathcal{D}_{c}}{\mathcal{D}_{c}\mathcal{D}_{c}^{-}},\mathbb{C}[\mathfrak{h}])$ as a ${{\cal R}}_{c}$-module.

2. Preliminaries

2.1. The rational Cherednik algebra

Let $W$ be a finite reflection group in a complex vector space $\mathfrak{h}$, and let $R\subset\mathfrak{h}^{*}$ be the corresponding set of roots. To each $W$-invariant function $c:R\to\mathbb{C},c\mapsto c_{\alpha},$ one attaches an associative algebra $H_{c}$, called the rational Cherednik algebra. Given $\alpha\in\mathfrak{h}^{*},$ write $\alpha^{\vee}\in\mathfrak{h}$ for the coroot, and $s_{\alpha}\in\operatorname{Gl}(\mathfrak{h})$ for the reflection corresponding to $\alpha.$ The rational Cherednik algebra $H_{c}$(which was denoted $H_{1,c}$ in [6]) is generated by the vector spaces $\mathfrak{h},\mathfrak{h}^{*},$ and the group algebra of $W,$ with defining relations given by

[TABLE]

[TABLE]

[TABLE]

Thus, the elements $x\in\mathfrak{h}^{*}$ generate a subalgebra $\mathbb{C}[\mathfrak{h}]\subset H_{c}$ of polynomial functions on $\mathfrak{h}$, the elements $y\in\mathfrak{h}$ generate a subalgebra $\mathbb{C}[\mathfrak{h}^{*}]\subset H_{c}$, and the elements $w\in W$ span a copy of the group algebra $\mathbb{C}[W]$ sitting naturally inside $H_{c}$. A basis of $H_{c}$ is given by the set $\displaystyle\{p\cdot q\cdot w:p$ is an ordered monomial in the $x_{i},\ q$ is an ordered monomial in the $y_{i},\ w\in W\}.$ In other words, multiplication on $H_{c}$ induces a vector space isomorphism (the PWB theorem see [6]):

[TABLE]

The case of the symmetric group $W=S_{n}$ acting on $\mathfrak{h}=\mathbb{C}^{n}$ by permuting coordinates is as follows. The reflections in this case are transpositions. Let $S$ be the set of all transpositions, as they are all conjugate, the function $c:S\to\mathbb{C}$ must be a constant and we identify its value in $\mathbb{C}$. Then the rational Cherednik algebra $H_{c}(S_{n},\mathfrak{h})$ is the quotient of $\mathbb{C}\langle x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\rangle\rtimes\mathbb{C}[S_{n}]$ by the relations

[TABLE]

where $\mathbb{C}\langle E\rangle$ denote the free algebra on a set $E$, and $s_{ij}$ is the transposition of $i$ and $j$.

2.1.1. The grading element

Let $W\subset\operatorname{GL}(\mathfrak{h}).$ Let $S$ be the set of reflections in a reflection group $W$, so $(W,S)$ is a coxeter system. For any reflection $s\in S,$ let $\lambda_{s}$ be the eigenvalue of $s$ on $\alpha_{s}\in\mathfrak{h}^{*}(\mbox{i.e.}\ s\alpha_{s}=\lambda_{s}\alpha_{s}),$ and let $\alpha_{s}^{\vee}\in\mathfrak{h}$ be an eigenvector such that $s\alpha_{s}^{\vee}=\lambda_{s}^{-1}\alpha_{s}^{\vee}.$ We normalize them in such a way that $\langle\alpha_{s},\alpha_{s}^{\vee}\rangle=2.$

Let $c:S\to\mathbb{C},s\mapsto c_{s}$ be a function invariant with respect to conjugation. Let

[TABLE]

where the $y_{i}$’s and $x_{i}$’s form basis of $\mathfrak{h}$ and $\mathfrak{h}^{*}$ respectively. We have

[TABLE]

If $W$ is a real reflection group, it follows that

$\displaystyle\mathbf{h}=\sum_{i}x_{i}y_{i}+\frac{1}{2}\dim\mathfrak{h}-\sum_{s\in S}c_{s}s$. Let $\displaystyle{\mathbf{E}}=-\frac{1}{2}\sum_{i}x_{i}^{2},{\mathbf{F}}=\frac{1}{2}\sum_{i}y_{i}^{2}.$ Then $\displaystyle\mathbf{h}=\frac{1}{2}\sum_{i}(x_{i}y_{i}+y_{i}x_{i})$ and $\{{\mathbf{E}},\mathbf{h},{\mathbf{F}}\}$ forms an $\mathfrak{sl}_{2}$-triple in $H_{c}$.

The eigenspace decomposition of $H_{c}$ under the adjoint action of $\mathbf{h}$ puts a grading on $H_{c}$: $W$ is in degree 0, $\mathfrak{h}^{*}$ is in degree 1 and $\mathfrak{h}$ is in degree -1.

2.2. The Polynomial representation

There is faithful action of $H_{c}$ on the polynomial ring $\mathbb{C}[\mathfrak{h}]$ using the Dunkl-Opdam operators [1]. To avoid confusion with the subring $\mathbb{C}[\mathfrak{h}]\subset H_{c},$ let

[TABLE]

denote the vector space on which this representation will be constructed.

Let $c:S\to\mathbb{C},s\mapsto c_{s}$ be a function invariant with respect to conjugation. For $y\in\mathfrak{h}$ the Dunkl-Opdam operator $D_{y}$ is defined by the formula

[TABLE]

where $\partial_{y}$ denotes the partial derivatives in the direction of $y$. Clearly, $D_{y}\in\mathcal{D}(\mathfrak{h}_{reg})\rtimes\mathbb{C}[W]$, where $\mathfrak{h}_{reg}$ is the set of regular points of $\mathfrak{h}$ and $\mathcal{D}(\mathfrak{h}_{reg})$ denotes the algebra of differential operators on $\mathfrak{h}_{reg}$.

There is an action of $H_{c}(W,\mathfrak{h})$ on $\mathbb{C}[\mathfrak{h}]v_{0}$ given by

[TABLE]

The rational Cherednik algebra comes with a natural embedding $\Phi_{c}:H_{c}\to\mathcal{D}(\mathfrak{h}_{reg})\rtimes\mathbb{C}[W]$ defined by the formula $x\mapsto x,y\mapsto D_{y},w\mapsto w$. This embedding is called the Dunkl operator embedding. Let $\mathbb{D}_{c}$ denote the subalgebra of $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[\mathfrak{h}])$ generated by $x,D_{y},w$, where $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[\mathfrak{h}])$ is the algebra of all endomorphisms on $\mathbb{C}[\mathfrak{h}]$. From now on we will identify the spherical $H_{c}$ with the ring of differential operators $\mathbb{D}_{c}$

2.3. The spherical subalgebra

Let ${\mathbf{e}}=\frac{1}{|W|}\sum_{w\in W}w\in\mathbb{C}[W]$ be the symmetrizer, the subalgebra ${\mathbf{e}}H_{c}{\mathbf{e}}$ of $H_{c}$ is called the spherical algebra.

The elements $\{{\mathbf{E}}{\mathbf{e}},\mathbf{h}{\mathbf{e}},{\mathbf{F}}{\mathbf{e}}\}$ form an $\mathfrak{sl}_{2}$-triple in the algebra ${\mathbf{e}}H_{c}{\mathbf{e}}$; in particular the adjoint $\mathbf{h}{\mathbf{e}}$-action gives a grading on ${\mathbf{e}}H_{c}{\mathbf{e}}$. Note that ${\mathbf{e}}(\mathcal{D}(\mathfrak{h}_{reg})\rtimes\mathbb{C}[W]){\mathbf{e}}=\mathcal{D}(\mathfrak{h}_{reg})^{W}.$ Therefore the restriction of the Dunkl embedding gives the embedding $\Phi_{c}:{\mathbf{e}}H_{c}{\mathbf{e}}\hookrightarrow\mathcal{D}(\mathfrak{h}_{reg})^{W}.$

The natural $\mathbb{C}^{*}$-action on $\mathfrak{h}_{reg}$ by dilations gives a ${\mathbb{Z}}$-grading on $\mathcal{D}(\mathfrak{h}_{reg})^{W}$ given by the weights on the $\mathbb{C}^{*}$-action, also there is a standard filtration on $\mathcal{D}(\mathfrak{h}_{reg})$ by the order of differential operators, let $\mathcal{D}(\mathfrak{h}_{reg})_{-}$ denote the subalgebra in $\mathcal{D}(\mathfrak{h}_{reg})$ spanned by all homogeneous differential operators $u\in\mathcal{D}(\mathfrak{h}_{reg})$ such that weight($u$) + order($u$) $\geq 0.$ Let

[TABLE]

be the Calogero-Moser operator, where $\Delta$ denotes the second-order Laplacian on $\mathfrak{h}$. We let ${\cal C}_{c}$ be the centraliser of $L_{c}$ in the algebra $\mathcal{D}(\mathfrak{h}_{reg})_{-}^{W}=\mathcal{D}(\mathfrak{h}_{reg})^{W}\cap\mathcal{D}(\mathfrak{h}_{reg})_{-},$ and we let $\mathcal{D}_{c}$ denote the subalgebra in $\mathcal{D}(\mathfrak{h}_{reg})^{W}$ generated by ${\cal C}_{c}$ and by $\mathbb{C}[\mathfrak{h}]^{W}$, the subalgebra of $W$-invariant polynomials. The image of the map $\Phi_{c}:{\mathbf{e}}H_{c}{\mathbf{e}}\hookrightarrow\mathcal{D}(\mathfrak{h}_{reg})^{W}$ equals to $\mathcal{D}_{c}$ [2]. From now on we will identify the spherical subalgebra ${\mathbf{e}}H_{c}{\mathbf{e}}$ with the ring of differential operators $\mathcal{D}_{c}$.

2.4. Notation

We will throughout this note assume that we have a finite reflection group $W$ acting faithfully on a finite complex dimensional vector space $\mathfrak{h}$ (contragredient action on the dual $\mathfrak{h}^{*}$ of $\mathfrak{h}$), hence inducing a graded action on the graded symmetric algebras $\mathbb{C}[\mathfrak{h}]=\operatorname{sym}(\mathfrak{h}^{*})$. Let $S$ be the set of reflection of $W$. Given a $W$-invariant function $c:S\to\mathbb{C}$, The rational Cherednik algebra associated to $W$ and $\mathfrak{h}$ is denoted by $\mathbb{D}_{c}=H_{c}(W,\mathfrak{h})$. In fact we will identify the Chedrenik algebra with subalgebra of differential operators under the Dunkl embedding. Therefore let us identify, using an invariant scalar product, $\mathfrak{h}$ and $\mathfrak{h}^{*}$, and let us choose a orthonormal basis $x_{1},\ldots,x_{n}$ in $\mathfrak{h}^{*}$. If $x\in\mathfrak{h}^{*}$ we will write $D_{x}$ for the Dunkl operator relative to the vector in $\mathfrak{h}$ corresponding to $x$ under our identification. A basis of $\mathbb{D}_{c}$ is given by \displaystyle\big{\{}p\cdot q\cdot w\big{\}} where $p$ is an ordered monomial in the $x_{i}$, $q$ is an ordered monomial in the $D_{x_{i}}$ , and $w\in W.$

In other words, the following multiplication map is an isomorphism of vectors spaces:

[TABLE]

where $x=(x_{1},\ldots,x_{n})$ is a basis of $\mathfrak{h}$, $D=(D_{x_{1}},\ldots,D_{x_{n}})$ its dual basis of $\mathfrak{h}^{*}$, and $w\in W.$ Note as a Lie algebra $\mathbb{D}_{c}$ contains the general Lie algebra $\mathfrak{h}\otimes\mathfrak{h}^{*}=\mathfrak{gl}(\mathfrak{h}),$ and this Lie subalgebra contains the corresponding grading element under our identification $\displaystyle\mathbf{h}=\sum_{i}x_{i}D_{x_{i}}+\frac{1}{2}\dim\mathfrak{h}-\sum_{s\in S}\frac{2c_{s}}{1-\lambda_{s}}s$. The adjoint action of $\mathbf{h}$ on $\mathbb{D}_{c}$ gives a decomposition $\mathbb{D}_{c}=\bigoplus_{k}\mathbb{D}_{c}(k)$, where $\mathbb{D}_{c}(k)=\{v\in\mathbb{D}_{c}|\ [\mathbf{h},v]=kv\}$; it gives $\mathbb{D}_{c}$ the same grading as the natural one that is induced by the identification (PWB), placing $\mathfrak{h}^{*}$ in degree -1, $\mathfrak{h}$ in degree 1 and $W$ in degree 0. The spherical subalgebra of $\mathbb{D}_{c}$ will be denoted by $\mathcal{D}_{c}$ and identified with its image under the Dunkl embedding.

2.5. Nabla operators

Let us denote by $\nabla=\sum_{i=1}^{n}x_{i}\partial_{x_{i}}$ the nabla operator, we have that

[TABLE]

where $C$ is a constant. In fact

[TABLE]

Lemma 2.1**.**

let $V$ be a finite-dimensional complex space and $W$ a finite subgroup of $\operatorname{GL}(V)$. Let ${\mathbf{e}}=\frac{1}{|W|}\sum_{w\in W}w\in\mathbb{C}[W]$ denote the symmetrising idempotent of $W.$ Then there is an algebra isomorphism

[TABLE]

defined by $z\mapsto z{\mathbf{e}}.$

Proof.

If $f\in\mathbb{C}[V]$ then {\mathbf{e}}f{\mathbf{e}}=\big{(}\frac{1}{|W|}\sum_{w\in W}w(f)\big{)}{\mathbf{e}}. ∎

Let $\mathfrak{h}=\mathbb{C}$ and $W$ a reflection group acting faithfully on $\mathbb{C}.$ Let $D$ be the unique Dunkl operator up to scaling. The subspace $\mathbb{D}_{c}^{0}$ of degree 0 has a basis $\{x^{i}D^{i}\}_{i\geq 0}$, where we in particular have the canonical element $\nabla=x\partial$. We get the following easy result concerned with $\nabla$.

Lemma 2.2**.**

We have the following facts:

1. (1)

$\mathbb{D}_{c}^{0}=\mathbb{C}[\nabla]\rtimes W.$* In particular there are polynomials $p_{k}\in\mathbb{C}[t]\rtimes W,k\in\mathbb{N},$ such that $x^{k}D^{k}=p_{k}(\nabla).$* 2. (2)

$[\nabla,x^{i}D^{j}]=(i-j)x^{i}D^{j}.$** 3. (3)

$\mathcal{D}_{c}^{0}\cong\mathbb{C}[\nabla]^{W}.$**

Proof.

By induction on $k.$ ∎

2.6. Representation of groups and $\mathcal{D}_{c}$-modules

.

Any $W$-module $\tau$ becomes a $\mathbb{C}[W]\ltimes\mathbb{C}[\mathfrak{h}^{*}]$-module by declaring that $\mathfrak{h}$ acts 0 on it. We may therefore construct an $H_{c}$-module

[TABLE]

This is the Verma module corresponding to $\tau$. Since $H_{c}\cong\mathbb{C}[\mathfrak{h}]\otimes(\mathbb{C}[W]\ltimes\mathbb{C}[\mathfrak{h}^{*}])$ as a right $\mathbb{C}[W]\ltimes\mathbb{C}[\mathfrak{h}^{*}]$-module, we have

[TABLE]

as a $\mathbb{C}[W]\ltimes\mathbb{C}[\mathfrak{h}]$-module.

The group algebra $\mathcal{D}_{c}[W]$ with coefficients in $\mathcal{D}_{c}\cong eH_{c}e$ consists of functions $\sum_{w\in W}P_{w}w:W\to\mathcal{D}_{c},w\mapsto P_{w},$ where the product is

[TABLE]

Then $B=\mathbb{C}[\mathfrak{h}]$ is a $\mathcal{D}_{c}[W]$-module. Recall also that if $M$ is a semisimple module over a ring $R,$ and $N$ is a simple $R$-module, then the isotopic component $M_{N}$ of $M$ associated to $N$ is the sum $\sum N^{\prime}\subset M$ of all $N^{\prime}\subset M$ such that $N^{\prime}\cong N.$ Let $\hat{W}$ denote the set of isomorphism classes of irreducible complex $W$-representations.

Let us first state some general fact about polynomial invariants of finite group, Let $G$ be a finite group and $B^{G}$ the subalgebra of $G$-invariant polynomials and consider $B$ as a $B^{G}$-module

Theorem 2.3** (Chevalley-Maschke).**

$B$* is a free $B^{G}$-module if only if $G$ is a finite complex reflection group. Then*

1. (1)

[TABLE]

where the $V_{\chi}$ run over all pairwise inequivalent irreducibles $G$-modules and $m_{\chi}$ is the multiplicity of $V_{\chi}.$ 2. (2)

$B^{G}$* is a polynomial algebra.*

Proof.

[3, Thm 4.1], [16] ∎

We have an analog result in the following proposition. We recall that $\mathcal{D}_{0}=\mathcal{D}(\mathfrak{h})^{W}$ the ring of invariant differential operators on $\mathfrak{h}.$

Proposition 2.4**.**

If $c$ is such that $B$ is generic as $\mathcal{D}_{c}[W]$-module, we have a decomposition into simple submodules

[TABLE]

where each simple $B_{\chi}$ occurs with multiplicity one.

1. (1)

This decomposition coincides with the decomposition of $B$ into isotopic components either as a representation of $W$ or as a $\mathcal{D}_{c}$-module. 2. (2)

If $B_{\chi}$ is the isotopic component of the irreducible $W$-representation $V_{\chi}$ and of the simple $\mathcal{D}_{c}$-module $N_{\chi}$, respectively, then, as a $\mathcal{D}_{c}$-module,

[TABLE] 3. (3)

In the situation of (2),

[TABLE]

as a $\mathcal{D}_{c}$-module, and

[TABLE]

as a representation of $W.$

Proof.

1. (1-2)

Let $V_{\lambda}\in\hat{W}$, we have

[TABLE]

then from [17, Prop 1.5, Thm 1.6] there corresponds an irreducible $\mathcal{D}_{c}$-module $N_{\lambda}$ such that: If $N_{\lambda}$ is equivalent to $B_{\mu}$ as a $\mathcal{D}_{c}$-module then $\lambda=\mu;$ and as a $(\mathcal{D}_{c},W)$-module $B$ is equal to $\bigoplus_{\chi\in\hat{W}}N_{\chi}\otimes V_{\chi}$, and it follows that as $\mathcal{D}_{c}[W]$-module

[TABLE] 2. (3)

If $c$ is generic we know $\mathcal{D}_{c}$ is simple and generated as an algebra by $\operatorname{sym}(\mathfrak{h})$ and $\operatorname{sym}(\mathfrak{h}^{*})$ [2, Thm 4.6]. Then by [17, Thm 2.6], for every $\lambda\in\hat{W},N_{\chi}$ is an irreducible module of $\mathcal{D}_{c}$ and $B=\bigoplus_{\chi\in\hat{W}}N_{\chi}\otimes V_{\chi},$ as a $\mathcal{D}_{c}[W]$-module.

∎

3. Equivalence between $\mathcal{D}_{c}$-modules and ${{\cal R}}_{c}$-modules

We present our main result, which is about studying $\mathcal{D}_{c}$-modules $M$ by their lowest weight space $\operatorname{Ann}_{\mathcal{D}_{c}^{-}}(M),$ where the latter is a module over ${{\cal R}}_{c}=\mathcal{D}_{c}^{0}/(\mathcal{D}_{c}^{0}\cap(\mathcal{D}_{c}\mathcal{D}_{c}^{-}))$

3.1. Abstract equivalence

We describe the equivalence first in a more general setting. If $M$ is an arbitrary module over a ring $R,$ then $\operatorname{Mod}_{R}(M)$ denotes the category with objects all $R$-submodules of $M$ and as morphisms all $R$-homomorphisms between these modules.

Assume that the element $\mathbf{h}{\mathbf{e}}\in\mathcal{D}_{c}$ has an adjoint action on the $\mathbb{C}$-algebra $\mathcal{D}_{c},v\mapsto[\mathbf{h}{\mathbf{e}},v],$ which is semisimple, and that the semisimple decomposition gives a grading $\mathcal{D}_{c}=\oplus D_{c}(k),$ where $v\in\mathcal{D}_{c}(k)$ if $[\mathbf{h}{\mathbf{e}},v]=kv.$ We make the triangular decomposition

[TABLE]

where $\mathcal{D}_{c}^{-}=\oplus_{k<0}\mathcal{D}_{c}(k),\mathcal{D}_{c}^{0}=\mathcal{D}_{c}(0),$ and $\mathcal{D}_{c}^{+}=\oplus_{k>0}\mathcal{D}_{c}(k)$. Define also the ring ${{\cal R}}_{c}=\mathcal{D}_{c}^{0}/\mathcal{D}_{c}^{0}\cap\mathcal{D}_{c}\mathcal{D}_{c}^{-}.$

Define the functor

[TABLE]

and the map

[TABLE]

Here $\mathcal{D}_{c}/\mathcal{D}_{c}\mathcal{D}_{c}^{-}$ is a $(\mathcal{D}_{c},{{\cal R}}_{c})$-bimodule, so that one gets the adjoints pair of functors $(\mathcal{D}_{c}/\mathcal{D}_{c}\mathcal{D}_{c}^{-}\otimes_{{{\cal R}}_{c}}\cdot,\operatorname{Hom}_{\mathcal{D}_{c}}(\mathcal{D}_{c}/\mathcal{D}_{c}\mathcal{D}_{c}^{-},\cdot),$ while $\delta$ in general does not give a functor on the category $\operatorname{Mod}_{{{\cal R}}_{c}}(\mathscr{L}(M)).$

In what follows by the support of a semi-simple $\mathbb{C}[\mathbf{h}{\mathbf{e}}]$-module we mean the set of nonzero eigenvalues of $\mathbf{h}{\mathbf{e}}.$

Lemma 3.1**.**

Let $V$ be a simple ${{\cal R}}_{c}$-module that is semisimple as $\mathbb{C}[\mathbf{h}{\mathbf{e}}]$-module, and which we also regard as a simple module over the ring $\mathcal{B}=\mathcal{D}_{c}^{0}+\mathcal{D}_{c}^{-}$ by the projection $\mathcal{D}_{c}^{0}\to{{\cal R}}_{c}$ and the trivial action of $\mathcal{D}^{-}.$ Then

1. (1)

The support of $W$ as a $\mathbb{C}[\mathbf{h}{\mathbf{e}}]$-module consists of one element. 2. (2)

$\mathcal{D}_{c}\otimes_{\mathcal{B}}V$* contains a unique maximal submodule.*

Proof.

1. (1)

is clear since ${{\cal R}}_{c}$ preserves any eigenspace of $\mathbf{h}{\mathbf{e}}.$ 2. (2)

Also $\mathbf{h}{\mathbf{e}}$ acts semi-simply on $\mathcal{D}_{c}\otimes_{\mathcal{B}}V$ as a derivation by $\mathbf{h}{\mathbf{e}}(Q\otimes v)=[\mathbf{h}{\mathbf{e}},Q]\otimes v+Q\otimes\mathbf{h}{\mathbf{e}}v.$ Since $V$ is simple, the support of any proper submodule of $\mathcal{D}_{c}\otimes_{\mathcal{B}}V,$ regarded as $\mathbb{C}[\mathbf{h}{\mathbf{e}}]$-module, is disjoint from the support of $V.$ The maximal proper submodule is then the sum of all proper submodules.

∎

Theorem 3.2**.**

Let $M$ be a semisimple $\mathcal{D}_{c}$-module which is semisimple over $\mathbf{h}{\mathbf{e}}$ and satisfying $\delta\circ\mathscr{L}(M)=M.$ Then $\mathscr{L}:\operatorname{Mod}_{\mathcal{D}_{c}}(M)\to\operatorname{Mod}_{{{\cal R}}_{c}}(\mathscr{L}(M))$ defines an isomorphism of categories, with inverse $\delta:\operatorname{Mod}_{{{\cal R}}_{c}}(\mathscr{L}(M))\to\operatorname{Mod}_{\mathcal{D}_{c}}(M).$

Proof.

All direct sums below are internal, and by an $\mathbf{h}{\mathbf{e}}$-isotypical component of $M$ associated to $\lambda$ we intend the subspace of $M$ consisting of elements $m$ such that $\mathbf{h}{\mathbf{e}}\cdot m=\lambda m.$ Let $V$ be a ${{\cal R}}_{c}$-submodule of $\mathscr{L}(M)$ and $N$ a submodule of $M.$

1. (a)

$\delta\circ\mathscr{L}(N)=N:$ if $N$ is a submodule of $M,$ by semisimplicity there exists a module $N_{1}$ such that $M=N\otimes N_{1},$ so that

[TABLE]

and $N=\delta(\mathscr{L}(N)).$ 2. (b)

$\mathscr{L}\circ\delta(V)=V:$ we note that it follows from the decomposition (T), that if $N\subset M$ is a simple $\mathcal{D}_{c}$-module, $\mathscr{L}(N)$ contains only one isotypical component with respect to the action $\mathbf{h}{\mathbf{e}}.$ Assume first that $V$ contains only a single $\mathbf{h}{\mathbf{e}}$-isotypical component, and that $\delta(V)=\oplus_{i\in I}N_{i},$ where $N_{i}$ are simple $\mathcal{D}_{c}$-submodules of $M.$

[TABLE]

Since $\delta(V)=V\oplus\mathcal{D}_{c}^{+}V,$ and $V\subset\mathscr{L}(\delta(V)),$ it is clear that $\mathscr{L}(\delta(V))=V\oplus V^{\prime},$ where $V$ and $V^{\prime}$ have different $\mathbf{h}$-isotypic components. Thus, there is a subset $I^{\prime}\subset I$ such that $V=\oplus_{i\in I}\mathscr{L}(N_{i}).$ Then

[TABLE]

where the second equality follows from(a). Therefore $I=I^{\prime}$ and so $\mathscr{L}\circ\delta(V)$ on the right side of (*) equals $V.$

Assume then $V=V_{1}\oplus V_{2},$ where $V_{1}$ and $V_{2}$ have no common $\mathbf{h}{\mathbf{e}}$-isotypical component, and satisfy that $\mathscr{L}\circ\delta(V_{i})=V_{i},i=1,2.$ Then $\mathscr{L}(\delta(V_{1})\cap\delta(V_{2}))\subset V_{1}\cap V_{2}=\{0\},$ hence by (a) $\delta(V_{1})\cap\delta(V_{2})=\{0\},$ so that

[TABLE]

and, by assumptiom,

[TABLE]

Since any $V$ may be decomposed as a $\mathcal{D}_{c}^{0}$-module into isotopic components for $\mathbf{h}{\mathbf{e}},$ it follows by induction that $\mathscr{L}\circ\delta(V)=V.$ 3. (c)

$\operatorname{Hom}_{\mathcal{D}_{c}}(N_{1},N_{2})=\operatorname{Hom}_{{{\cal R}}_{c}}(\mathscr{L}(N_{1}),\mathscr{L}(N_{2})):$ since $\mathscr{L}$ is additive and the category $\operatorname{Mod}_{\mathcal{D}_{c}}(M)$ is semisimple, it suffice to prove this when $N_{1}$ and $N_{2}$ are simple. If $V_{1}\subset\mathscr{L}(N_{1}),\ V_{1}\neq\{0\},$ then $\delta(V_{1})=N_{1};$ hence by (b) $V_{1}=\mathscr{L}(N_{1});$ therefore $\mathscr{L}(N_{1})$ is simple and for the same reason $\mathscr{L}(N_{2})$ is also simple. It is obvious that an isomorphism $\phi:N_{1}\to N_{2}$ induces an isomorphism $\mathscr{L}(N_{1})\to\mathscr{L}(N_{2}).$ Conversely, let $\psi:\mathscr{L}(N_{1})\to\mathscr{L}(N_{2})$ be a non-zero homomorphism, hence it is an isomorphism. There is a canonical inclusion homomorphism of $\mathcal{D}_{c}^{0}$-modules $f:\mathscr{L}(N_{2})\to N_{2},$ so that we get a map of $\mathcal{D}_{c}^{0}$-modules $f\circ\psi:\mathscr{L}(N_{1})\to N_{1}.$ Hence we get a non-zero homomorphism of $\mathcal{D}_{c}$-modules $h:\mathcal{D}_{c}\otimes_{\mathcal{B}}\mathscr{L}(N_{1})\to N_{2},$ which is surjective since $N_{2}$ is simple by Schur’s Lemma. We moreover have a surjective map $\mathcal{D}_{c}\otimes_{\mathcal{B}}\mathscr{L}(N_{1})\to N_{1}.$ Since $\mathscr{L}(N_{1})$ is simple, by Lemma 3.1 $\mathcal{D}_{c}\otimes_{\mathcal{B}}\mathscr{L}(N_{1})$ has a unique maximal proper submodule. Therefore we get a unique isomorphism $N_{1}\to N_{2}$ that extends $\psi.$ 4. (d)

$\operatorname{Hom}_{{{\cal R}}_{c}}(V_{1},V_{2})=\operatorname{Hom}_{\mathcal{D}_{c}}(\delta(V_{1}),\delta(V_{2})):$ putting $N_{i}=\delta(V_{i})$ we have by (b) that $\mathscr{L}(N_{i})=V_{i},\ i=1,2.$ Hence by (c):

[TABLE]

∎

The semi-simple adjoint action of the Euler deformed operator $\mathbf{h}{\mathbf{e}}$ induces a $\mathbb{Z}$-grading, we have the decomposition (T), and by Proposition 2.4 $B$ is semisimple $\mathcal{D}_{c}$-module. If $N\subset B$ a simple $\mathcal{D}_{c}$-submodule, the vector space $N^{a}$ of lowest degree homogeneous elements in $N$ will be annihilated by $\mathcal{D}_{c}^{-},$ hence $N^{a}\subset\mathscr{L}(B),$ so that $N=D_{c}N^{a}\subset\mathcal{D}_{c}\mathscr{L}(B).$ Since this is true for any simple submodule of the semisimple module $B,$ we have $B=\mathcal{D}_{c}\mathscr{L}(B).$ Hence the condition of Theorem 3.2 are obtained for $\mathcal{D}_{c}$ and $M=B.$ We can therefore immediately conclude most of the following result.

Corollary 3.3**.**

Suppose that $W$ is a finite reflection group acting faithfully on $B=\operatorname{sym}(\mathfrak{h}^{*})$ and $H_{c}$. Put $\mathcal{D}_{c}={\mathbf{e}}H_{c}{\mathbf{e}}$ and ${{\cal R}}_{c}=\mathcal{D}_{c}^{0}/\mathcal{D}_{c}^{0}\cap\mathcal{D}_{c}\mathcal{D}_{c}^{-}$

1. (1)

$\mathscr{L}(B)$ is a finite-dimensional semisimple ${{\cal R}}_{c}$-module. 2. (2)

There is an isomorphism of categories between the category of $\mathcal{D}_{c}$-submodules of $B$ and the category of ${{\cal R}}_{c}$-submodules of $\mathscr{L}(B)$, and its inverse is $V\mapsto\mathcal{D}_{c}V,$ where $V$ is an ${{\cal R}}_{c}$-submodule of $\mathscr{L}(B).$ 3. (3)

Simple $\mathcal{D}_{c}$-submodules of $B$ correspond to simple ${{\cal R}}_{c}$-submodules of $\mathscr{L}(M).$ 4. (4)

Each simple ${{\cal R}}_{c}$-submodule of $\mathscr{L}(B)$ is concentrated in a single degree.

Proof.

We know that $\dim_{\mathbb{C}}(V_{\chi})<\infty$ and by Proposition 2.4 we have

[TABLE]

since $N_{\chi}$ lies in one degree then $\dim_{\mathbb{C}}\mbox{Ann}_{\mathcal{D}_{c}^{-}}(N_{\chi})<\infty$,

hence $\dim_{\mathbb{C}}(\mathscr{L}(B))<\infty.$ ∎

Example 3.4**.**

1. (1)

The case $n=2$ may be related to the 1-dimensional case ($\mathfrak{h}=\mathbb{C}$) by restricting functions $\mathbb{C}^{2}\to\mathbb{C}$ to the space $x_{1}+x_{2}=0$. Take $W=\langle s\rangle$ acting on $\mathfrak{h}=\mathbb{C}\xi$ and put $\mathfrak{h}^{*}=\mathbb{C}x$ where $\langle\xi,x\rangle=1$. Then $H_{c}$ is the $\mathbb{C}[c]$-algebra with generators $s,x,\xi$ and relations

[TABLE]

We have $e=\frac{1}{2}(1+s),$ the spherical algebra $eH_{c}e$ has a basis $\displaystyle(ex^{i}\xi^{j})_{{i,j\geq 0},{i+j\ {even}}}$. In fact one has

[TABLE]

and $ex^{i}\xi^{j}e=\pm ex^{i}\xi^{j}e.$ $\mathcal{D}_{c}$ is generated as a $\mathbb{C}[c]$-algebra by $u=\frac{1}{2}ex^{2}e,v=-\frac{1}{2}e\xi^{2}e$ and $w=ex\xi e.$ The decomposition of $\mathbb{C}[\mathfrak{h}]$ as a $\mathcal{D}_{c}$-module is given by:

[TABLE] 2. (2)

We may also consider $S_{2}$ acting on $\mathfrak{h}=\mathbb{C}^{2}$ by permuting coordinates.

[TABLE] 3. (3)

For $n=3$, let $\mathfrak{h}=\bigoplus\mathbb{C}\xi_{i}$ and $\mathfrak{h}^{*}=\bigoplus\mathbb{C}x_{i}$ with the dual basis. By the Dunkl operator embedding [5], we may consider the rational Cherednik algebra as a subalgebra of $\mathcal{D}(\mathfrak{h}_{\mbox{reg}})\rtimes\mathbb{C}[S_{3}]$ where $\mathcal{D}(\mathfrak{h}_{\mbox{reg}})$ is the algebra of differential operators on $\mathfrak{h}_{\mbox{reg}}=\{(\xi_{1},\xi_{2},\xi_{3})\in\mathfrak{h}\ |\ \xi_{i}\neq\xi_{j}\ \mbox{for}\ i\neq j\}$. Then $H_{c}$ is generated by $x_{i}\in\mathfrak{h}^{*},g\in S_{3}$ and the Dunkl operator associated to $\xi_{i}$;

[TABLE]

In the same way, the spherical algebra $\mathcal{D}_{c}=eH_{c}e$ is embedded in the algebra of invariant differential operators $\mathcal{D}(\mathfrak{h}_{\mbox{reg}})^{S_{3}}$ and is generated by $\mathbb{C}[\mathfrak{h}]^{S_{3}}$, and the invariant differential operators $L_{1},L_{2},$ and $L_{3}$ (see [6], [14], [2, Prop 4.10]) where $L_{1}=\sum\partial_{\xi_{i}},$ $L_{2}=\sum\partial_{\xi_{i}}^{2}-c\sum_{i\neq j}\frac{1}{(x_{i}-x_{j})^{2}},L_{3}=\sum\partial_{\xi_{i}}^{3}+\mbox{lower order terms}.$ We claim that

[TABLE]

3.2. Computation of $\mathcal{D}_{c}^{-},\mathcal{D}_{c}^{0}$ and ${{\cal R}}_{c}$

3.2.1. General procedure using basic invariants

Assume that $\{f_{i}\},\{g_{i}\}$ are homogeneous generators of $\operatorname{sym}(\mathfrak{h})^{W}$ and $\operatorname{sym}(\mathfrak{h}^{*})^{W}$, respectively, where one observe that $\{g_{i}\}\subset\mathcal{D}_{c}^{-}.$ Let $\mathfrak{a}$ be a graded Lie subalgebra of $\mathcal{D}_{c}$ which contains the Lie algebra $\langle f_{i},g_{i}\rangle$ that is generated by $\{f_{i}\}\cup\{g_{i}\}.$

Letting ${\cal U}(\mathfrak{a})$ be the envelopping algebra of $\mathfrak{a}$ we have a canonical homomorphism

[TABLE]

Proposition 3.5**.**

1. (1)

The homomorphism $j$ is surjective. 2. (2)

Let $\{r_{k}\}$ be a homogeneous basis of $\mathfrak{a}$ and $\mathfrak{a}^{-}=\sum_{\deg(r_{k})<0}\mathbb{C}r_{k}$ be the set of elements of negative degree. Then

$\mathfrak{a}^{-}\subset\mathcal{D}_{c}\subset\mathcal{D}_{c}\mathfrak{a}^{-}$. 3. (3)

Let $\mathfrak{a}^{0}$ be the subalgebra of degree 0 in $\mathfrak{a}$. Then we have a surjective homomorphism

[TABLE]

In particular, if $\mathfrak{a}^{0}$ is commutative, then ${{\cal R}}_{c}$ is commutative.

It follows from (2) that $\mathscr{L}(M)=\operatorname{Ann}_{\mathfrak{a}^{-}}(M)$ and from (3) that the ${{\cal R}}_{c}$-module structure of $\mathscr{L}(M)$ comes from its structure of $\mathfrak{a}^{0}$-module.

Proof.

1. (1)

: This follows from the generelization of Levasseur-Stafford for $eH_{c}e$ [2, Thm 4.6] and [4, Thm 3.21], stating that the sets $\{f_{i}\},\{g_{i}\}$ together generate $\mathcal{D}_{c}.$ 2. (2-3)

: Provide the homogeneous basis $\{r_{k}\}$ with a total ordering that is compatible with the degrees in the sense that $\deg(r_{i})\leq\deg(r_{j}),$ when $i\geq j.$ It follows from the PWB Theorem, using the homomorphism above, that any element $P\in\mathcal{D}_{c}$ can be expressed (non-uniquely) in the form

[TABLE]

where $i_{1}\leq\cdots\leq i_{r}.$ Hence the factors in each term $r_{i_{1}}\cdots r_{i_{s}}$ have descending degree $\deg(r_{i_{1}})\geq\cdots\geq\deg(r_{i_{s}}).$ If $P\in\mathcal{D}_{c}^{-}$ then the last factor in each term has $\deg(r_{i_{s}})<0$ and so $P\in\mathcal{D}_{c}\mathfrak{a}^{-}.$ This gives (2). Furthermore, when $P\in\mathcal{D}^{0}$ we can write

[TABLE]

where $\deg(r_{i_{1}})=\cdots=\deg(r_{i_{s}})=0$ and $\alpha_{i_{1}\ldots i_{s}}\in\mathbb{C}.$ This gives (3).

∎

3.2.2. ${{\cal R}}_{c}$ for $G(m,e,n)$

As an example we will consider the irreducible imprimitive complex reflections groups $G=G(m,e,n)$, where $e$ and $m$ are positive integers such that $e|m,$ and determine generators of ${{\cal R}}_{c}$ when $e=1.$ Let $\mathfrak{h}$ be a complex vector space dimension $n$. Then

[TABLE]

where $S_{n}$ is realized as permutation matrices and $A(m,e,n)$ as diagonal matrices whose entries belong to $\mu_{m},$ the group of $m$-roots of unity, such that their determinant belongs to $\mu_{d}\mu_{m},$ where $d=\frac{m}{e},$ and in the semi-direct product $S_{n}$ acts on $A(m,e,n)$ by permutation. This means that $G$ can be realized as permutation matrices with entries in $\mu_{m},$ such that the product of non-zero entries belongs to $\mu_{d}\subset\mu_{m};$ see [3].

Put $\bar{A}=A(m,1,n),A=A(m,e,n)$ and $\bar{G}=\bar{A}\rtimes S_{n}=G(m,1,n)$ so that $A\subset\bar{A}$ and $G\subset\bar{G}.$ Let $\mathcal{D}_{c}(G)={\mathbf{e}}H_{c}(\mathfrak{h},G){\mathbf{e}}$ be the spherical algebra attached to the group $G$ , $\operatorname{sym}{\mathfrak{h}^{*}}=\mathbb{C}[x_{1},\ldots,x_{n}],$ and $\operatorname{sym}{\mathfrak{h}}=\mathbb{C}[\xi_{1},\ldots,\xi_{n}]$. Define the $S_{n}$-invariant element $\Theta=(\prod_{i=1}^{n}x_{i})^{d},\Psi=(\prod_{i=1}^{n}\xi_{i})^{d},$ and $h_{i}(x)=\sum_{j=1}^{n}x_{j}^{i}.$

Proposition 3.6**.**

1. (1)

[TABLE]

where

[TABLE] 2. (2)

[TABLE]

Proof.

1. (1)

: This is of course well known, but let us at least sketch the argument. We have (as detailed below)

[TABLE]

The second equality can be seen by first noting that if a polynomial is $A(m,e,n)$-invariant, then each of its monomial terms is invariant, and these are exactly given by powers of the monomials $x_{1}^{m},\ldots,x_{n}^{m},(x_{1}\cdots x_{n})^{d}.$ The $n$ monomials $x_{i}^{m}$ are algebraically independent, while the last is $S_{n}$-invariant and algebraically dependent on the other ones. Therefore the second equality on the second line follows from the well-known fact that $\{f_{i}\}_{i=1}^{n}$ is an algebraically independent set of generators of $\mathbb{C}[x_{1}^{m},\ldots,x_{n}^{m}]^{S_{n}}.$ When $e>1,$ then $f_{n}\in\mathbb{C}[f_{1},\ldots,f_{n-1},\Theta],$ while if $e=1$ ( so that $m=d),$ then $\Theta\in\mathbb{C}[f_{1},\ldots,f_{n}],$ so that $\operatorname{sym}(\mathfrak{h}^{*})=\mathbb{C}[f_{1},\ldots,f_{n}].$ 2. (2)

: This follows from the theorem of Levasseur and Stafford for the Spherical algebra [2, Thm 4.6].

∎

Proposition 3.6 means that we have good control of the invariants $\{f_{i}\}$ for the ring $\operatorname{sym}(\mathfrak{h}^{*})^{G(m,e,n)}$ which are needed in Proposition 3.5. However, it is only in the case $e=1$ that we obtain a really useful description of the Lie subalgebra $\mathfrak{a}^{\prime}=Lie\langle f_{i}(x),f_{i}(\xi)\rangle$ of $\mathcal{D}_{c}(\bar{G})$, using the basic invariants $f_{i}(x)=h_{i}(x_{1}^{m},\ldots,x_{n}^{m})$ for $\operatorname{sym}(\mathfrak{h}^{*})$ and $f_{i}(\xi)$ its isomorphic ring $\operatorname{sym}(\mathfrak{h}).$

Acknowledgments

I am deeply thankful to Professor Rikard Bøgvad for instructive comments during the writing of this note.