# Two invariant subalgebras of rational Cherednik algebras

Gwyn Bellamy  ,  Misha Feigin  and  Niall Hird  School of Mathematics and
Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Originally motivated by connections to integrable systems, two natural
subalgebras of the rational Cherednik algebra have been considered in the
literature. The first is the subalgebra generated by all degree zero elements
and the second is the Dunkl angular momentum subalgebra.

In this article, we study the ring-theoretic and homological properties of
these algebras. Our approach is to realise them as rings of invariants under
the action of certain reductive subgroups of $\mathrm{SL}_{2}$. This allows us
to describe their centres. Moreover, we show that they are
Auslander–Gorenstein and Cohen–Macaulay and, at $t=0$, give rise to prime PI-
algebras whose PI-degree we compute.

Since the degree zero subalgebra can be realized as the ring of invariants for
the maximal torus $\mathrm{T}\subset\mathrm{SL}_{2}$ and the action of this
torus on the rational Cherednik algebra is Hamiltonian, we also consider its
(quantum) Hamiltonian reduction with respect to $\mathrm{T}$. At $t=1$, this
is a filtered quantization of the quotient of the minimal nilpotent orbit
closure $\overline{\mathcal{O}}_{\min}$ in $\mathfrak{gl}(n)$ by the
reflection group $W$. At $t=0$, we get a graded Poisson deformation of the
symplectic singularity $\overline{\mathcal{O}}_{\min}/W$.

##  1\. Introduction

One key motivation for the introduction of rational Cherednik algebras by
Etingof and Ginzburg [18] was their connection with the generalized
Calogero–Moser integrable systems. For instance, the algebras can be used to
give a uniform proof of the complete integrability of these systems [27]. In
the case of a real reflection group, the second author together with Hakobyan
introduced in [19] the Dunkl angular momentum algebra
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ as the subalgebra of a rational
Cherednik algebra $H_{t,\mathbf{c}}=H_{t,\mathbf{c}}(W)$ generated by the
(Dunkl deformed) angular momentum operators $x_{i}y_{j}-x_{j}y_{i}$ and the
reflection group $W$. Part of the motivation for introducing this algebra is
the fact that it contains in its centre essentially the angular generalized
Calogero–Moser Hamiltonian. This makes the angular part of generalized
Calogero–Moser systems amenable to study via representation theory. At the
same time, they also introduced the closely related degree zero subalgebra
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$, generated by the elements $x_{i}y_{j}$
and the reflection group. This algebra, in turn, is related to Calogero–Moser
systems in the harmonic confinement.

Another interesting context where the algebras
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ and
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ arise is the deformed Howe duality. Thus
the algebra $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ plays a role in the Dunkl
deformed Howe dual pair $(\mathfrak{so}(n),\mathfrak{sl}(2))$ [14] while the
algebra $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ relates to the Dunkl deformation
of $(\mathfrak{gl}(n),\mathfrak{gl}(1))$ duality [21].

In this article, we study the ring-theoretic and homological properties of
these two subalgebras. Our primary focus is on understanding their properties
in the case $t=0$. Since the algebras are defined in terms of a set of
generators, it can be difficult to prove directly any of their abstract (e.g.
homological) properties. Our key observation is that both of these algebras
can be realised as rings of invariants under certain groups of automorphisms
of the rational Cherednik algebra $H_{t,\mathbf{c}}$. Since this is obvious in
the case of $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$, we will assume for the
remainder of the introduction that $W$ is an irreducible real reflection
group. In this case, $\mathrm{SL}_{2}$ acts as a group of algebra
automorphisms on $H_{t,\mathbf{c}}$. Let $\mathrm{T}$ be the maximal torus of
diagonal matrices in $\mathrm{SL}_{2}$ so that
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}=H_{t,\mathbf{c}}^{\mathrm{T}}$. We show
in Theorem 4.9 that
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}=H_{t,\mathbf{c}}^{\mathrm{SL}_{2}}$.
Realizing both algebras as rings of invariants under a connected reductive
group $\Gamma\subset\mathrm{SL}_{2}$ allows us to give a remarkably uniform
treatment of these algebras; this we do in Sections 6 and 7.

### The centre of the algebras

Our original motivation was to give an explicit description of the centre of
the algebras $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ and
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ when $t=0$. Let
$P=\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{*}]$ and recall that $H_{0,0}$
is the skew group ring $P\rtimes W$. As above, $\Gamma$ can be either
$\mathrm{T}$ or $\mathrm{SL}_{2}$. Let $Z(A)$ be the centre of an algebra $A$.

######  Theorem.

(7.6) For any $\mathbf{c}$,
$\operatorname{\mathrm{gr}}Z(H_{0,\mathbf{c}}^{\Gamma})=Z(P^{\Gamma}\rtimes
W)$.

This theorem, together with the simple fact that if $A\subset B$ and
$\operatorname{\mathrm{gr}}A=\operatorname{\mathrm{gr}}B$ then $A=B$, leads to
the following corollary.

######  Corollary.

(7.7) There is equality of commutative rings

$Z(H_{0,\mathbf{c}}^{\Gamma})=Z(H_{0,\mathbf{c}})^{\Gamma}\otimes\mathbb{C}Z(W),$

where $\mathbb{C}Z(W)$ is the group algebra of the centre $Z(W)$ of the group
$W$. In particular, as a $Z(H_{0,\mathbf{c}})^{\Gamma}$-module,

$Z(H_{0,\mathbf{c}}^{\Gamma})=\bigoplus_{i=0}^{\ell-1}Z(H_{0,\mathbf{c}})^{\Gamma}\mathbf{a}_{i}$

where $\mathbf{a}_{0},\dots,\mathbf{a}_{\ell-1}$ are the primitive idempotents
in $\mathbb{C}Z(W)$.

If we set $\mathbf{a}=\mathbf{a}_{0}$ to be the symmetrising idempotent then
it follows that the centre of $\mathbf{a}H_{0,\mathbf{c}}^{\Gamma}\mathbf{a}$
equals $\mathbf{a}Z(H_{0,\mathbf{c}}^{\Gamma})\cong
Z(H_{0,\mathbf{c}})^{\Gamma}$. In fact, each summand
$Z(H_{0,\mathbf{c}})^{\Gamma}\mathbf{a}_{i}$ of $Z(H_{0,\mathbf{c}}^{\Gamma})$
is isomorphic to $Z(H_{0,\mathbf{c}})^{\Gamma}$. Being a ring of invariants,
the latter is very well-behaved (despite being singular) as illustrated by the
following result.

######  Theorem.

(7.5) The affine scheme
$Y_{\mathbf{c}}:=\operatorname{\mathrm{Spec}}Z(H_{0,\mathbf{c}})^{\Gamma}$ is
a normal irreducible variety. It is Gorenstein with rational singularities.

### Ring-theoretic and homological properties

Since the idempotents $\mathbf{a}_{i}$ are central in
$H_{t,\mathbf{c}}^{\Gamma}$, there is a decomposition of algebras
$H_{t,\mathbf{c}}^{\Gamma}=\bigoplus_{i=0}^{\ell-1}\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$.
If $\overline{W}$ is the quotient $W/Z(W)$ then associated to each idempotent
$\mathbf{a}_{i}$ is a twisted group algebra $\mathbb{C}_{f_{i}}\overline{W}$
and we show that the associated graded of
$\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ equals
$P^{\Gamma}\rtimes\mathbb{C}_{f_{i}}\overline{W}$. This implies that each of
the rings $\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ is prime.
Our main result on the general properties of these algebras is summarized
below.

######  Theorem.

(6.14) The algebras $H_{t,\mathbf{c}}^{\Gamma}$ and
$\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ are
Auslander–Gorenstein and (GK) Cohen–Macaulay. These algebras have
Gelfand–Kirillov dimension $2\dim\mathfrak{h}-\dim\Gamma$.

Our main tool is to compute the associated graded of these algebras, and their
centres and lift ring-theoretic and homological properties from the associated
graded. Our approach follows closely that of Brown and Changtong [9].

When $t=0$, Corollary 7.7 implies that both $H_{0,\mathbf{c}}^{\Gamma}$ and
$\mathbf{a}H_{0,\mathbf{c}}^{\Gamma}\mathbf{a}$ are finite modules over their
respective centres. In particular, they are Polynomial Identity (PI) rings. We
study the simple modules over the prime PI ring
$\mathbf{a}H_{0,\mathbf{c}}^{\Gamma}\mathbf{a}$. Our main result is summarized
as follows.

######  Theorem.

(7.12) For all parameters $\mathbf{c}$,

  1. (i)

$H_{0,\mathbf{c}}^{\Gamma}$ and
$\mathbf{a}H_{0,\mathbf{c}}^{\Gamma}\mathbf{a}$ are Polynomial Identity (PI)
algebras and the PI degree of the prime PI-algebra
$\mathbf{a}H_{0,\mathbf{c}}^{\Gamma}\mathbf{a}$ equals $|\overline{W}|$.

  2. (ii)

Let $L$ be a simple $\mathbf{a}H_{0,\mathbf{c}}^{\Gamma}\mathbf{a}$-module
whose support is contained in the regular locus of $Y_{\mathbf{c}}$. Then
$L|_{\overline{W}}\cong\mathbb{C}\overline{W}$.

  3. (iii)

The regular locus of $Y_{\mathbf{c}}$ is contained in the Azumaya locus.

We do not know if the two loci in Theorem 7.12(iii) are actually equal.

We also examine the centre of the degree zero subalgebra for $t\neq 0$. Since
the centre of the rational Cherednik algebra is trivial in this case, one
would expect that the centre of $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ is also
small when $t\neq 0$. This is indeed the case (cf. [19], where one central
generator was identified in the case of real reflection groups).

######  Proposition.

(6.20) For $t\neq 0$, the centre of $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$
equals $\mathbb{C}[\mathrm{eu}_{\mathbf{c}}]\otimes\mathbb{C}Z(W)$, where
$\mathrm{eu}_{\mathbf{c}}\in H_{t,\mathbf{c}}$ is the Euler element.

### (Quantum) Hamiltonian reduction

The action of $\mathrm{T}$ on the generalized Calogero–Moser space
$\operatorname{\mathrm{Spec}}Z(H_{0,\mathbf{c}})$ is Hamiltonian and we have
seen that the centre of $H_{0,\mathbf{c}}^{\Gamma}$ is essentially given by
the ring of invariants $Z(H_{0,\mathbf{c}})^{\Gamma}$. In applications to
symplectic singularities, it is natural to consider instead the Hamiltonian
reduction of $\operatorname{\mathrm{Spec}}Z(H_{0,\mathbf{c}})$ with respect to
$\mathrm{T}$. This is done in Section 8. For
$\zeta\in\mathbb{C}=(\mathrm{Lie}\,\mathrm{T})^{*}$, we define the algebra of
quantum Hamiltonian reduction to be the quotient

$A_{t,\mathbf{c},\zeta}=A_{t,\mathbf{c},\zeta}(W)=\mathbf{a}H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}\mathbf{a}/\langle\mathbf{a}\mathrm{eu}_{\mathbf{c}}-\zeta\rangle,$

whereas the Hamiltonian reduction of
$\operatorname{\mathrm{Spec}}Z(H_{0,\mathbf{c}})$ is

$\mu^{-1}(\zeta)/\\!\\!/\\!\,\mathrm{T}=\operatorname{\mathrm{Spec}}\left(Z(H_{0,\mathbf{c}})^{\mathrm{T}}/\langle\mathrm{eu}_{\mathbf{c}}-\zeta\rangle\right),$

where $\mu$ is the moment map. As a consequence of the fact that the double
centralizer property holds for the
$(A_{t,\mathbf{c},\zeta},eA_{t,\mathbf{c},\zeta}e)$-bimodule
$A_{t,\mathbf{c},\zeta}e$, where $e=\frac{1}{|W|}\sum_{w\in W}w$ (see Theorem
8.11), we show that:

######  Corollary.

(8.14) The centre of $A_{0,\mathbf{c},\zeta}$ is isomorphic to
$Z(H_{0,\mathbf{c}})^{\mathrm{T}}/\langle\mathrm{eu}_{\mathbf{c}}-\zeta\rangle$.

Rational Cherednik algebras provide a means to study Poisson deformations of
the symplectic quotient singularity $(\mathfrak{h}\times\mathfrak{h}^{*})/W$
because the centre of $H_{0,0}$ can be identified with the ring of functions
on this quotient. Let $\mathcal{O}_{\mathrm{min}}\subset\mathfrak{gl}(n)$ be
the minimal nilpotent orbit with respect to the adjoint action of
$\operatorname{\mathrm{GL}}(n)$. The above corollary implies that the centre
of $A_{0,0,0}$ can be identified with the ring of functions on the symplectic
singularity $\overline{\mathcal{O}}_{\mathrm{min}}/W$ and hence
$A_{t,\mathbf{c},\zeta}$ can be used to study Poisson deformations of this
singularity. Our first result in this direction is the following.

######  Theorem.

(8.9) For $t\neq 0$, the algebra $eA_{t,\mathbf{c},\zeta}e$ is a filtered
quantization of $\overline{\mathcal{O}}_{\mathrm{min}}/W$.

The algebra $A_{t,\mathbf{c},\zeta}$ can also be identified with a summand of
the global sections of a sheaf of Cherednik algebras on the projective space
$\mathbb{P}(\mathfrak{h})$, as studied in [4]. Since the latter sheaf of
algebras has finite global dimension and localization holds for Weil generic
parameters, we deduce that:

######  Theorem.

(8.10) For Weil generic $(t,\mathbf{c},\zeta)$, $A_{t,\mathbf{c},\zeta}$ has
finite global dimension.

At $t=0$, we can consider the commutative algebras $Z(A_{0,\mathbf{c},\zeta})$
as forming a flat family $Z(A_{0})\to\mathfrak{c}$ over the parameter space
$\mathfrak{c}$ of all pairs $(\mathbf{c},\zeta)$. As one would hope, this
gives rise to Poisson deformations of the symplectic singularity
$\overline{\mathcal{O}}_{min}/W$.

######  Theorem.

(8.16) $\operatorname{\mathrm{Spec}}Z(A_{0})\to\mathfrak{c}$ is a graded
Poisson deformation of the symplectic singularity
$\overline{\mathcal{O}}_{min}/W$.

This result should provide a representation-theoretic way to enumerate the
projective $\mathbb{Q}$-factorial terminalizations of the singularity
$\overline{\mathcal{O}}_{min}/W$.

### Outline of the article.

Section 2 contains some basic results on graded and filtered algebras that we
will use later. Then Section 3 recalls the definition of rational Cherednik
algebras and outlines their basic properties. In Section 4, we begin the study
of the two subalgebras. We give explicit presentations of these algebras and
show that they are rings of invariants. The behaviour of
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ is exceptional when $n=2$ and we study
this case separately in Section 5, giving an explicit description of the
centre of the algebra (which does not depend on the parameters
$t,\mathbf{c}$). In Section 6 we study the properties of the invariant rings
$H_{t,\mathbf{c}}^{\Gamma}$. The case $t=0$ is considered in detail in Section
7. Finally, in Section 8 we study the quantum Hamiltonian reduction of
$H_{t,\mathbf{c}}$ with respect to the action of $\mathrm{T}$.

### Conventions

We work throughout over the complex numbers $\mathbb{C}$. Unadorned tensor
product will mean $\otimes_{\mathbb{C}}$.

### Acknowledgements

The first author was partially supported by a Research Project Grant from the
Leverhulme Trust and by the Engineering and Physical Sciences Research Council
grants EP/W013053/1 and EP/R034826/1. The second author was partially
supported by the Engineering and Physical Sciences Research Council grant
EP/W013053/1.

##  2\. Filtered and graded algebras

In this section, we recall basic facts about filtered and graded algebras that
will be required later. The centre of an algebra $A$ is denoted $Z(A)$.

###  2.1. Graded algebras

For a $\mathbb{Z}$-graded vector space $M$, $M_{i}$ will denote the degree $i$
part. Let $B$ be a $\mathbb{C}$-algebra. A ($\mathbb{Z}$-)grading on $B$ is a
direct sum decomposition $B=\bigoplus_{i\in\mathbb{Z}}B_{i}$ such that
$B_{i}\cdot B_{j}\subset B_{i+j}$. The fact that $B$ is $\mathbb{Z}$-graded
can be interpreted as an action of $\mathbb{C}^{\times}$ on $B$ by algebra
automorphisms such that

$B_{i}:=\\{b\in B\,|\,\lambda\cdot b=\lambda^{i}b,\
\forall\lambda\in\mathbb{C}^{\times}\\}.$

Then $B^{\mathbb{C}^{\times}}=B_{0}$.

###  2.2. Filtered algebras

Let $A$ be a $\mathbb{C}$-algebra. A filtration
$(\mathcal{F}_{i})_{i\in\mathbb{Z}}$ of $A$ is a nested collection of
subspaces $\cdots\subset\mathcal{F}_{i}\subset\mathcal{F}_{i+1}\subset\cdots$
such that $\mathcal{F}_{i}\mathcal{F}_{j}\subset\mathcal{F}_{i+j}$. The
associated graded algebra of a filtered algebra $A$ is

$\operatorname{\mathrm{gr}}A=\operatorname{\mathrm{gr}}_{\mathcal{F}}A=\bigoplus_{i\in\mathbb{Z}}\,(\operatorname{\mathrm{gr}}_{\mathcal{F}}A)_{i},\quad\textrm{
with
}\quad(\operatorname{\mathrm{gr}}_{\mathcal{F}}A)_{i}=\mathcal{F}_{i}/\mathcal{F}_{i-1}.$

The following summarizes basic results on filtered algebras that we need.

######  Lemma 2.1.

Let $A$ be a $\mathbb{C}$-algebra equipped with a filtration
$(\mathcal{F}_{i})_{i\in\mathbb{Z}}$ such that $\mathcal{F}_{-1}=0$ and
$\bigcup_{i}\mathcal{F}_{i}=A$.

  1. (i)

Let $B\subset A$ be a subalgebra filtered by restriction. Then
$\operatorname{\mathrm{gr}}_{\mathcal{F}}B$ is a subalgebra of
$\operatorname{\mathrm{gr}}_{\mathcal{F}}A$ and
$\operatorname{\mathrm{gr}}_{\mathcal{F}}B=\operatorname{\mathrm{gr}}_{\mathcal{F}}A$
implies that $B=A$.

  2. (ii)

$\operatorname{\mathrm{gr}}_{\mathcal{F}}Z(A)\subset
Z(\operatorname{\mathrm{gr}}_{\mathcal{F}}A)$.

  3. (iii)

Assume that there exists a reductive group $G$ acting on $A$ such that each
$\mathcal{F}_{i}$ is $G$-stable. Then $G$ acts on
$\operatorname{\mathrm{gr}}_{\mathcal{F}}A$ and
$\operatorname{\mathrm{gr}}_{\mathcal{F}}(A^{G})=(\operatorname{\mathrm{gr}}_{\mathcal{F}}A)^{G}$.

Recall that an element $u\in U$ is normal if $uU=Uu$ is a two-sided ideal in
$U$. Also, $u$ is said to be regular if $ur=0$ or $ru=0$ implies $r=0$.

######  Lemma 2.2.

Let $U$ be a filtered algebra and $u\in U$ a normal element such that its
symbol $\sigma(u)\in\operatorname{\mathrm{gr}}U$ is regular. Then
$\operatorname{\mathrm{gr}}(U/\langle
u\rangle)=(\operatorname{\mathrm{gr}}U)/\langle\sigma(u)\rangle$.

###### Proof.

Let $I=\langle u\rangle$ and assume $u$ has degree $d$. By definition of the
quotient filtration, the exact sequence $0\to I\to U\to U/I\to 0$ is strictly
filtered. This means that the associated graded sequence
$0\to\operatorname{\mathrm{gr}}I\to\operatorname{\mathrm{gr}}U\to\operatorname{\mathrm{gr}}(U/I)\to
0$ is also exact. Therefore, it suffices to show that
$\operatorname{\mathrm{gr}}I=\langle\sigma(u)\rangle$. Since $u$ is normal, an
element of $\operatorname{\mathrm{gr}}I$ is of the form $\sigma(ur)$ for some
$r\in U$. If $r\in\mathcal{F}_{i}U$ then

$0\neq\sigma(u)\sigma(r)=ur\,\textrm{mod}\,\mathcal{F}_{i+d-1}\in\operatorname{\mathrm{gr}}_{i+d}U.$

This implies that $ur\in\mathcal{F}_{i+d}\setminus\mathcal{F}_{i+d-1}$ and
hence $\sigma(ur)=\sigma(u)\sigma(r)$. In particular,
$\sigma(ur)\in\langle\sigma(u)\rangle$. Hence
$\operatorname{\mathrm{gr}}I\subset\langle\sigma(u)\rangle$. The opposite
inclusion is obvious. ∎

##  3\. The rational Cherednik algebra

In this section we recall the definition of rational Cherednik algebras
associated to complex reflection groups, following [18], [16], [24].

Let $\mathfrak{h}$ be a complex vector space. Recall that a complex reflection
is an invertible linear map $s\colon\mathfrak{h}\to\mathfrak{h}$ of finite
order that fixes a hyperplane $\operatorname{\mathrm{Ker}}\alpha_{s}$
pointwise for some $\alpha_{s}\in\mathfrak{h}^{*}$. That is, the rank of
$(s-1)$ is one. Let $\alpha_{s}^{\vee}\in\mathfrak{h}$ be an eigenvector of
$s$ with the eigenvalue different from 1. Let $\lambda_{s}\in\mathbb{C}$ be
the non-trivial eigenvalue of $s$ on $\mathfrak{h}^{*}$:
$s(\alpha_{s})=\lambda_{s}\alpha_{s}$.

######  Definition 3.1.

A complex reflection group $W$ is a finite subgroup of $GL(\mathfrak{h})$
generated by complex reflections. The set of complex reflections in $W$ is
denoted $\mathcal{S}$.

We will also be denoting a complex reflection group as a pair
$(W,\mathfrak{h})$. A complex reflection group $(W,\mathfrak{h})$ is said to
be irreducible if $\mathfrak{h}$ is an irreducible representation of $W$.

Assume that $(W,\mathfrak{h})$ is irreducible. Note that
$Z(W)=\mathbb{Z}/{\ell}\mathbb{Z}$ because
$Z(W)\subset\mathbb{C}^{\times}\mathrm{Id}_{\mathfrak{h}}$, using the fact
that $\mathfrak{h}$ is irreducible. The order of the centre of an irreducible
complex reflection group is the greatest common divisor of the degrees of a
set of homogeneous fundamental invariants of $W$ [15, Theorem 1.2]. In
particular, $Z(G(m,p,n))=\mathbb{Z}/(m/p)\mathbb{Z}$.

######  Definition 3.2.

Fix a complex reflection group $(W,\mathfrak{h})$, $t\in\mathbb{C}$ and a
conjugation invariant function
$\mathbf{c}\colon\mathcal{S}\rightarrow\mathbb{C}$. The associated rational
Cherednik algebra $H_{t,\mathbf{c}}=H_{t,\mathbf{c}}(W)$ is the quotient of
the skew-group ring $T(\mathfrak{h}\oplus\mathfrak{h}^{*})\rtimes W$ by the
relations

$x\otimes x^{\prime}-x^{\prime}\otimes x,\ y\otimes
y^{\prime}-y^{\prime}\otimes y,\ y\otimes x-x\otimes
y-\kappa(x,y),\quad\forall\,x,x^{\prime}\,\in\mathfrak{h}^{*},y,y^{\prime}\,\in\mathfrak{h},$

where $T(\mathfrak{h}\oplus\mathfrak{h}^{*})$ is the tensor algebra on
$\mathfrak{h}\oplus\mathfrak{h}^{*}$ and

$\kappa(x,y)=tx(y)-\sum_{s\in\mathcal{S}}\frac{2\mathbf{c}(s)}{\alpha_{s}(\alpha_{s}^{\vee})}x(\alpha_{s}^{\vee})\alpha_{s}(y)s\in\mathbb{C}W.$

The space of parameters for the rational Cherednik algebra is
$\mathbb{C}\oplus\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})$. For
ease of notation, we write $H_{t,\mathbf{c}}(W)$ as $H_{t,\mathbf{c}}$
whenever $W$ is clear from the context.

###  3.1. The Euler element

Let $n=\dim\mathfrak{h}$, and let us choose a basis
$y_{1},\ldots,y_{n}\in\mathfrak{h}$, and a dual basis
$x_{1},\ldots,x_{n}\in\mathfrak{h}^{*}$. The Euler element
$\mathrm{eu}_{\mathbf{c}}\in H_{t,\mathbf{c}}(W)$ is defined to be

$\mathrm{eu}_{\mathbf{c}}=\sum_{i=1}^{n}x_{i}y_{i}-\sum_{s\in\mathcal{S}}\frac{2c_{s}}{1-\lambda_{s}}s.$

It does not depend on a basis, and it satisfies

$[\mathrm{eu}_{\mathbf{c}},x]=tx,\quad[\mathrm{eu}_{\mathbf{c}},y]=-ty,\quad[\mathrm{eu}_{\mathbf{c}},w]=0,$
(3.1)

for all $x\in\mathfrak{h}^{*},y\in\mathfrak{h}$ and $w\in W$. Indeed, in order
to check the first equality we consider

$[\mathrm{eu}_{\mathbf{c}},x]=\sum_{i=1}^{n}x_{i}[y_{i},x]-\sum_{s\in\mathcal{S}}\frac{2c_{s}}{1-\lambda_{s}}[s,x],$
(3.2)

where

$[y_{i},x]=tx(y_{i})-\sum_{s\in\mathcal{S}}\frac{2c_{s}}{\alpha_{s}(\alpha_{s}^{\vee})}\alpha_{s}(y_{i})x(\alpha_{s}^{\vee})s.$
(3.3)

Then substitution of (3.3) into (3.2) gives

$[\mathrm{eu}_{\mathbf{c}},x]=tx-\sum_{s\in\mathcal{S}}\frac{2c_{s}}{\alpha_{s}(\alpha_{s}^{\vee})}x(\alpha_{s}^{\vee})\alpha_{s}s-\sum_{s\in\mathcal{S}}\frac{2c_{s}}{1-\lambda_{s}}(s(x)-x)s.$

Let us choose $\alpha_{s}$ and $\alpha_{s}^{\vee}$ so that
$s(x)=x-x(\alpha_{s}^{\vee})\alpha_{s}$. Then
$\lambda_{s}=1-\alpha_{s}(\alpha_{s}^{\vee})$ and we get that
$[\mathrm{eu}_{\mathbf{c}},x]=0$. Similarly,

$[\mathrm{eu}_{\mathbf{c}},y]=-ty+\sum_{s\in\mathcal{S}}\frac{2c_{s}}{\alpha_{s}(\alpha_{s}^{\vee})}\alpha_{s}(y)s(\alpha_{s}^{\vee})s-\sum_{s\in\mathcal{S}}\frac{2c_{s}}{1-\lambda_{s}}(s(y)-y)s=-ty$

since $s(\alpha_{s}^{\vee})=\lambda_{s}^{-1}\alpha_{s}^{\vee}$. The remaining
relation in (3.1) can also be checked.

The algebra $H_{t,\mathbf{c}}(W)$ is $\mathbb{Z}$-graded with $\deg y=-1,\deg
x=1$ and $\deg w=0$ for $x\in\mathfrak{h}^{*},y\in\mathfrak{h}$ and $w\in W$.
We see from the above equations that this grading is inner when $t=1$, given
by the adjoint action of $\mathrm{eu}_{\mathbf{c}}$. As explained in Section
2.1, this can be though of as equipping with an action of
$\mathbb{C}^{\times}$. Since we wish to consider $\mathbb{C}^{\times}$ as a
maximal torus in $\mathrm{SL}_{2}$ (at least when $W$ is a real reflection
group) we write $\mathrm{T}:=\mathbb{C}^{\times}$ from now on. Then
$H_{t,\mathbf{c}}(W)_{0}=H_{t,\mathbf{c}}(W)^{\mathrm{T}}$.

###  3.2. Automorphisms

We have explained that $\mathrm{T}$ acts on $H_{t,\mathbf{c}}$ by algebra
automorphisms. When $W$ is a real reflection group, there is an action of
$\mathrm{SL}_{2}$ on $H_{t,\mathbf{c}}$, realising $\mathrm{T}$ as the maximal
torus of diagonal matrices. This action was first defined in [18, Corollary
5.3] and we recall it here. Since $W$ is real, we can fix a $W$-equivariant
isomorphism $\mathfrak{h}{\;\stackrel{{{}_{\sim}}}{{\to}}\;}\mathfrak{h}^{*}$
given by $v\mapsto v^{*}$. Then $\mathrm{SL}_{2}$ acts trivially on $W$ and

$g\cdot v=av+cv^{*},\quad g\cdot v^{*}=bv+dv^{*},\quad\textrm{for
}g=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{SL}_{2},\textrm{ and }v\in\mathfrak{h}.$

##  4\. Two subalgebras: the presentations

In this section we introduce two subalgebras of a rational Cherednik algebra
$H_{t,\mathbf{c}}(W)$, and we find their defining relations. We denote these
subalgebras as $H_{t,\mathbf{c}}^{\mathfrak{g}}(W)$, where
$\mathfrak{g}=\mathfrak{gl}(n)$ or $\mathfrak{g}=\mathfrak{so}(n)$. In the
latter case $W$ is assumed to be a real reflection group. The algebra
$H_{t,\mathbf{c}}^{\mathfrak{g}}(W)$ at $t=1$ is a flat deformation of the
skew product of a quotient of the universal enveloping algebra
$U({\mathfrak{g}})$ with the group $W$, which is established in [19] for the
case of real reflection groups $W$.

###  4.1. The degree zero subalgebra

Let $(W,\mathfrak{h})$ be an irreducible complex reflection group. Let
$x_{1},\dots,x_{n}$ be a basis of $\mathfrak{h}^{*}$ and let
$y_{1},\dots,y_{n}\in\mathfrak{h}$ be the dual basis.

######  Definition 4.1.

The degree zero subalgebra
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}=H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}(W)$
of $H_{t,\mathbf{c}}(W)$ is the subalgebra generated by all
$E_{ij}=x_{i}y_{j}$ and the group $W$.

Note that the $E_{ij}$ are a basis of $\mathfrak{h}\otimes\mathfrak{h}^{*}$ in
$H_{t,\mathbf{c}}$. Thus, $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ can
equivalently be described as the subalgebra generated by
$\mathfrak{h}\otimes\mathfrak{h}^{*}$ and $W$.

The algebra $H_{t,\mathbf{c}}$ admits a filtration $\mathcal{F}_{i}$ with
$\mathfrak{h},\mathfrak{h}^{*}$ in degree one and $W$ in degree zero. We give
all subalgebras of $H_{t,\mathbf{c}}$ a filtration by restriction, also
denoted $\mathcal{F}_{i}$. In particular,
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ is a filtered algebra. Let
$P=\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{*}]$. The terminology of
Definition 4.1 is justified by the following lemma.

######  Lemma 4.2.

$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}=H_{t,\mathbf{c}}^{\mathrm{T}}$ is the
subalgebra of all elements of degree zero.

###### Proof.

The inclusion $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}\subset
H_{t,\mathbf{c}}^{\mathrm{T}}$ is immediate since
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ is generated by elements of degree zero.
For the opposite inclusion, we equip both algebras with the subalgebra
filtration coming from $H_{t,\mathbf{c}}$. Then Lemma 2.1(i) implies that it
suffices to show equality of associated graded algebras. The filtration
$\mathcal{F}_{i}$ on $H_{t,\mathbf{c}}$ is $\mathrm{T}$-stable. Therefore
Lemma 2.1(iii) says that
$\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{\mathrm{T}}=(\operatorname{\mathrm{gr}}H_{t,\mathbf{c}})^{\mathrm{T}}$
and it is a consequence of the PBW Theorem for rational Cherednik algebras
[18, Theorem 1.3] that

$(\operatorname{\mathrm{gr}}H_{t,\mathbf{c}})^{\mathrm{T}}=(P\rtimes
W)^{\mathrm{T}}=P^{\mathrm{T}}\rtimes W.$

On the other hand,
$\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ certainly
contains the $E_{ij}$ and $W$. Therefore, it suffices to note that the
$E_{ij}\in P$ generate $P^{\mathrm{T}}$ and hence $P^{\mathrm{T}}\rtimes
W\subset\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$. ∎

######  Corollary 4.3.

$\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}=P^{\mathrm{T}}\rtimes
W=(\operatorname{\mathrm{gr}}H_{t,\mathbf{c}})^{\mathrm{T}}$.

Recall that

$S_{jk}:=[y_{j},x_{k}]=t\delta_{jk}-\sum_{s\in\mathcal{S}}\frac{2c_{s}}{\alpha_{s}(\alpha_{s}^{\vee})}\alpha_{s}(y_{j})x_{k}(\alpha_{s}^{\vee})s.$
(4.1)

Let us rescale elements $\alpha_{s}\in\mathfrak{h}^{*}$,
$\alpha_{s}^{\vee}\in\mathfrak{h}$ so that a reflection
$s\colon{\mathfrak{h}}\to\mathfrak{h}$ has the form
$s(v)=v-\alpha_{s}(v)\alpha_{s}^{\vee}$ for any $v\in{\mathfrak{h}}$.

The next statement generalises a result from [19] to arbitrary $t$ and complex
reflection groups.

######  Theorem 4.4.

The algebra $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ is isomorphic to the
quotient of the skew product of the tensor algebra
$T(\mathfrak{h}^{*}\otimes\mathfrak{h})\rtimes{W}$ by the relations

$\displaystyle[E_{ij},E_{kl}]=t\delta_{jk}E_{il}-t\delta_{il}E_{kj}+\sum_{s\in\mathcal{S}}\frac{2c_{s}}{\alpha_{s}(\alpha_{s}^{\vee})}\bigl{(}\alpha_{s}(y_{l})x_{i}(\alpha_{s}^{\vee})E_{kj}-\alpha_{s}(y_{j})x_{k}(\alpha_{s}^{\vee})E_{il}\bigr{)}s$
$\displaystyle+\sum_{s\in\mathcal{S}}\frac{2c_{s}}{\alpha_{s}(\alpha_{s}^{\vee})}\alpha_{s}(y_{j})\alpha_{s}(y_{l})\sum_{r=1}^{n}x_{r}(\alpha_{s}^{\vee})\bigl{(}x_{k}(\alpha_{s}^{\vee})E_{ir}-x_{i}(\alpha_{s}^{\vee})E_{kr}\bigr{)}s,$
(4.2) $\displaystyle
E_{ij}E_{kl}-E_{il}E_{kj}=t\delta_{jk}E_{il}-t\delta_{lk}E_{ij}+\sum_{s\in\mathcal{S}}\frac{2c_{s}}{\alpha_{s}(\alpha_{s}^{\vee})}x_{k}(\alpha_{s}^{\vee})\bigl{(}\alpha_{s}(y_{l})E_{ij}-\alpha_{s}(y_{j})E_{il}\bigr{)}s.$
(4.3)

A basis of the algebra $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ is given by the
elements

$E_{i_{1}j_{1}}^{m_{1}}\ldots E_{i_{k}j_{k}}^{m_{k}}w,$ (4.4)

where $k\in\mathbb{Z}_{\geq 0}$, $m_{s}\in\mathbb{N}$, $w\in W$ and

$1\leq i_{1}\leq\ldots\leq i_{k}\leq n,\quad 1\leq j_{1}\leq\ldots\leq
j_{k}\leq n,\text{ and }(i_{s},i_{s+1})\neq(j_{s},j_{s+1})\text{ for }1\leq
s\leq n-1.$ (4.5)

###### Proof.

For the elements $E_{ij}=x_{i}y_{j}\in H_{t,\mathbf{c}}$ we have that

$[E_{ij},E_{kl}]=x_{i}S_{jk}y_{l}-x_{k}S_{li}y_{j},$

where $S_{jk}=[y_{j},x_{k}]$ has the form (4.1). This implies relation (4.2)
in $H_{t,\mathbf{c}}^{\mathfrak{gl}_{n}}\subset H_{t,\mathbf{c}}$. Similarly,
relation (4.3) can be checked by making use of the equality

$E_{ij}E_{kl}-E_{il}E_{kj}=x_{i}S_{jk}y_{l}-x_{i}S_{lk}y_{j}.$

It is also clear that the algebra $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ is
generated by the elements $E_{ij}$ ($1\leq i,j\leq n$) and $w\in W$. Note that
elements $\eqref{basisgl}$ with the specified restrictions (4.5) span the
algebra $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$. Indeed, by moving elements of
the group to the right and by the relation (4.2) together with an induction on
the degree in $E_{ij}$, any element of the algebra can be represented as a
linear combination of monomials (4.4), where restriction $1\leq
i_{1}\leq\ldots\leq i_{k}\leq n$ holds. Then by employing relations (4.3) and
induction on the degree in $E_{ij}$ one can assume that all the restrictions
(4.5) hold.

We are left to show that elements (4.4) from the algebra
$H_{t,\mathbf{c}}^{\mathfrak{gl}_{n}}$ with the specified restrictions (4.5)
are linearly independent. Observe that any element

$\prod_{i=1}^{n}x_{i}^{a_{i}}y_{i}^{b_{i}}w\in
H_{t,\mathbf{c}}^{\mathfrak{gl}_{n}},\quad a_{i},b_{i}\in\mathbb{Z}_{\geq
0},\,w\in W$ (4.6)

from the standard PBW basis of the algebra $H_{t,\mathbf{c}}$ can be uniquely
represented as a linear combination of elements (4.4), (4.5). Indeed, there
will be a unique term $h$ in the linear combination of degree
$\sum_{k=1}^{n}a_{k}=\sum_{k=1}^{n}b_{k}$ in elements $E_{ij}$. If $a_{1}$ and
$b_{1}$ are both positive then the first factor of $h$ has to be
$E_{11}^{\min(a_{1},b_{1})}$. If $a_{2}$ and $b_{2}$ are both positive then
the second factor will be $E_{21}^{\min(a_{2},b_{1}-a_{1})}$ or
$E_{12}^{\min(a_{1}-b_{1},b_{2})}$ or $E_{22}^{\min(a_{2},b_{2})}$ in the
cases $b_{1}>a_{1}$ or $a_{1}>b_{1}$ or $a_{1}=b_{1}$, respectively. This
process can be continued to determine $h$ completely. In the cases when some
$a_{i}$ or $b_{i}$ vanish, the corresponding variable $x_{i}$ or $y_{i}$ is
not present in the monomial (4.6) and the index $i$ does not appear in the
first or second place, respectively, in the factors of $h$, otherwise the
process of building the element $h$ is similar to the above.

This proves that elements (4.4), (4.5) form a basis of the algebra, which also
implies that relations (4.2), (4.3) are complete. ∎

###  4.2. The Dunkl angular momentum subalgebra

Now let $W$ be a finite real reflection group. Let $y_{1},\ldots,y_{n}$ be an
orthonormal basis of $\mathfrak{h}$ for a fixed $W$-invariant inner product
$(\cdot,\cdot)$. Let $x_{1},\ldots,x_{n}$ be the dual basis of
${\mathfrak{h}}^{*}$. Note that $\mathfrak{h}^{*}\cong\mathfrak{h}$ as
$W$-modules. Define elements $M_{ij}=x_{i}y_{j}-x_{j}y_{i}\in
H_{t,\mathbf{c}}(W)$, where $1\leq i,j\leq n$. Note that $M_{ij}=-M_{ji}$.

######  Definition 4.5.

(cf. [19]) The Dunkl angular momentum subalgebra
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}=H_{t,\mathbf{c}}^{\mathfrak{so}(n)}(W)$
of $H_{t,\mathbf{c}}(W)$ is the subalgebra generated by all $M_{ij}$ and the
group $W$.

In the real case we have $\alpha_{s}(\alpha_{s}^{\vee})=2$ and formula (4.1)
takes the form

$S_{jk}=[y_{j},x_{k}]=t\delta_{jk}-\sum_{s\in\mathcal{S}}c_{s}\alpha_{s}(y_{j})x_{k}(\alpha_{s}^{\vee})s.$
(4.7)

Consider elements $m_{ij}\in\Lambda^{2}\mathfrak{h}$ given by
$m_{ij}=y_{i}\wedge y_{j}$. The next statement was established in [19] for
$t=1$. The case of general $t$ is similar and we give details below.

######  Theorem 4.6.

The algebra $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ is isomorphic to the
quotient $B$ of the skew product of the tensor algebra
$T(\Lambda^{2}\mathfrak{h})\rtimes{W}$ by the relations

$\displaystyle[m_{ij},m_{kl}]=m_{il}S_{jk}+m_{jk}S_{il}-m_{ik}S_{jl}-m_{jl}S_{ik},$
(4.8) $\displaystyle
m_{ij}m_{kl}+m_{jk}m_{il}+m_{ki}m_{jl}=m_{ij}S_{kl}+m_{jk}S_{il}+m_{ki}S_{jl}.$
(4.9)

The isomorphism is given by the map $\rho\colon B\to
H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ such that $\rho(m_{ij})=M_{ij}$ and
$\rho(w)=w$ for $w\in W$. A basis of the algebra
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ is given by the elements

$M_{i_{1}j_{1}}^{r_{1}}\ldots M_{i_{k}j_{k}}^{r_{k}}w,$ (4.10)

where $k\in\mathbb{Z}_{\geq 0}$, $r_{s}\in\mathbb{N}$, $w\in W$ and

$i_{s}<j_{s},\quad 1\leq i_{1}\leq\ldots\leq i_{k}\leq n,\quad
i_{s}=i_{s+1}\implies j_{s}<j_{s+1},\quad i_{s}<i_{s^{\prime}}<j_{s}\implies
j_{s^{\prime}}\leq j_{s}.$ (4.11)

###### Proof.

Consider an isomorphism $\varphi\colon\mathfrak{h}\to\mathfrak{h}^{*}$ given
by $\varphi(v)(u)=(v,u)$ for any $v,u\in\mathfrak{h}$. Under this isomorphism
$\varphi(y_{i})=x_{i}$ and
$\varphi(\alpha_{s}^{\vee})=\frac{1}{2}(\alpha_{s}^{\vee},\alpha_{s}^{\vee})\alpha_{s}$
since $\alpha_{s}(\alpha_{s}^{\vee})=2$ and by orthogonality of the reflection
$s\in S$. We also have $\beta(v)=\varphi(v)(\varphi^{-1}(\beta))$ for any
$v\in\mathfrak{h},\beta\in\mathfrak{h}^{*}$. Therefore

$\alpha_{s}(y_{j})x_{i}(\alpha_{s}^{\vee})=x_{j}\left(\frac{2\alpha_{s}^{\vee}}{(\alpha_{s}^{\vee},\alpha_{s}^{\vee})}\right)\frac{(\alpha_{s}^{\vee},\alpha_{s}^{\vee})}{2}\alpha_{s}(y_{i})=\alpha_{s}(y_{i})x_{j}(\alpha_{s}^{\vee}).$
(4.12)

Note that $M_{ij}=E_{ij}-E_{ji}$, and elements $E_{ij}$ satisfy relations
given in Theorem 4.4. We apply the first relation of Theorem 4.4 to

$[M_{ij},M_{kl}]=[E_{ij},E_{kl}]-[E_{ji},E_{kl}]-[E_{ij},E_{lk}]+[E_{ji},E_{lk}],$

and we simplify the result by making use of relation (4.12). After collecting
terms at each $M_{ij}$ and cancellations of the terms coming from the second
sum in the right-hand side of equality (4.2) we get

$[M_{ij},M_{kl}]=M_{il}S_{jk}+M_{jk}S_{il}-M_{ik}S_{jl}-M_{jl}S_{ik}.$ (4.13)

Consider now

$M_{ij}M_{kl}+M_{jk}M_{il}+M_{ki}M_{jl}=(E_{ij}E_{kl}-E_{kj}E_{il})-(E_{ij}E_{lk}-E_{ik}E_{lj})-(E_{ji}E_{kl}-E_{ki}E_{jl})\\\
+(E_{ji}E_{lk}-E_{jk}E_{li})+(E_{jk}E_{il}-E_{ik}E_{jl})+(E_{kj}E_{li}-E_{ki}E_{lj})\\\
=-(x_{i}S_{jl}y_{k}-x_{i}S_{kl}y_{j})+(x_{j}S_{il}y_{k}-x_{j}S_{kl}y_{i})+(x_{k}S_{jl}y_{i}-x_{k}S_{il}y_{j}),$
(4.14)

where we used the relation $S_{ij}=S_{ji}$ which follows from (4.12). By
Jacobi identity $[S_{jl},y_{k}]=[S_{kl},y_{j}]$, which implies that relation
(4.14) can be rearranged to the form

$M_{ij}M_{kl}+M_{jk}M_{il}+M_{ki}M_{jl}=M_{ij}S_{kl}+M_{jk}S_{il}+M_{ki}S_{jl}.$
(4.15)

Consider the map $\rho_{0}\colon\Lambda^{2}\mathfrak{h}\to\langle M_{ij}|1\leq
i<j\leq n\rangle$ given by $\rho(m_{ij})=M_{ij}$, where in the codomain we
have the linear span. The map $\rho_{0}$ is $W$-equivariant. It follows from
relations (4.13) and (4.15) that $\rho_{0}$ extends to a well-defined
homomorphism of algebras $\rho\colon B\to H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$
by defining $\rho(w)=w$. It is also clear that $\rho$ is an epimorphism.
Furthermore, monomials (4.10) with restrictions (4.11) linearly generate the
algebra $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}(W)$. This follows from a
noncrossing visualisation of these monomials and since application of the
relation (4.15) allows to reduce the number of crossings in the monomials from
the algebra $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}(W)$ (see [19]). Consider now
the highest symbol of the monomials (4.10) in the Dunkl embedding, it has the
form $L_{i_{1}j_{1}}^{r_{1}}\ldots L_{i_{k}j_{k}}^{r_{k}}w$, where
$L_{ij}=x_{i}p_{j}-x_{j}p_{i}$. These highest symbols are linearly independent
(see e.g. [20]), which implies that elements (4.10) with restrictions (4.11)
form a basis of the algebra $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}(W)$.

Similarly, by applying relations (4.8), (4.9) we get that the linear span of
the corresponding monomials $m_{i_{1}j_{1}}^{r_{1}}\ldots
m_{i_{k}j_{k}}^{r_{k}}w\in B$ gives the entire algebra $B$. These monomials
are linearly independent since $\rho$ maps them to a basis of
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}(W)$, and it follows that $\rho$ is an
isomorphism. ∎

As a corollary we get the following statement.

######  Proposition 4.7.

There is an isomorhism of algebras

$H_{0,0}^{\mathfrak{so}(n)}\cong\mathrm{gr}\,H_{t,\mathbf{c}}^{\mathfrak{so}(n)}.$

###### Proof.

The isomorphism $H_{0,0}\cong\mathrm{gr}\,H_{t,\mathbf{c}}$ restricts to an
injective morphism

$H_{0,0}^{\mathfrak{so}(n)}\hookrightarrow\mathrm{gr}\,H_{t,\mathbf{c}}.$

It is sufficient to show that the image of this map equals
$\mathrm{gr}\,H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$.

Theorem 4.6 says that a basis of $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ is
given by the monomials (4.10) satisfying restrictions (4.11). Consider a basis
element $b=M_{i_{1}j_{1}}^{r_{1}}\cdots M_{i_{k}j_{k}}^{r_{k}}w$ and let
$r=2\sum_{i=1}^{k}r_{i}$. Under the standard filtration $b\in\mathcal{F}_{r}$
and $b\not\in\mathcal{F}_{r-1}$. Then the symbol of $b$ in
$\mathrm{gr}\,H_{t,\mathbf{c}}=H_{0,0}$ is also $M_{i_{1}j_{1}}^{r_{1}}\cdots
M_{i_{k}j_{k}}^{r_{k}}w$. Since Theorem 4.6 again says that all such monomials
are a basis of $H_{0,0}^{\mathfrak{so}(n)}$ the equality
$H_{0,0}^{\mathfrak{so}(n)}=\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$
holds.

∎

An immediate consequence of Proposition 4.7 is that:

######  Corollary 4.8.

$H^{\mathfrak{so}(n)}$ is a flat family of algebras over the parameter space
$\mathbb{C}\oplus\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})$.

The next result is the key to understanding the algebra
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ since rings of invariants are often much
easier to study than algebras given by generators and relations.

######  Theorem 4.9.

There is equality
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}=H_{t,\mathbf{c}}^{\mathrm{SL}_{2}}$ of
subalgebras of $H_{t,\mathbf{c}}$.

###### Proof.

First we show that there is an inclusion

$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}\subset
H_{t,\mathbf{c}}^{\mathrm{SL}_{2}}.$

Written explicitly the action of any $g\in\mathrm{SL}_{2}$ is
$g(x_{i})=ax_{i}+cy_{i}$, $g(y_{j})=bx_{j}+dy_{j}$, where $ad-bc=1$. The group
$\mathrm{SL}_{2}$ acts trivially on $W$ by definition. It suffices then to
show that $g$ fixes any generator $x_{i}y_{j}-x_{j}y_{i}$. We perform the
calculation explicitly

$g(x_{i}y_{j}-x_{j}y_{i})=adx_{i}y_{j}+bcy_{i}x_{j}-adx_{j}y_{i}-bcy_{j}x_{i}.$

As $W$ is a real reflection group $[x_{i},y_{j}]=[x_{j},y_{i}]$ and therefore
$y_{i}x_{j}-y_{j}x_{i}=x_{j}y_{i}-x_{i}y_{j}$ and so the above becomes

$adx_{i}y_{j}+bcx_{j}y_{i}-adx_{j}y_{i}-bcx_{i}y_{j}=(ad-
bc)(x_{i}y_{j}-x_{j}y_{i})=x_{i}y_{j}-x_{j}y_{i}.$

Identifying $\mathfrak{h}\oplus\mathfrak{h}^{*}$ with the space of $(2\times
n)$-matrices, it is a classical result in invariant theory [22, Proposition
9.2] that the subalgebra $A$ of $P$ generated by the determinants
$x_{i}y_{j}-x_{j}y_{i}$ is the invariant ring $P^{\mathrm{SL}_{2}}$. Since
$H_{0}^{\mathfrak{so}(n)}=A\rtimes W$ and
$H_{0}^{\mathrm{SL}_{2}}=P^{\mathrm{SL}_{2}}\rtimes W$, we deduce that
$H_{0}^{\mathfrak{so}(n)}=H_{0}^{\mathrm{SL}_{2}}$.

We have shown that $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}\subset
H_{t,\mathbf{c}}^{\mathrm{SL}_{2}}$. If we can show that
$\mathrm{gr}\,H_{t,\mathbf{c}}^{\mathfrak{so}(n)}=\mathrm{gr}\,H_{\mathbf{c}}^{\mathrm{SL}_{2}}$
the result will follow by Lemma 2.1(i). As $\mathrm{SL}_{2}$ is a reductive
group we can apply Lemma 2.1(iii) to get
$\mathrm{gr}\,H_{\mathbf{c}}^{\mathrm{SL}_{2}}=(\mathrm{gr}\,H_{\mathbf{c}})^{\mathrm{SL}_{2}}=H_{0}^{\mathrm{SL}_{2}}$
and as we have shown in Proposition 4.7,
$H_{0}^{\mathfrak{so}(n)}=\mathrm{gr}\,H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$.
Hence,

$\mathrm{gr}\,H_{t,\mathbf{c}}^{\mathfrak{so}(n)}=H_{0}^{\mathfrak{so}(n)}=H_{0}^{\mathrm{SL}_{2}}=\mathrm{gr}\,H_{\mathbf{c}}^{\mathrm{SL}_{2}}.$

∎

######  Remark 4.10.

If $W$ is not a real reflection group then it does not preserve the linear
span of the $M_{ij}$ and the algebra generated by the $M_{ij}$ and $W$ is
larger than expected; in general this algebra will equal
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$.

######  Remark 4.11.

One can generalise the construction of the degree zero and the Dunkl angular
momentum subalgebras as follows. Let
$\mathfrak{g}\subset\operatorname{\mathrm{End}}_{\mathbb{C}}(\mathfrak{h})\cong\mathfrak{h}\otimes\mathfrak{h}^{*}$
be a reductive Lie subalgebra. Pick $W\subset
N_{GL(\mathfrak{h})}(\mathfrak{g})$ a (finite) reflection group. Then one can
consider the subalgebra of $H_{t,\mathbf{c}}$ generated by $\mathfrak{g}$ and
$W$. It would be interesting to classify all pairs $(\mathfrak{g},W)$ such
that one gets a flat family of quotients of $U(\mathfrak{g})\rtimes W$ when
$t=1$.

###  4.3. Subgroups of $\mathrm{SL}_{2}$

One can consider $H_{t,\mathbf{c}}^{\Gamma}$ for other reductive subgroups
$\Gamma\subset\mathrm{SL}_{2}$ and ask if there exists
$\mathfrak{g}\subset\mathfrak{h}\otimes\mathfrak{h}^{*}$ such that
$H_{t,\mathbf{c}}^{\Gamma}=H_{t,\mathbf{c}}^{\mathfrak{g}}$. We do not
consider finite $\Gamma$ here. Other than $\mathrm{T}$ and $\mathrm{SL}_{2}$,
this leaves just the normalizer $N$ of $\mathrm{T}$ in $\mathrm{SL}_{2}$.

Let $F$ be the automorphism of $H_{t,\mathbf{c}}$ defined by
$F(x_{i})=y_{i},F(y_{i})=-x_{i}$ and $F(w)=w$. Then $F^{2}=-1\in\mathrm{T}$
and we can consider the action of the group $N=\mathrm{T}\rtimes F$ on
$H_{t,\mathbf{c}}$.

Recall that $M_{ij}:=x_{i}y_{j}-x_{j}y_{i}$ and
$H^{\mathfrak{so}(n)}_{t,\mathbf{c}}$ is generated by the $M_{ij}$ and $W$.

######  Lemma 4.12.

There is a proper inclusion $H^{\mathfrak{so}(n)}_{t,\mathbf{c}}\subset
H^{N}_{t,\mathbf{c}}$.

###### Proof.

It is easy to check that $H_{t,\mathbf{c}}^{\mathfrak{so}(n)}\subset
H_{t,\mathbf{c}}^{N}$. Since this is an inclusion of filtered algebras, it
suffices to show that the inclusion is proper for the associated graded
algebras. By Lemma 2.1(iii) and Theorem 4.9 it is sufficient to establish that
$H_{0,0}^{\mathfrak{so}(n)}\neq H_{0,0}^{N}$. We have

$H^{N}_{0,0}=(\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^{*}]\rtimes
W)^{N}=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^{*}]^{N}\rtimes W.$

Then it suffices to show that the subalgebra $B$ of
$\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^{*}]$ generated by the $M_{ij}$ is
a proper subalgebra of $\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^{*}]^{N}$.
Consider the elements $C_{ij,kl}:=x_{i}x_{j}y_{k}y_{l}+x_{k}x_{l}y_{i}y_{j}$.
They belong to $\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^{*}]^{N}$. We claim
that they do not belong to $B$. For this, we define
$\Delta\colon\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^{*}]^{N}\to\mathbb{C}[x,y]^{N}=\mathbb{C}[x^{2}y^{2}]$
by $\Delta(x_{i})=x,\Delta(y_{i})=y$. Notice that $\Delta(M_{ij})=0$ but
$\Delta(C_{ij,kl})=2x^{2}y^{2}\neq 0$. The claim follows. ∎

Consider now the degree two component of $H_{0,0}^{N}=P^{N}\rtimes W$. Notice
that $P_{2}^{\mathrm{T}}=\mathfrak{h}\otimes\mathfrak{h}^{*}\supset P_{2}^{N}$
since $\mathrm{T}\subset N$. Under the identification
$\mathfrak{h}\otimes\mathfrak{h}^{*}=\operatorname{\mathrm{End}}(\mathfrak{h})$,
the automorphism $F$ corresponds to the map $A\mapsto-A^{T}$. Therefore,
$P_{2}^{N}=P_{2}^{\mathrm{SL}_{2}}=\mathfrak{so}(n)$ and hence
$(\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{N})_{2}=\mathfrak{so}(n)\otimes\mathbb{C}W$.
This, combined with Lemma 4.12, implies that there is no subalgebra
$\mathfrak{g}$ of $\mathfrak{gl}(n)$ such that
$H^{N}_{t,\mathbf{c}}=H^{\mathfrak{g}}_{t,\mathbf{c}}$.

###  4.4. The centralizer of $\mathfrak{so}(n)$ in $W$

Assume now that $(W,\mathfrak{h})$ be a complex reflection group. Each element
$m\in W$ acts as a linear transformation of $\mathfrak{h}$, denote this
corresponding map $M\in\operatorname{\mathrm{GL}}(\mathfrak{h})$. Recall that
$W$ has a natural action on the dual space $\mathfrak{h}^{*}$, given by
$(m\cdot y)(x)=y(m^{-1}x)$ and therefore the element $m\in W$ corresponds to
$(M^{-1})^{\mathrm{T}}\in\operatorname{\mathrm{GL}}(\mathfrak{h}^{*})$ [23].

We can identify the space $\mathfrak{h}\otimes\mathfrak{h}^{*}$ with
$\mathrm{End}(\mathfrak{h})$ via the map $x_{i}\otimes y_{j}\rightarrow
x_{i}y_{j}\colon\mathfrak{h}\rightarrow\mathfrak{h}$. Equip
$\mathrm{End}(\mathfrak{h})$ with a $W$-action by letting $w\cdot M=wMw^{-1}$.
With this action the aforementioned map becomes an isomorphism of $W$-modules.

######  Proposition 4.13.

If $w\in W$ commutes with all elements $x_{i}y_{j}-x_{j}y_{i}$ and
$\mathrm{dim}\,\mathfrak{h}\geq 3$ then $w$ acts as a scalar multiple of the
identity. If $n=2$ then $w$ acts as the sum of a scalar multiple of the
identity and a skew-symmetric matrix.

###### Proof.

Using the identification with $\mathrm{End}(\mathfrak{h})$, the pair
$x_{i}y_{j}-x_{j}y_{i}$ corresponds to the matrix with $1$ in the $i^{th}$ row
and $j^{th}$ column, $-1$ in the $j^{th}$ row and $i^{th}$ column and $0$
everywhere else. Let us denote this matrix by $P_{ij}$. Then for $m$ to
commute with $x_{i}y_{j}-x_{j}y_{i}$ we require $m\cdot
P_{ij}=MP_{ij}M^{-1}=P_{ij}$. Let $m_{ij}$ denote the entry in the $i^{th}$
row and $j^{th}$ column of $M$. Calculating $MP_{ij}M^{-1}=P_{ij}$ explicitly
gives the relations $m_{ij}=-m_{ji}$, $m_{ii}=m_{jj}$ and $0$ otherwise. There
are two cases to consider.

If $\dim\,\mathfrak{h}\geq 3$ we must have that $m_{k\ell}=0$ for all
$k\neq\ell$. This is because the commutation relation for
$x_{1}y_{2}-x_{2}y_{1}$ implies that $m_{k\ell}=0$ for all $(k,\ell)\neq(1,2)$
or $(2,1)$. But then the commutation relations for $x_{1}y_{3}-x_{3}y_{1}$
force $m_{k\ell}=0$ for all $(k,\ell)\neq(1,3)$ or $(3,1)$. The only non-zero
terms of $M$ are the diagonal entries which are all equal.

If $\dim\,\mathfrak{h}=2$ then we need only consider the element
$x_{1}y_{2}-x_{2}y_{1}$ and we have $m_{11}=m_{22}$ and $m_{12}=-m_{21}$.
Hence the matrix associated to the action of $m$ has the form

$M=\begin{bmatrix}a&b\\\ -b&a\end{bmatrix}$

for $a,\,b\in\mathbb{C}$, as required. ∎

######  Corollary 4.14.

Suppose that $W\subset\mathrm{O}(n)$. Then the centraliser
$Z_{W}(\mathfrak{so}(n))$ of $\mathfrak{so}(n)$ equals $Z(W)$ for $n\geq 3$,
and $Z_{W}(\mathfrak{so}(2))=W\cap\mathrm{SO}(2)$ (the case $n=2$).

If $W$ is a real reflection group then we identify
$\mathfrak{so}(n)\subset\operatorname{\mathrm{End}}(\mathfrak{h})=\mathfrak{h}\otimes\mathfrak{h}^{*}$
with $\wedge^{2}\mathfrak{h}$ using the $W$-equivariant isomorphism
$\mathfrak{h}{\;\stackrel{{{}_{\sim}}}{{\to}}\;}\mathfrak{h}^{*}$.

######  Lemma 4.15.

Assume that $W$ is an irreducible real reflection group and
$\dim\mathfrak{h}\geq 3$. Then the kernel of $W\to GL(\wedge^{2}\mathfrak{h})$
equals $Z(W)$.

###### Proof.

Since $Z(W)$ is either trivial, or equals $\\{\pm 1\\}$, the kernel clearly
contains $Z(W)$. Let $g\in W$ act trivially on $\wedge^{2}\mathfrak{h}$. Since
$g$ has finite order, it is diagonalizable. If $n=\dim\mathfrak{h}$ and
$N=n(n-1)/2=\dim\wedge^{2}\mathfrak{h}$ then we write
$\omega_{1},\dots,\omega_{n}$ for the eigenvalues of $g$ on $\mathfrak{h}$.
Recall that each $\omega_{i}$ is a root of unity. Then in a suitable basis $g$
acts on $\wedge^{2}\mathfrak{h}$ as a diagonal matrix with diagonal entries
$\omega_{1}\omega_{2},\dots,\omega_{n-1}\omega_{n}$. If this is the identity
matrix then $\omega_{i}\omega_{j}=1$ for $i\neq j$. If $\omega_{1}\notin\\{\pm
1\\}$ say then $\omega_{2}=\omega_{3}=\omega_{1}^{-1}$ and
$\omega_{3}=\omega_{2}^{-1}$ is a contradiction. Thus, $\omega_{i}\in\\{\pm
1\\}$ for all $i$. If $\omega_{i}=-1$ for some $i$ then this forces
$\omega_{i}=-1$ for all $i$. Thus, $g=-\mathrm{Id}$ in this case. ∎

##  5\. The centre of the angular momentum algebra in rank two

In this section we assume $\dim\mathfrak{h}=2$. Then $W=D_{\ell}$ is the
dihedral group of order $2\ell$, for $\ell\geq 3$, and we write
$D_{\ell}=\\{1,\sigma,\dots,\sigma^{\ell-1},s_{1},\dots,s_{\ell}\\}$, where
$\sigma$ is a rotation and the $s_{i}$ are the reflections. Note that
$s_{i}\sigma s_{i}=\sigma^{-1}$. Let
$a_{0},\dots,a_{\ell-1}\in\mathbb{C}\langle\sigma\rangle$ be the basis of the
group algebra consisting of primitive idempotents. Explicitly, we have
$a_{i}=\frac{1}{\ell}\sum_{k=0}^{\ell-1}\zeta^{-ki}\sigma^{k}$, where $\zeta$
is a fixed primitive $\ell$th root of unity. Then $\sigma
a_{i}=\zeta^{i}a_{i}$. In $\mathbb{C}D_{\ell}$ we have
$s_{i}a_{j}=a_{-j}s_{i}$ where we define $a_{i+\ell}:=a_{i}$ for
$i\in\mathbb{Z}$. We set

$b_{i}=\left\\{\begin{array}[]{ll}a_{i}+a_{-i}&2i\neq 0\,\textrm{mod}\,\ell\\\
a_{i}&2i=0\,\textrm{mod}\,\ell\end{array}\right.\quad\textrm{for
$i=0,\dots,N:=\lfloor\ell/2\rfloor$}.$

We also set $d_{i}=a_{i}-a_{-i}$ for
$i=1,\dots,\widetilde{N}:=\lfloor\frac{\ell-1}{2}\rfloor$. Note that
$N+\widetilde{N}=\ell-1$. Moreover, $d_{i}^{2}=b_{i}$ and $d_{i}b_{i}=d_{i}$.

######  Lemma 5.1.

The spaces

$Z_{+}=\\{z\in\mathbb{C}\langle\sigma\rangle\,|\,s_{i}z=zs_{i}\,\forall\,i\\},\quad
Z_{-}=\\{z\in\mathbb{C}\langle\sigma\rangle\,|\,s_{i}z=-zs_{i}\,\forall\,i\\},$

have basis $b_{0},\dots,b_{N}$ and $d_{1},\dots,d_{\widetilde{N}}$
respectively.

###### Proof.

Since $\langle\sigma\rangle$ is a normal subgroup of $D_{\ell}$, the latter
acts by conjugation on $\mathbb{C}\langle\sigma\rangle$ and this action
factors through the quotient $D_{\ell}/\langle\sigma\rangle=\langle
s_{1}\rangle$. The $a_{i}$ are a basis of $\mathbb{C}\langle\sigma\rangle$.
Therefore, we get a basis of $Z_{+}$ and $Z_{-}$ respectively by applying the
averaging operator $(1/2)(1+s_{1})$ (and rescaling if necessary), respectively
the operator $(1/2)(1-s_{1})$. The result follows. ∎

######  Proposition 5.2.

The centre of $H_{t,\mathbf{c}}^{\mathfrak{so}(2)}(D_{\ell})$ is a direct sum
of $N+1$ polynomial rings. More specifically, when $\ell$ is even,

$Z(H_{t,\mathbf{c}}^{\mathfrak{so}(2)}(D_{\ell}))=b_{0}\mathbb{C}[M_{1,2}^{2}]\oplus\left(\bigoplus_{i=1}^{N-1}b_{i}\mathbb{C}[M_{1,2}d_{i}]\right)\oplus
b_{N}\mathbb{C}[M_{1,2}^{2}],$

and when $\ell$ is odd,

$Z(H_{t,\mathbf{c}}^{\mathfrak{so}(2)}(D_{\ell}))=b_{0}\mathbb{C}[M_{1,2}^{2}]\oplus\left(\bigoplus_{i=1}^{N}b_{i}\mathbb{C}[M_{1,2}d_{i}]\right).$

###### Proof.

The subalgebra $H_{t,\mathbf{c}}^{\mathfrak{so}(2)}(D_{\ell})\subset
H_{t,\mathbf{c}}(D_{\ell})$ is generated by the element $M_{1,2}$ and elements
$w\in D_{\ell}$. It has a basis $\\{M_{1,2}^{j}w\\}$, where
$j\in\mathbb{Z}_{\geq 0}$. The subalgebra is graded with $\deg M_{1,2}=2$ and
$\deg w=0$. Therefore, its centre is also graded. We consider each graded
piece in turn. In even degrees, we are looking for elements $M_{1,2}^{2k}z$,
$k\in\mathbb{Z}_{\geq 0}$, with $z\in\mathbb{C}D_{\ell}$ acting trivially on
$M_{1,2}$ and commuting with $D_{\ell}$. By Corollary 4.14, the first
condition forces $z\in\mathbb{C}\langle\sigma\rangle$ since
$\dim\mathfrak{h}=2$. Then, the second condition in turn means that $z$
belongs to $Z_{+}$. Hence, $z$ is a linear combination of the $b_{i}$ (which
are all in the centre). In odd degrees, we have elements of the form
$M_{1,2}^{2k+1}z$, again for some $z\in\mathbb{C}D_{\ell}$. For this to
commute with $M_{1,2}$, we must have $[M_{1,2},z]=0$ forcing
$z\in\mathbb{C}\langle\sigma\rangle$. To commute with $D_{\ell}$ it suffices
to check that $[s_{1},M_{1,2}^{2k+1}z]=0$. This forces $s_{1}z=-zs_{1}$; that
is, $z\in Z_{-}$. Thus, the element is a linear combination of the elements
$M_{1,2}^{2k+1}d_{i}$. Since $d_{i}d_{j}=\delta_{i,j}b_{i}$ the result
follows. ∎

We note that for $t\neq 0$ the centre of the algebra
$H_{t,\mathbf{c}}^{\mathfrak{so}(n)}$ is described in [19], [20] for $n\geq
3$. However, it is missed in the references that the case $n=2$ requires
different considerations as above.

##  6\. Invariant subalgebras

In order to give a uniform treatment of the rings of invariants under both
$\mathrm{T}$ and $\mathrm{SL}_{2}$, we make a number of general assumptions,
which we check hold for both of these groups.

###  6.1. Setup

First we note that the rational Cherednik algebra can be considered as an
algebra $H$ over the polynomial ring
$R=\mathbb{C}[\mathbb{C}\oplus\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})]$
such that specializing the parameters to some
$(t,\mathbf{c})\in\mathbb{C}\oplus\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})$
gives $H_{t,\mathbf{c}}$. We will consider automorphisms of $H$ that are
$R$-linear and hence induce automorphisms of each $H_{t,\mathbf{c}}$.

Throughout, we fix $\Gamma$ to be a group of automorphisms of $H$ satisfying

  1. (I)

$\Gamma$ is reductive; and

  2. (II)

$\Gamma$ acts trivially on $W$ and $R$ and linearly on
$\mathfrak{h}\oplus\mathfrak{h}^{*}$, preserving the symplectic form.

The group $W$ acts by conjugation as inner automorphisms on $H$. The action of
$Z(W)$ is trivial on $W$ and $R$ and linear on
$\mathfrak{h}\oplus\mathfrak{h}^{*}$. Therefore, we may also impose that

  1. (III)

the image of $Z(W)\to\mathrm{Aut}(H)$ lies in $\Gamma$.

In view of (II) we may write $H^{\Gamma\times
W}=(H^{W})^{\Gamma}=(H^{\Gamma})^{W}$ without ambiguity. The following
elementary observation will be used repeatedly.

######  Lemma 6.1.

The group $\Gamma$ preserves the filtration on $H_{t,\mathbf{c}}$ and hence
acts as graded automorphisms of $H_{0,0}=P\rtimes W$.

###### Proof.

Notice that since $\mathcal{F}_{0}H_{t,\mathbf{c}}=\mathbb{C}W$,
$\mathcal{F}_{1}H_{t,\mathbf{c}}=\mathbb{C}W\oplus\mathfrak{h}\oplus\mathfrak{h}^{*}$
and

$\mathcal{F}_{k}H_{t,\mathbf{c}}=(\mathcal{F}_{k-1}H_{t,\mathbf{c}})(\mathcal{F}_{1}H_{t,\mathbf{c}}(W))$

the claim follows from (II). ∎

######  Remark 6.2.

When considering rational Cherednik algebras associated to an arbitrary
complex reflection group $W$ one can usually reduce to the case where
$\mathfrak{h}$ is irreducible. However, this is not the case when considering
the rings of invariants $H^{\Gamma}_{t,\mathbf{c}}$. Moreover, we use in an
essential way several times the fact that $\mathfrak{h}$ is assumed
irreducible. In the case where $\mathfrak{h}$ is reducible, the “correct”
approach is to consider the action of $\Gamma^{k}$, where $k$ is the number of
irreducible summands of $\mathfrak{h}$.

###  6.2. GIT quotients

Recall that a representation $V$ of an algebraic group $\Gamma$ is said to be
stable if a generic $\Gamma$-orbit is closed in $V$.

######  Lemma 6.3.

Let $\Gamma$ be $\mathrm{T}$, respectively $\mathrm{SL}_{2}$, acting on
$V=\mathfrak{h}\oplus\mathfrak{h}^{*}$, respectively on
$\mathfrak{h}\otimes\mathbb{C}^{2}$. Then $V$ is a stable representation.

###### Proof.

For $\mathrm{T}$, a generic orbit in $V$ is defined by the system of equations
$x_{i}y_{i}=\alpha_{i}$ for some $\alpha_{i}\neq 0$. In particular, it is
closed and hence $V$ is stable.

For $\mathrm{SL}_{2}$, we identify $V$ with the space of $2\times n$ matrices,
with $\mathrm{SL}_{2}$ acting by left multiplication; here
$n=\dim\mathfrak{h}$. Then $\mathbb{C}[V]^{\Gamma}$ is generated by the column
determinants $\mathrm{det}_{i,j}$ for $1\leq i<j\leq n$. Let $\pi\colon V\to
V/\\!\\!/\\!\,\Gamma$ be the quotient map and choose any $p\in
V/\\!\\!/\\!\,\Gamma$ other than $0$. Then there exists some $i<j$ such that
$\mathrm{det}_{i,j}(A)\neq 0$ for all $A\in\pi^{-1}(p)$. Since
$\mathrm{SL}_{2}$ acts freely (by left multiplication) on the space of all
$2\times 2$ matrices of fixed non-zero determinant, it implies that
$\mathrm{SL}_{2}$ acts freely on $\pi^{-1}(p)$. Now if $\pi^{-1}(p)$ was not a
single $\mathrm{SL}_{2}$-orbit, then the (unique) closed
$\mathrm{SL}_{2}$-orbit in $\pi^{-1}(p)$ would have dimension less than
$\dim\mathrm{SL}_{2}$. In particular, it is not a free orbit. We deduce that
$\pi^{-1}(p)$ is a single orbit and hence $V$ is stable. ∎

######  Proposition 6.4.

Let $\Gamma$ be $\mathrm{T}$, respectively $\mathrm{SL}_{2}$, acting on
$V=\mathfrak{h}\oplus\mathfrak{h}^{*}$, respectively on
$\mathfrak{h}\otimes\mathbb{C}^{2}$. Then $V/\\!\\!/\\!\,\Gamma$ and
$V/\\!\\!/\\!\,(\Gamma\times W)$ are Gorenstein.

###### Proof.

We consider first $V/\\!\\!/\\!\,\Gamma$. In the case of $\mathrm{T}$, the
fact that the quotient is Gorenstein follows from standard results e.g. [13,
Corollary 6.4.3]. But we can also check in both cases directly from the
general result of Knop [29] that says that $V/\\!\\!/\\!\,\Gamma$ is
Gorenstein if the following three conditions hold.

  1. (a)

$V$ is a stable representation.

  2. (b)

$V_{s}:=\\{v\in V\,|\,\dim\Gamma_{v}>0\\}$ has codimension at least two.

  3. (c)

$\lambda_{V}=\lambda_{\mathrm{ad}}$.

Here, for a representation $U$, $\lambda_{U}$ is the linear character of
$\Gamma$ given by
$\Gamma\to\operatorname{\mathrm{GL}}(U)\stackrel{{\scriptstyle\det}}{{\longrightarrow}}\mathbb{C}^{\times}$.
We note that for both $\mathrm{T}$ and $\mathrm{SL}_{2}$,
$\lambda_{V}=\lambda_{\mathrm{ad}}$ is the trivial character. Therefore (c)
holds. Lemma 6.3 says that (a) holds. Thus, we only need to check (b).

For $T$, the space $V_{s}$ equals $\\{0\\}$. Therefore, (b) is satisfied.

For $\Gamma=\mathrm{SL}_{2}$, we make the identification of $V$ with the space
of $2\times n$ matrices, as in the proof of Lemma 6.3. It follows from what is
written there that $V_{s}\subset\pi^{-1}(0)$. In fact, we claim
$V_{s}=\pi^{-1}(0)$. Notice that $\pi^{-1}(0)$ is precisely the set of
$2\times n$ matrices of rank at most $1$. Let $A$ have rank one. Then $gA=A$
for $g\in\mathrm{SL}_{2}$ if and only if $g$ acts trivially on
$\mathrm{Im}\,A$. The subgroup of $\mathrm{SL}_{2}$ fixing a line pointwise is
one-dimensional. Hence $A\in V_{s}$. The space of $2\times n$ matrices of rank
at most $1$ is $n+1$-dimensional. Therefore, we see that (b) holds if $n\geq
3$. When $n=1$, $V/\\!\\!/\\!\,\Gamma=\\{0\\}$ and when $n=2$,
$V/\\!\\!/\\!\,\Gamma=\mathbb{A}^{1}$ since
$\mathbb{C}[V]^{\Gamma}=\mathbb{C}[\mathrm{det}_{1,2}]$. In both cases,
$V/\\!\\!/\\!\,\Gamma$ is Gorenstein.

Now the argument for $V/\\!\\!/\\!\,(\Gamma\times W)$ is similar. Since $W$ is
finite, properties (a) and (b) follow immediately from the fact that they hold
for $V$ as a $\Gamma$-module. For (c), $\lambda_{V}|_{W}$ and
$\lambda_{\mathrm{ad}}|_{W}$ are both the trivial character (the former
because $W\subset\mathrm{Sp}(V)$) so once again (c) follows from the fact that
it holds for $V$ as a $\Gamma$-module. ∎

######  Lemma 6.5.

Assume $\dim\mathfrak{h}>1$ for $\Gamma=\mathrm{T}$ and $\dim\mathfrak{h}>2$
for $\Gamma=\mathrm{SL}_{2}$. The action of $\overline{W}$ on
$V/\\!\\!/\\!\,\Gamma$ is generically free.

###### Proof.

Let $\Gamma=\mathrm{T}$ and $w\in W\setminus Z(W)$. Then we can choose
$v\in\mathfrak{h}$ and $\lambda\in\mathfrak{h}^{*}$ such that neither is an
eigenvector for $w$. Since $v,\lambda\neq 0$, the $\mathrm{T}$-orbit
$\mathcal{O}_{(v,\lambda)}$ of $(v,\lambda)$ is closed but free. Thus, to show
that $w\cdot\pi(v,\lambda)\neq\pi(v,\lambda)$, it suffices to show that
$w(\mathcal{O}_{(v,\lambda)})\neq\mathcal{O}_{(v,\lambda)}$. But this follows
from the fact that $(w(v),w(\lambda))$ does not belong to the line spanned by
$(v,\lambda)$.

Now let $\Gamma=\mathrm{SL}_{2}$, in which case there is a $W$-equivariant
isomorphism $\mathfrak{h}{\;\stackrel{{{}_{\sim}}}{{\to}}\;}\mathfrak{h}^{*}$,
which we write $v\mapsto v^{*}$. Again, if $w\in W\setminus Z(W)$ then we wish
to construct a point $(v,\lambda)\in\mathfrak{h}\oplus\mathfrak{h}^{*}$ such
that the $\mathrm{SL}_{2}$-orbit $\mathcal{O}_{(v,\lambda)}$ of $(v,\lambda)$
is closed but free and
$w(\mathcal{O}_{(v,\lambda)})\neq\mathcal{O}_{(v,\lambda)}$. Since
$\dim\mathfrak{h}>2$ and $w$ is a diagonal matrix but not a scalar matrix,
there exists a $2$-dimensional subspace $U\subset\mathfrak{h}$ such that
$w(U)\not\subset U$. Let $v_{1},v_{2}$ be a basis of $U$. We take
$(v,\lambda)=(v_{1},v_{2}^{*})$. This corresponds to a $2\times n$ matrix of
rank $2$ and hence its $\mathrm{SL}_{2}$-orbit is closed and free (see the
proof of Lemma 6.3). But since $w(v_{2}^{*})=(w(v_{2}))^{*}$ and the span of
$v_{1},v_{2}$ does not equal $U$, it must be the case that
$w(v_{1},v_{2}^{*})$ does not belong to $U\oplus U^{*}$, within which lies the
$\mathrm{SL}_{2}$-orbit of $(v_{1},v_{2}^{*})$. ∎

######  Remark 6.6.

If $\dim\mathfrak{h}=2$ then $W\cap SO(2)$ acts trivially on
$V/\\!\\!/\\!\,\mathrm{SL}_{2}$ and so Lemma 6.5 is false in that case.

######  Proposition 6.7.

Assume $\dim\mathfrak{h}>1$ when $\Gamma=\mathrm{T}$ and $\dim\mathfrak{h}>2$
when $\Gamma=\mathrm{SL}_{2}$. The group $\overline{W}$ acts freely on an open
subset of $V/\\!\\!/\\!\,\Gamma$ with complement of codimension at least two.

###### Proof.

As in the proof of Proposition 6.4, let $\pi\colon V\to V/\\!\\!/\\!\,\Gamma$
be the quotient map. For $\Gamma=\mathrm{T}$,
$\pi^{-1}(0)=\mathfrak{h}\times\\{0\\}\cup\\{0\\}\times\mathfrak{h}^{*}$ has
dimension $\dim\mathfrak{h}$ and $\pi^{-1}(\overline{u})$ is a single closed
free $\mathrm{T}$-orbit for any $\overline{u}\neq 0$. If we assume
$\dim\mathfrak{h}>1$ then it suffices to show that $\overline{W}$ acts freely
on an open subset of $(V/\\!\\!/\\!\,\Gamma)\setminus\\{0\\}$ whose complement
has codimension at least two (in $(V/\\!\\!/\\!\,\Gamma)\setminus\\{0\\}$ or
in $V/\\!\\!/\\!\,\Gamma$). Similarly, for $\Gamma=\mathrm{SL}_{2}$, it is
shown in the proof of Proposition 6.4 that $\pi^{-1}(\overline{u})$ is a
single closed free $\mathrm{SL}_{2}$-orbit for any $\overline{u}\neq 0$.
Therefore, if we assume $\dim\mathfrak{h}>2$ then it suffices to show that
$\overline{W}$ acts freely an open subset of
$(V/\\!\\!/\\!\,\Gamma)\setminus\\{0\\}$ whose complement has codimension at
least two. Let $U=V\setminus\pi^{-1}(0)$.

For $u\in U$, let $\overline{u}=\pi(u)$. If $w\in W$ fixes $\overline{u}$ then
$w(u)\in\Gamma\cdot u$ so there must exist $z\in\Gamma$ such that $zw(u)=u$.
The element $z$ is unique because $\Gamma_{u}=\\{1\\}$. Moreover, since $w$
has finite order, $z$ must also have finite order. In particular, it is a
semi-simple element of $\Gamma$. We must show that $\dim
T_{\overline{u}}(V/\\!\\!/\\!\,\Gamma)-\dim(T_{\overline{u}}(V/\\!\\!/\\!\,\Gamma))^{w}$
is at least two. If this difference is zero then $w$ acts trivially on
$V/\\!\\!/\\!\,\Gamma$ since the variety is irreducible; in this case $w\in
Z(W)$ by Lemma 6.5 and can be ignored. Thus, it suffices to show that the
difference is always even. Equivalently, the number of non-trivial eigenvalues
(counted with multiplicity) of $w$ acting on
$T_{\overline{u}}(V/\\!\\!/\\!\,\Gamma)$ is even.

If $\mathfrak{k}$ is the Lie algebra of $\Gamma$ then

$T_{\overline{u}}(V/\\!\\!/\\!\,\Gamma)\cong(T_{u}V)/(\mathfrak{k}\cdot
u)=V/(\mathfrak{k}\cdot u).$

and the moment map for the Hamiltonian action of $\Gamma$ on $V$ is given by
$\mu(u)(X)=\omega(X\cdot u,u)$, for $X\in\mathfrak{k}$. Then
$(d_{u}\mu)(v)(X)=2\omega(X\cdot u,v)$ which implies that
$\operatorname{\mathrm{Ker}}(d_{u}\mu)=\mathfrak{k}^{\perp}$ (this is a
general property of moment maps). Identifying $\mathfrak{k}$ with the Lie
algebra of $\Gamma\times W$, there is a $\langle wz\rangle$-equivariant short
exact sequence

$0\to\mathfrak{k}^{\perp}/\mathfrak{k}\to V/(\mathfrak{k}\cdot
u)=T_{\overline{u}}(V/\\!\\!/\\!\,\Gamma)\stackrel{{\scriptstyle
d_{u}\mu}}{{\longrightarrow}}\mathfrak{k}^{*}\to 0.$

Now $\mathfrak{k}^{\perp}/\mathfrak{k}$ is a symplectic vector space on which
$zw$ acts symplectically, so it is certainly the case that
$\dim\mathfrak{k}^{\perp}/\mathfrak{k}-\dim(\mathfrak{k}^{\perp}/\mathfrak{k})^{wz}$
is even. The action of $zw$ on $\mathfrak{k}^{*}$ equals the action of $z$.
Since $z$ is semi-simple, we can choose a maximal torus
$\mathrm{T}\subset\Gamma$ containing $z$. The fact that every root in the root
space decomposition of $\mathfrak{k}$ with respect to $\mathrm{T}$ appears
with its negative implies that
$\dim\mathfrak{k}^{*}-\dim(\mathfrak{k}^{*})^{z}$ is even too.

∎

For the remainder of this article, we assume:

$\dim\mathfrak{h}>1$ when $\Gamma=\mathrm{T}$ and $\dim\mathfrak{h}>2$ when
$\Gamma=\mathrm{SL}_{2}$.

###  6.3. Arbitrary parameters

In this section we assume that the parameters $t,\mathbf{c}$ are arbitrary
scalars. Recall that $Z(W)$ is cyclic (of order $\ell$) since $\mathfrak{h}$
is assumed to be irreducible. Let
$\mathbf{a}=\mathbf{a}_{0},\dots,\mathbf{a}_{\ell-1}\in\mathbb{C}Z(W)$ be the
idempotent basis of this group algebra.

In the case of a real reflection group $W$ we have $\ell=1$ unless $W$ has an
element acting as $-\text{Id}$ on $\mathfrak{h}$, in which case $\ell=2$. Then
the idempotents are $\mathbf{a}=\frac{1}{2}(\text{Id}+(-\text{Id}))$ and
$\mathbf{a}_{1}=\frac{1}{2}(\text{Id}-(-\text{Id}))$.

By assumption (III), the conjuation action of $Z(W)$ on
$H_{t,\mathbf{c}}^{\Gamma}$ is trivial and hence $\mathbb{C}Z(W)$ is central
in $H_{t,\mathbf{c}}^{\Gamma}$.

The following is an immediate consequence of this fact.

######  Lemma 6.8.

$H_{t,\mathbf{c}}^{\Gamma}=\bigoplus_{i=0}^{\ell-1}\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$
as an algebra and hence

$H_{t,\mathbf{c}}^{\Gamma}/\langle
1-\mathbf{a}_{i}\rangle\cong\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}.$

Let $\overline{W}=W/Z(W)$. Recall [12] that a $2$-cocycle on $\overline{W}$
with values in the trivial $\overline{W}$-module $\mathbb{C}^{\times}$ is a
map $f\colon\overline{W}\times\overline{W}\to\mathbb{C}^{\times}$ such that

$f(g_{2},g_{3})f(g_{1},g_{2}g_{3})=f(g_{1}g_{2},g_{3})f(g_{1},g_{2}).$

We have $f=d\alpha$ for some $1$-cocycle $\alpha$ if
$f(g,h)=\alpha(h)\alpha(gh)^{-1}\alpha(g)$. For any $2$-cocycle of
$\overline{W}$, we can form a twisted group algebra
$\mathbb{C}_{f}\overline{W}$ with multiplication $u\cdot w=f(u,w)uw$. Choose
coset representatives $w_{1},\dots,w_{\ell}$ of $Z(W)$ in $W$.

######  Lemma 6.9.

Define $f_{i}\colon\overline{W}\times\overline{W}\to\mathbb{C}^{\times}$ by

$w_{a}w_{b}\mathbf{a}_{i}=f_{i}(\bar{w}_{a},\bar{w}_{b})w_{c}\mathbf{a}_{i}$

if $w_{a}w_{b}=z_{ab}w_{c}$ in $W$ with $z_{ab}\in Z(W)$. Then $f_{i}$ is a
$2$-cocycle and
$(\mathbb{C}W)\mathbf{a}_{i}\cong\mathbb{C}_{f_{i}}\overline{W}$.

###### Proof.

We will check directly that $f_{i}$ is a $2$-cocycle, but then explain
conceptually why this is the case. Note that if $\chi_{i}\colon
Z(W)\to\mathbb{C}^{\times}$ is the character defined by $\mathbf{a}_{i}$ then
$f_{i}(\bar{w}_{a},\bar{w}_{b})=\chi_{i}(z_{ab})$. Let
$w_{a}w_{b}=z_{ab}w_{x}$ and $w_{b}w_{c}=z_{bc}w_{y}$. Note that

$z_{ay}w_{u}:=w_{a}w_{y}=w_{a}(z_{bc}^{-1}w_{b}w_{c})=z_{ab}z_{bc}^{-1}w_{x}w_{c}.$

Then
$f_{i}(\overline{w}_{b},\bar{w}_{c})f_{i}(\bar{w}_{a},\bar{w}_{b}\bar{w}_{c})=\chi_{i}(z_{bc})\chi_{i}(z_{ay})$
and

$f_{i}(\bar{w}_{a}\bar{w}_{b},\bar{w}_{c})f_{i}(\bar{w}_{a},\bar{w}_{b})=\chi_{i}(z_{ay}z_{bc}z_{ab}^{-1})\chi_{i}(z_{ab})=\chi_{i}(z_{bc})\chi_{i}(z_{ay}).$

∎

As explained in [12], there is a canonical bijection between elements of
$H^{2}(\overline{W},Z(W))$ and isomorphism classes of central extensions $1\to
Z(W)\to G\to\overline{W}\to 1$. In particular, $W$ itself corresponds to a
class $[f]\in H^{2}(\overline{W},Z(W))$. The explicit construction given in
[12] shows that $f_{i}=\chi_{i}\circ f$. It follows directly that $f_{i}$ is a
$2$-cocycle. Note that if we choose new representatives
$w_{a}^{\prime}=z_{a}w_{a}$ and define
$\alpha_{i}\colon\overline{W}\to\mathbb{C}^{\times}$ by
$\alpha_{i}(\bar{w}_{a})=\chi_{i}(z_{a})$ then
$f_{i}^{\prime}(\bar{w}_{a},\bar{w}_{b})=f_{i}(\bar{w}_{a},\bar{w}_{b})(d\alpha_{i})(\bar{w}_{a},\bar{w}_{b})$.
This shows that $[f]=[f^{\prime}]$ in $H^{2}(\overline{W},Z(W))$.

######  Lemma 6.10.

The ring $\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ contains
$\mathbb{C}_{f_{i}}\overline{W}$ and
$\operatorname{\mathrm{gr}}(\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i})\cong
P^{\Gamma}\rtimes\mathbb{C}_{f_{i}}\overline{W}$.

###### Proof.

The first claim follows from Lemma 6.9.

Note that
$0\to\mathcal{F}_{j-1}H_{t,\mathbf{c}}^{\Gamma}\to\mathcal{F}_{j}H_{t,\mathbf{c}}^{\Gamma}\to\operatorname{\mathrm{gr}}_{j}H_{t,\mathbf{c}}^{\Gamma}\to
0$ is a sequence of $\mathbb{C}Z(W)$-modules. This implies that
$\operatorname{\mathrm{gr}}(\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i})\cong\mathbf{a}_{i}(P^{\Gamma}\rtimes
W)\mathbf{a}_{i}$. Then, once again by Lemma 6.9,
$\mathbf{a}_{i}(P^{\Gamma}\rtimes
W)\mathbf{a}_{i}=P^{\Gamma}\rtimes\mathbb{C}_{f_{i}}\overline{W}$. ∎

######  Corollary 6.11.

The ring $\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ is prime.

###### Proof.

By Lemma 6.10, it suffices to prove that
$P^{\Gamma}\rtimes\mathbb{C}_{f_{i}}\overline{W}$ is a prime ring. But this is
precisely the statement of [32, Corollary 12.6] since the group $\overline{W}$
acts by outer automorphisms on $P^{\Gamma}$. ∎

We will usually focus on $\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a}$
rather than $H^{\Gamma}_{t,\mathbf{c}}$. The main reason for this is that
Corollary 6.11 says that $\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a}$ is
prime. The ring $H^{\Gamma}_{t,\mathbf{c}}$ is prime if and only if
$H^{\Gamma}_{t,\mathbf{c}}=\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a}$,
which is the case precisely when $Z(W)=1$.

We recall that a noetherian algebra $A$ is Auslander–Gorenstein if it has both
finite left and right injective dimensions and for each finitely generated
(left or right) $A$-module $M$ the following holds: for every integer $i\geq
0$ and every submodule $N$ of $\operatorname{\mathrm{Ext}}^{i}_{A}(M,A)$, we
have $\operatorname{\mathrm{Ext}}^{j}_{A}(N,A)=0$ for all $j<i$. As noted in
[9], by a theorem of Zaks, if a noetherian algebra has finite right and left
injective dimensions, then these are equal.

The grade of a finitely generated left $A$-module $M$ is defined to be

$j_{A}(M)=\min\\{i\in\mathbb{Z}_{\geq
0}\,|\,\operatorname{\mathrm{Ext}}^{i}_{A}(M,A)\neq 0\\}.$

Assume that $A$ has finite Gelfand–Kirillov dimension. Then $A$ is said to be
(GK) Cohen–Macaulay if

$j_{A}(M)+\mathrm{GK}\textrm{-}\mathrm{dim}(M)=\mathrm{GK}\textrm{-}\mathrm{dim}(A),$

for all finitely generated left $A$-modules.

Now let $A=P^{\Gamma}\rtimes W$. Consider $A$ as the left $P^{\Gamma}$-module.
Then the space $\operatorname{\mathrm{Hom}}_{P^{\Gamma}}(A,P^{\Gamma})$ is a
left $A$-module via $(a\star\phi)(b)=\phi(ba)$ for $a,b\in A$ and
$\phi\in\operatorname{\mathrm{Hom}}_{P^{\Gamma}}(A,P^{\Gamma})$.

######  Lemma 6.12.

There is an isomorphism of left $A$-modules
$\sigma\colon{}_{A}A{\;\stackrel{{{}_{\sim}}}{{\to}}\;}\operatorname{\mathrm{Hom}}_{P^{\Gamma}}({}_{P^{\Gamma}}A,P^{\Gamma})$
given by

$\sigma(f\otimes u)\left(\sum_{w\in W}b_{w}\otimes
w\right)=b_{u^{-1}}u^{-1}(f),\quad\forall\ f\in P^{\Gamma},u\in W.$

###### Proof.

It is immediate that $\sigma(f\otimes
u)\in\operatorname{\mathrm{Hom}}_{P}({}_{P}A,P)$. We check that it is a
morphism of left $A$-modules. Let $f_{i}\in P^{\Gamma},u_{i}\in W$. Then
$(f_{1}\otimes u_{1})(f_{2}\otimes u_{2})=f_{1}u_{1}(f_{2})\otimes
u_{1}u_{2}$. So

$\displaystyle\sigma((f_{1}\otimes u_{1})(f_{2}\otimes u_{2}))(b)$
$\displaystyle=\sigma(f_{1}u_{1}(f_{2})\otimes u_{1}u_{2})(b)$
$\displaystyle=b_{u_{2}^{-1}u_{1}^{-1}}u_{2}^{-1}(u_{1}^{-1}(f_{1}u_{1}(f_{2})))$
$\displaystyle=b_{u_{2}^{-1}u_{1}^{-1}}u_{2}^{-1}(u_{1}^{-1}(f_{1})f_{2})$
$\displaystyle=b_{u_{2}^{-1}u_{1}^{-1}}(u_{2}^{-1}u_{1}^{-1}(f_{1}))u_{2}^{-1}(f_{2}),$

and

$\displaystyle(f_{1}\otimes u_{1})\star\sigma(f_{2}\otimes u_{2})(b)$
$\displaystyle=\sigma(f_{2}\otimes u_{2})(b(f_{1}\otimes u_{1}))$
$\displaystyle=\sigma(f_{2}\otimes u_{2})\left(\sum_{w\in
W}b_{w}w(f_{1})\otimes wu_{1}\right)$
$\displaystyle=b_{u_{2}^{-1}u_{1}^{-1}}(u_{2}^{-1}u_{1}^{-1}(f_{1}))u_{2}^{-1}(f_{2}).$

∎

######  Lemma 6.13.

Let $M$ be a finitely generated left $A$-module. Then
$j_{A}(M)=j_{P^{\Gamma}}(M)$.

###### Proof.

Since $A$ is free of finite rank over $P^{\Gamma}$, the Eckmann-Shapiro Lemma,
together with Lemma 6.12, says that

$\operatorname{\mathrm{Ext}}^{i}_{P^{\Gamma}}(M,P^{\Gamma})=\operatorname{\mathrm{Ext}}^{i}_{A}(M,\operatorname{\mathrm{Hom}}_{P^{\Gamma}}(A,P^{\Gamma}))=\operatorname{\mathrm{Ext}}^{i}_{A}(M,A).$

The lemma follows. ∎

######  Theorem 6.14.

The algebras $H_{t,\mathbf{c}}^{\Gamma}$ and
$\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ are
Auslander–Gorenstein and (GK) Cohen–Macaulay. These algebras have
Gelfand–Kirillov dimension $2\dim\mathfrak{h}-\dim\Gamma$.

###### Proof.

By Proposition 6.4, the ring $P^{\Gamma}$ is a commutative noetherian
Gorenstein domain. Therefore it is Auslander–Gorenstein. It follows from [34]
that $\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{\Gamma}=P^{\Gamma}\rtimes W$
is Auslander–Gorenstein. Hence $H_{t,\mathbf{c}}^{\Gamma}$ is also
Auslander–Gorenstein by [5, Theorem 3.9]. This implies that each summand
$\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ is also
Auslander–Gorenstein.

Since the generic $\Gamma$-orbit in $\mathfrak{h}\times\mathfrak{h}^{*}$ is
closed by Lemma 6.3, the algebra

$P^{\Gamma}=\mathbb{C}\left[(\mathfrak{h}\times\mathfrak{h}^{*})/\\!\\!/\\!\,\Gamma\right]$

has Krull (and hence Gelfand–Kirillov) dimension
$2\dim\mathfrak{h}-\dim\Gamma$. By [30, Proposition 8.1.14], it follows that
both $P^{\Gamma}\rtimes W$ and $H_{t,\mathbf{c}}^{\Gamma}$ also have GK-
dimension $2\dim\mathfrak{h}-\dim\Gamma$.

Finally, we check that $H_{t,\mathbf{c}}^{\Gamma}$ and
$\mathbf{a}_{i}H_{t,\mathbf{c}}^{\Gamma}\mathbf{a}_{i}$ are Cohen–Macaulay. By
[30, Proposition 8.6.5], it suffices to show that $A=P^{\Gamma}\rtimes W$ and
$\mathbf{a}_{i}A\mathbf{a}_{i}$ are Cohen–Macaulay. Note that they have finite
GK-dimension by the previous paragraph. The commutative ring $P^{\Gamma}$ is
Cohen–Macaulay by Hochster’s Theorem [13, Theorem 6.4.2]. Thus, if $M$ is a
finitely generated $A$-module, it is finitely generated over $P^{\Gamma}$ and
applying Lemma 6.13 gives

$\mathrm{GK}\textrm{-}\mathrm{dim}(A)=\mathrm{GK}\textrm{-}\mathrm{dim}(P^{\Gamma})=\mathrm{GK}\textrm{-}\mathrm{dim}_{P^{\Gamma}}(M)+j_{P^{\Gamma}}(M)=\mathrm{GK}\textrm{-}\mathrm{dim}_{A}(M)+j_{A}(M).$

Thus, $A$ is Cohen–Macaulay. Each $\mathbf{a}_{i}$ is central in $A$ and
$\mathbf{a}_{i}A\mathbf{a}_{i}$ has the same GK-dimension as $A$. Therefore,
if $M_{i}$ is a finitely generated $\mathbf{a}_{i}A\mathbf{a}_{i}$-module,
$\operatorname{\mathrm{Ext}}_{A}^{r}(M_{i},A)=\operatorname{\mathrm{Ext}}^{r}_{\mathbf{a}_{i}A\mathbf{a}_{i}}(M_{i},\mathbf{a}_{i}A\mathbf{a}_{i})$
and
$\mathrm{GK}\textrm{-}\mathrm{dim}_{\mathbf{a}_{i}A\mathbf{a}_{i}}(M_{i})=\mathrm{GK}\textrm{-}\mathrm{dim}_{A}(M_{i})$.
Thus,

$j_{\mathbf{a}_{i}A\mathbf{a}_{i}}(M_{i})+\mathrm{GK}\textrm{-}\mathrm{dim}_{\mathbf{a}_{i}A\mathbf{a}_{i}}(M_{i})=j_{A}(M_{i})+\mathrm{GK}\textrm{-}\mathrm{dim}_{A}(M_{i})=\mathrm{GK}\textrm{-}\mathrm{dim}(A)=\mathrm{GK}\textrm{-}\mathrm{dim}(\mathbf{a}_{i}A\mathbf{a}_{i}),$

showing that $\mathbf{a}_{i}A\mathbf{a}_{i}$ is Cohen–Macaulay too. ∎

We note for later use:

######  Corollary 6.15.

$\mathbf{a}H^{\Gamma}_{\mathbf{c}}e$ is a Cohen–Macaulay
$eH^{\Gamma}_{\mathbf{c}}e$-module.

###### Proof.

The proof is identical to the proof of [18, Theorem 1.5(ii)], using the fact
that $P^{\Gamma}$ is a Cohen–Macaulay ring and a finitely generated module
over the Gorenstein ring $P^{\Gamma\times W}$ (see Proposition 6.4). ∎

###  6.4. Double centralizer property

Let $e\in\mathbb{C}W$ denote the trivial idempotent. Note that

$\operatorname{\mathrm{End}}_{\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a}}(\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}e)\cong
eH^{\Gamma}_{t,\mathbf{c}}e.$

Indeed, consider a map $\varphi\colon
eH^{\Gamma}_{t,\mathbf{c}}e\to\operatorname{\mathrm{End}}_{\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a}}(\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}e)$
given by $\varphi(u)=f_{u}$, where the element
$f_{u}\in\operatorname{\mathrm{End}}_{\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a}}(\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}e)$
is determined by $f_{u}(e)=u$, $u\in eH^{\Gamma}_{t,\mathbf{c}}e$. It is easy
to see that $\varphi$ is an isomorphism. The next theorem establishes the
double centralizer property for the bimodule
$\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}e$.

######  Theorem 6.16.

$\operatorname{\mathrm{End}}_{eH^{\Gamma}_{t,\mathbf{c}}e}(\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}e)\cong(\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a})^{\mathrm{op}}$.

###### Proof.

First, we note that
$\mathbf{a}H^{\Gamma}_{0,0}\mathbf{a}=\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{*}]^{\Gamma}\rtimes\overline{W}$.
Since $(\mathfrak{h}\times\mathfrak{h}^{*})/\\!\\!/\\!\,\Gamma$ is a normal
variety and the group $\overline{W}$ acts freely on a dense open subset of
$(\mathfrak{h}\times\mathfrak{h}^{*})/\\!\\!/\\!\,\Gamma$ with complement of
codimension at least two (Proposition 6.7), the double centralizer property
holds for $\mathbf{a}H^{\Gamma}_{0,0}\mathbf{a}$ by [6, Proposition E.2.2(2)].
Therefore the proof of [18, Theorem 1.5(vi)] shows that it holds for all
parameters $(t,\mathbf{c})$. ∎

We note that the Satake isomorphism holds in this situation. The proof is
identical to that of the usual Satake isomorphism for symplectic reflection
algebras; see [3, Theorem 1.7.3].

######  Corollary 6.17.

The map $z\mapsto ez$ is an isomorphism
$Z(\mathbf{a}H^{\Gamma}_{t,\mathbf{c}}\mathbf{a}){\;\stackrel{{{}_{\sim}}}{{\to}}\;}Z(eH^{\Gamma}_{t,\mathbf{c}}e)$.

When $t=0$, it follows from Corollary 6.17 that
$Z(\mathbf{a}H^{\Gamma}_{0,\mathbf{c}}\mathbf{a}){\;\stackrel{{{}_{\sim}}}{{\to}}\;}eH^{\Gamma}_{0,\mathbf{c}}e$.

###  6.5. The centre of $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ at $t\neq 0$

We note that it is not true that the centre of
$H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ is trivial when $t\neq 0$. Indeed, it
was shown in [19, Section 7] that
$Z(H_{1,\mathbf{c}}^{\mathfrak{gl}(n)}(\mathfrak{S}_{n}))=\mathbb{C}[\mathrm{eu}_{\mathbf{c}}]$.
We show that a similar result holds for any complex reflection group.

The centre $P^{\mathrm{T}\times W}\otimes\mathbb{C}Z(W)$ of
$P^{\mathrm{T}}\rtimes W$ becomes a Poisson algebra because
$H_{t,\mathbf{c}}^{\mathrm{T}}$ is a quantization of $P^{\mathrm{T}}\rtimes W$
for any $(t,\mathbf{c})$ with $t\neq 0$. The Poisson structure on
$P^{\mathrm{T}\times W}\otimes\mathbb{C}Z(W)$ is independent of the choice of
$(t,\mathbf{c})$.

If $B$ is a Poisson algebra then $\mathrm{Cas}(B)\subset B$ denotes the
Poisson centre, consisting of all elements $b$ such that $\\{b,-\\}=0$
($\mathrm{Cas}$ stands for ”Casimirs”).

######  Lemma 6.18.

Let $A,B$ be $\mathbb{C}$-algebras. Assume $B$ is a Possion domain and
$A\subset B$ a noetherian Poisson subalgebra such that $B$ is a finite
$A$-module. Then $\mathrm{Cas}(A)=\mathrm{Cas}(B)\cap A$.

###### Proof.

It is clear that $\mathrm{Cas}(B)\cap A\subset\mathrm{Cas}(A)$. Let
$a\in\mathrm{Cas}(A)$ and $\delta=\\{a,-\\}$ the corresponding derivation of
$B$. Then $\delta|_{A}=0$ by assumption and we wish to show that $\delta=0$.
Since our algebras are over $\mathbb{C}$, the field of fractions
$\mathrm{Frac}(B)$ is a finite separable extension of $\mathrm{Frac}(A)$ and
hence the extension is generically étale. Thus, there exists $f\in A$ such
that the extension $A[f^{-1}]\subset B[f^{-1}]$ is étale. The derivation
$\delta$ extends (uniquely) to $B[f^{-1}]$ with $\delta|_{A[f^{-1}]}=0$. But,
since the extension is étale, we have

$\mathrm{Der}(A[f^{-1}])\subset\mathrm{Der}(B[f^{-1}])=B[f^{-1}]\otimes_{A[f^{-1}]}\mathrm{Der}(A[f^{-1}]).$

Thus, $\delta|_{A[f^{-1}]}=0$ implies that $\delta=0$. ∎

######  Lemma 6.19.

The Poisson centre of $P^{\mathrm{T}\times W}\otimes\mathbb{C}Z(W)$ equals
$\mathbb{C}[\mathrm{eu}_{0}]\otimes\mathbb{C}Z(W)$.

###### Proof.

The idempotents $\mathbf{a}_{i}\in\mathbb{C}Z(W)$ are central in
$H_{1,\mathbf{c}}^{\mathrm{T}}(W)$. This means that they are Poisson central
in $P^{T\times W}\otimes\mathbb{C}Z(W)$. Hence the map
$(f_{i})\mapsto\sum_{i}f_{i}\otimes\mathbf{a}_{i}$ is an isomorphism of
Poisson algebras $(P^{\mathrm{T}\times W})^{\oplus\ell}\cong
P^{\mathrm{T}\times W}\otimes\mathbb{C}Z(W)$. Therefore, it suffices to show
that $\mathrm{Cas}(P^{\mathrm{T}\times W})=\mathbb{C}[\mathrm{eu}_{0}]$.

The Poisson structure on $P^{\mathrm{T}\times W}$ comes by restriction from
the Possion structure on $P^{\mathrm{T}}$. In other words,
$P^{\mathrm{T}\times W}\subset P^{\mathrm{T}}$ is a (finite) extension of
Poisson algebras. Both these algebras are noetherian domains since they are
subrings of invariants in $P$. Therefore, Lemma 6.18 says that
$\mathrm{Cas}(P^{\mathrm{T}\times W})$ is the intersection of
$\mathrm{Cas}(P^{\mathrm{T}})$ with $P^{\mathrm{T}\times W}$. Hence, it
suffices to argue that
$\mathrm{Cas}(P^{\mathrm{T}})=\mathbb{C}[\mathrm{eu}_{0}]$. This is precisely
the content of the proof of [19, Lemma 1]. ∎

######  Proposition 6.20.

For $t\neq 0$, the centre of $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$ equals
$\mathbb{C}[\mathrm{eu}_{\mathbf{c}}]\otimes\mathbb{C}Z(W)$.

###### Proof.

It is clear that $\mathbb{C}[\mathrm{eu}_{\mathbf{c}}]\otimes\mathbb{C}Z(W)$
belongs to the centre of $H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}$. Now assume
$f\in Z(H_{t,\mathbf{c}}^{\mathfrak{gl}(n)})$. Then the symbol $\sigma(f)$ of
$f$ belongs to the Poisson centre of

$Z(\operatorname{\mathrm{gr}}H_{t,\mathbf{c}}^{\mathfrak{gl}(n)})=Z(P^{\mathrm{T}\rtimes
W})=P^{\mathrm{T}\times W}\otimes\mathbb{C}Z(W).$

The Poisson centre of $P^{\mathrm{T}}\rtimes W$ equals
$\mathbb{C}[\mathrm{eu}_{0}]\otimes\mathbb{C}Z(W)$ by Lemma 6.19. We argue by
induction on the degree of $\sigma(f)$ that this means
$f\in\mathbb{C}[\mathrm{eu}_{\mathbf{c}}]\otimes\mathbb{C}Z(W)$. If
$\deg\sigma(f)=0$ then $f\in\mathcal{F}_{0}=\mathbb{C}Z(W)$ since
$\mathcal{F}_{-1}=0$. Therefore, we may assume $\deg\sigma(f)=k>0$ and
$g\in\mathbb{C}[\mathrm{eu}_{0}]\otimes\mathbb{C}Z(W)$ for all $g\in
Z(H_{t,\mathbf{c}}^{\mathfrak{gl}(n)})$ of degree less than $k$. Since
$\sigma(f)$ is homogeneous, $\sigma(f)=\sum_{i}\mathrm{eu}_{0}^{k}\otimes
d_{i}\mathbf{a}_{i}$ for some $d_{i}\in\mathbb{C}$. Then
$f-\sum_{i}\mathrm{eu}_{\mathbf{c}}^{k}\otimes
d_{i}\mathbf{a}_{i}\in\mathcal{F}_{k-1}$ is still central. By induction,
$f-\sum_{i}\mathrm{eu}_{\mathbf{c}}^{k}\otimes
d_{i}\mathbf{a}_{i}\in\mathbb{C}[\mathrm{eu}_{\mathbf{c}}]\otimes\mathbb{C}Z(W)$
and hence so too is $f$. ∎

Proposition 6.20, together with Lemma 6.8, implies that the centre of
$\mathbf{a}H_{1,\mathbf{c}}^{\mathfrak{gl}(n)}\mathbf{a}$ equals
$\mathbb{C}[\mathbf{a}\mathrm{eu}_{\mathbf{c}}]$.

##  7\. Invariants at $t=0$

In this section we consider the properties of the invariant rings at $t=0$.
For brevity, we write $H_{\mathbf{c}}=H_{\mathbf{c}}(W)$ for the rational
Cherednik algebra at $t=0$ and some fixed $\mathbf{c}$.

###  7.1. The centre at $t=0$

We begin by considering the case $\mathbf{c}=0$. Let
$U=(\mathfrak{h}\otimes\mathfrak{h}^{*})^{\Gamma}$ and note that $W$ acts on
$U$ by (II).

######  Proposition 7.1.

If $\operatorname{\mathrm{Ker}}(W\to\operatorname{\mathrm{GL}}(U))=Z(W)$ then
$Z(P^{\Gamma}\rtimes W)=P^{\Gamma\times W}\otimes\mathbb{C}Z(W)$

###### Proof.

If we identify degree $2$ part of the $\mathbb{N}$-graded algebra $P$ with
$\mathfrak{h}\otimes\mathfrak{h}^{*}$ then $U\subset P^{\Gamma}$. Let
$a=\sum_{w}f_{w}\otimes w$ be in the centre of $P^{\Gamma}\rtimes W$. Then,
for $u\in U$,

$[a,u]=\sum_{w}f_{w}(w^{-1}(u)-u)\otimes w.$

For this to be zero, we must have $w^{-1}(u)=u$ for all $w$ with $f_{w}\neq 0$
because $P^{\Gamma}$ is a domain. Since
$\operatorname{\mathrm{Ker}}(W\to\operatorname{\mathrm{GL}}(U))=Z(W)$, it
follows that $a=\sum_{w\in Z(W)}f_{w}\otimes w$. Moreover, $[a,u]=0$ for all
$u\in W$ implies that $f_{w}\in P^{\Gamma\times W}$.

On the other hand, it is clear that $a=\sum_{w\in Z(W)}f_{w}\otimes w$ with
$f_{w}\in P^{\Gamma\times W}$ belongs to the centre.

∎

######  Remark 7.2.

In the case $\Gamma=\mathrm{T}$, the space $U$ equals
$\mathfrak{h}\otimes\mathfrak{h}^{*}=\mathfrak{gl}(n)$ and
$\operatorname{\mathrm{Ker}}(W\to\operatorname{\mathrm{GL}}(\mathfrak{gl}(n)))$
equals $Z(W)$. This follows from the fact that
$W\cap\mathbb{C}^{\times}\mathrm{Id}_{\mathfrak{h}}=Z(W)$ in
$\operatorname{\mathrm{GL}}(\mathfrak{h})$. In the case
$\Gamma=\mathrm{SL}_{2}$, we identify
$U=\mathfrak{so}(n)\subset\mathfrak{gl}(n)$ and Corollary 4.14 says that
$\operatorname{\mathrm{Ker}}(W\to\operatorname{\mathrm{GL}}(\mathfrak{so}(n)))=Z(W)$
when $\dim\mathfrak{h}>2$.

Proposition 7.1 implies that $Z(P^{\Gamma}\rtimes W)=P^{\Gamma\times
W}\otimes\mathbb{C}Z(W)$ has basis
$\mathbf{a}=\mathbf{a}_{0},\dots,\mathbf{a}_{\ell-1}$ as a module over the
algebra $P^{\Gamma\times W}$.

######  Lemma 7.3.

The ring $Z(H_{\mathbf{c}}^{\Gamma})$ is a
$Z(H_{\mathbf{c}})^{\Gamma}$-algebra and $Z(W)\subset
Z(H_{\mathbf{c}}^{\Gamma})$.

###### Proof.

For the first claim, it suffices to note that the inclusion
$Z(H_{\mathbf{c}})\subset H_{\mathbf{c}}$ restricts to an inclusion of
algebras $Z(H_{\mathbf{c}})^{\Gamma}\subset Z(H_{\mathbf{c}}^{\Gamma})$. The
inclusion $Z(W)\subset Z(H_{\mathbf{c}}^{\Gamma})$ follows from the fact that
the image of the conjugation action of $Z(W)$ on $H_{\mathbf{c}}$ is assumed
by (III) to be in $\Gamma$. This means that for $z\in Z(W)$ and $h\in
H^{\Gamma}_{\mathbf{c}}$, $zhz^{-1}=h$. ∎

######  Lemma 7.4.

$\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}})^{\Gamma}=P^{\Gamma\times W}$.

###### Proof.

Since $\Gamma$ is reductive Lemma 2.1(iii) applies and so
$\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}})^{\Gamma}=(\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}}))^{\Gamma}$.
Then [18, Theorem 3.3] states that
$\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}})=Z(H_{0})$ completing the proof. ∎

We define
$Y_{\mathbf{c}}(W)=\operatorname{\mathrm{Spec}}\,Z(H_{\mathbf{c}})^{\Gamma}$.

######  Theorem 7.5.

The affine scheme $Y_{\mathbf{c}}(W)$ is a normal irreducible variety. It is
Gorenstein with rational singularities.

###### Proof.

The ring $Z(H_{\mathbf{c}})$ is an integrally closed domain since its
associated graded is $\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{*}]^{W}$,
which is integrally closed. It is well-known that taking invariants with
respect to a reductive group preserves these properties.

Since $\operatorname{\mathrm{Spec}}Z(H_{\mathbf{c}})$ has rational
singularities (see the proof of [25, Proposition 4.5]) and $Y_{\mathbf{c}}(W)$
is the categorical quotient of $\operatorname{\mathrm{Spec}}Z(H_{\mathbf{c}})$
by a reductive group, [8] shows that $Y_{\mathbf{c}}(W)$ has rational
singularities.

The algebra $Z(H_{\mathbf{c}})^{\Gamma}$ is a filtered algebra with associated
graded equal to $\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{*}]^{\Gamma\times
W}$. Therefore it follows from Proposition 6.4 and [5, Theorem 3.9] that
$Y_{\mathbf{c}}(W)$ is Gorenstein. ∎

The following result was the main motivation for this work.

######  Theorem 7.6.

For any $\mathbf{c}$,
$\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}}^{\Gamma})=Z(P^{\Gamma}\rtimes W)$.

###### Proof.

First, Corollary 4.3 and Proposition 7.1 imply that

$Z(\operatorname{\mathrm{gr}}H_{\mathbf{c}}^{\Gamma})=Z(P^{\Gamma}\rtimes
W)=P^{\Gamma\times W}\otimes\mathbb{C}Z(W).$

Next, Lemma 2.1(ii) says that $\mathrm{gr}\,Z(H_{\mathbf{c}}^{\Gamma})$ is
contained in $Z(\operatorname{\mathrm{gr}}H_{\mathbf{c}}^{\Gamma})$.
Therefore, the claim follows if we can show that $Z(P^{\Gamma}\rtimes W)$ is
contained in $\mathrm{gr}\,Z(H_{\mathbf{c}}^{\Gamma})$; we need to show that
$P^{\Gamma\times W}$ and $Z(W)$ are contained in
$\mathrm{gr}\,Z(H_{\mathbf{c}}^{\Gamma})$.

Lemma 7.3 implies that $Z(W)\subset\mathcal{F}_{0}Z(H_{\mathbf{c}}^{\Gamma})$
and hence $Z(W)\subset\mathrm{gr}\,Z(H_{\mathbf{c}}^{\Gamma})$ because
$\mathcal{F}_{-1}Z(H_{\mathbf{c}}^{\Gamma})=0$ (by definition,
$\mathcal{F}_{-1}H_{\mathbf{c}}=0$). Similarly, Lemma 7.3 says that
$Z(H_{\mathbf{c}})^{\Gamma}$ is contained in $Z(H_{\mathbf{c}}^{\Gamma})$.
Hence, Lemma 2.1(i) implies that
$\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}})^{\Gamma}$ is contained in
$\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}}^{\Gamma})$. By Lemma 7.4, this
means that $P^{\Gamma\times W}$ also belongs to
$\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}}^{\Gamma})$ as required. ∎

######  Corollary 7.7.

There is equality of commutative rings

$Z(H_{\mathbf{c}}^{\Gamma})=Z(H_{\mathbf{c}})^{\Gamma}\otimes\mathbb{C}Z(W).$

In particular, as a $Z(H_{\mathbf{c}})^{\Gamma}$-module,
$Z(H_{\mathbf{c}}^{\Gamma})=\bigoplus_{i=0}^{\ell-1}Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i}$.

###### Proof.

Since the $\mathbf{a}_{i}$ are a basis of $\mathbb{C}Z(W)$, it suffices to
show that $Z(H_{\mathbf{c}}^{\Gamma})$ equals the direct sum
$\bigoplus_{i=0}^{\ell-1}Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i}$, with each
$Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i}$ free of rank one over
$Z(H_{\mathbf{c}})^{\Gamma}$.

At $\mathbf{c}=0$, the space $Z(H_{0})^{\Gamma}\mathbf{a}_{i}=P^{\Gamma\times
W}\mathbf{a}_{i}$ is clearly free of rank one over $P^{\Gamma\times W}$. Since

$\operatorname{\mathrm{gr}}(Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i})=P^{\Gamma\times
W}\mathbf{a}_{i}$

(as in the proof of Lemma 6.10) it follows that
$Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i}$ is free of rank one over
$Z(H_{\mathbf{c}})^{\Gamma}$. The inclusion
$\bigoplus_{i=0}^{\ell-1}Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i}\subset
Z(H_{\mathbf{c}}^{\Gamma})$ gives rise to an inclusion

$P^{\Gamma\times
W}\otimes_{\mathbb{C}}\mathbb{C}Z(W)=\bigoplus_{i=0}^{\ell-1}P^{\Gamma\times
W}\mathbf{a}_{i}=\operatorname{\mathrm{gr}}\left(\bigoplus_{i=0}^{\ell-1}Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i}\right)\subset\operatorname{\mathrm{gr}}Z(H_{\mathbf{c}}^{\Gamma})=P^{\Gamma\times
W}\otimes_{\mathbb{C}}\mathbb{C}Z(W).$

We deduce from Lemma 2.1(i) that
$\bigoplus_{i=0}^{\ell-1}Z(H_{\mathbf{c}})^{\Gamma}\mathbf{a}_{i}=Z(H_{\mathbf{c}}^{\Gamma})$.
∎

Corollary 7.7 implies that $Z(H_{\mathbf{c}}^{\Gamma})$ is a domain if and
only if $Z(W)=1$. Each irreducible component of
$\operatorname{\mathrm{Spec}}Z(H_{\mathbf{c}}^{\Gamma})$ is isomorphic to
$\operatorname{\mathrm{Spec}}Z(H_{\mathbf{c}})^{\Gamma}$ since
$z\to\mathbf{a}_{i}z$ is a ring homomorphism. Thus,
$\operatorname{\mathrm{Spec}}Z(H_{\mathbf{c}}^{\Gamma})$ is the disjoint union
of $\ell$ copies of $Y_{\mathbf{c}}$.

###  7.2. Simple modules

In this section we consider the structure of a generic simple
$\mathbf{a}H_{\mathbf{c}}^{\Gamma}\mathbf{a}$-module. Let $k$ be an
algebraically closed field, $A$ a prime affine $k$-algebra that is a finite
module over its centre $Z(A)$. Let $M\subset A$ be a maximal ideal and
$\mathfrak{m}=M\cap Z(A)$. We recall that $M$ is said to be regular if one of
the following equivalent conditions hold

  1. (a)

$M$ is the unique maximal ideal of $A$ such that $M\cap Z(A)=\mathfrak{m}$

  2. (b)

there is a unique simple $A$-module $L$ whose annihilator in $Z(A)$ equals
$\mathfrak{m}$ (and thus, $\mathrm{ann}_{A}L=M$);

  3. (c)

$A_{\mathfrak{m}}$ is Azumaya over $Z(A)_{\mathfrak{m}}$.

For the proof and other equivalent characterizations of regular, see [11, III,
Theorem 1.6].

The following lemma says that given a flat family of PI-algebras, where the
centre of the algebras also varies flatly, the PI-degree of the algebras is
constant. Hence, it need only be computed for one fibre.

######  Lemma 7.8.

Let $R\subset Z(A)$ be a subalgebra and $\mathfrak{p}\lhd R$ a (prime) ideal
such that $A/\mathfrak{p}A$ is prime and the map $Z(A)\to Z(A/\mathfrak{p}A)$
is surjective. Then

$\mathrm{P.I.deg}(A)=\mathrm{P.I.deg}(A/\mathfrak{p}A).$

###### Proof.

Let $\overline{M}$ be regular maximal ideal of $A/\mathfrak{p}A$ and $M$ its
preimage in $A$. It suffices to show that $M$ is a regular (maximal) ideal in
$A$. Let $\overline{\mathfrak{m}}=Z(A/\mathfrak{p}A)\cap\overline{M}$ and
write $\mathfrak{m}$ for the preimage of $\overline{\mathfrak{m}}$ in $Z(A)$.
Then $\overline{M}=\overline{\mathfrak{m}}(A/\mathfrak{p}A)$ by [11, Theorem
III.1.6]. We claim that $\mathfrak{m}=M\cap Z(A)$ and $M=\mathfrak{m}A$. If
this is indeed the case then $M$ is regular in $A$, again by [11, Theorem
III.1.6].

Let $z\in Z(A)$ such that $\overline{z}\in\overline{\mathfrak{m}}$. Then
$\overline{z}\in\overline{M}$ and $\overline{z}\in Z(A/\mathfrak{p}A)$. Hence
$z\in M$ and $M\cap Z(A)\subset\mathfrak{m}$. But both are maximal ideals of
$Z(A)$, thus $M\cap Z(A)=\mathfrak{m}$.

Finally, we argue that $M=\mathfrak{m}A$. Clearly, $\mathfrak{m}A\subset M$.
If $x\in M$ then there exist $\overline{z}_{i}\in\overline{\mathfrak{m}}$ and
$\overline{x}_{i}\in A/\mathfrak{p}A$ such that
$\overline{x}=\sum_{i}\overline{z}_{i}\overline{x}_{i}$ in $A/\mathfrak{p}A$.
We choose lifts $z_{i}\in\mathfrak{m}$ and $x_{i}\in A$. Then
$x-\sum_{i}z_{i}x_{i}\in\mathfrak{p}A$. But
$\mathfrak{p}A\subset\mathfrak{m}A$ since $\mathfrak{p}\subset\mathfrak{m}$.
Thus, $x\in\mathfrak{m}A$. ∎

######  Corollary 7.9.

Let $A$ be a prime affine $k$-algebra, finite over $Z(A)$, and $R\subset Z(A)$
such that for all $\mathfrak{m}\lhd R$ maximal the map $Z(A)\to
Z(A(\mathfrak{m}))$ is surjective, where $A(\mathfrak{m}):=A/\mathfrak{m}A$.
Then

$\mathrm{P.I.deg}(A)=\mathrm{P.I.deg}(A/\mathfrak{m}A),\quad\forall\,\mathfrak{m}.$

Recall from Section 6.1 that
$(t,\mathbf{c})\in\mathbb{C}\oplus\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})$,
$R=\mathbb{C}[\mathbb{C}\oplus\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})]$
and $H$ denotes the rational Cherednik algebra defined over $R$. We wish to
consider the case where $t=0$ but $\mathbf{c}$ is a variable, so we set
$R_{0}=R/(t)$ and $H_{0}=H/tH$. The following lemma shows that the hypothesis
of Corollary 7.9 hold for $\mathbf{a}H^{\Gamma}_{0}\mathbf{a}$. Recall that a
$\mathbb{C}$-algebra is affine if it is finitely generated as an algebra over
$\mathbb{C}$.

######  Lemma 7.10.

Let $\mathbf{c}\in\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})$.

  1. (i)

The algebra $H_{0}^{\Gamma}$ is a finite $Z(H_{0}^{\Gamma})$-module and
$H_{0,\mathbf{c}}^{\Gamma}$ is a finite $Z(H_{0,\mathbf{c}}^{\Gamma})$-module.

  2. (ii)

The algebra $\mathbf{a}H^{\Gamma}_{0}\mathbf{a}$ is a prime affine
$\mathbb{C}$-algebra, finite over $Z(\mathbf{a}H^{\Gamma}_{0}\mathbf{a})$, and
$\mathbf{a}H^{\Gamma}_{0,\mathbf{c}}\mathbf{a}$ is a prime affine algebra,
finite over $Z(\mathbf{a}H^{\Gamma}_{0,\mathbf{c}}\mathbf{a})$.

  3. (iii)

For all $\mathfrak{m}\lhd R$ maximal the map
$Z(\mathbf{a}H^{\Gamma}_{0}\mathbf{a})\to
Z(\mathbf{a}H^{\Gamma}_{0}\mathbf{a}/\langle\mathfrak{m}\rangle)$ is
surjective.

###### Proof.

Part (i). Since $H_{0}$ is a finite $Z(H_{0})$-module, invariant theory says
that $H_{0}^{\Gamma}$ is a finite $Z(H_{0})^{\Gamma}$-module. Then
$Z(H_{0})^{\Gamma}\subset Z(H_{0}^{\Gamma})$ implies that $H_{0}^{\Gamma}$ is
finite over $Z(H_{0}^{\Gamma})$ too. The proof for $H^{\Gamma}_{0,\mathbf{c}}$
is identical.

Part (ii). Corollary 6.11 says that
$\mathbf{a}H^{\Gamma}_{0,\mathbf{c}}\mathbf{a}$ is prime. It also applies
verbatim to $\mathbf{a}H^{\Gamma}_{0}\mathbf{a}$, showing that it is prime. By
Theorem 4.4 and Theorem 4.6, $H_{0}^{\Gamma}$ is an affine $R_{0}$-algebra;
alternatively one can use that $\operatorname{\mathrm{gr}}H_{0}^{\Gamma}\cong
R_{0}\otimes(P^{\Gamma}\rtimes W)$ is an affine $R_{0}$-algebra to deduce
finite generation for $H_{0}^{\Gamma}$. Since $\mathbf{a}$ is central in
$H_{0}^{\Gamma}$, it follows that $\mathbf{a}H^{\Gamma}_{0}\mathbf{a}$ is an
affine $R_{0}$-algebra. The fact that $R_{0}$ is a finitely generated
$\mathbb{C}$-algebra then implies that $\mathbf{a}H^{\Gamma}_{0}\mathbf{a}$ is
a finitely generated $\mathbb{C}$-algebra i.e. it is affine. This implies that
$\mathbf{a}H^{\Gamma}_{0,\mathbf{c}}\mathbf{a}$ is affine too since it is a
quotient of $\mathbf{a}H^{\Gamma}_{0}\mathbf{a}$. The fact that these algebras
are finite over their centres is a consequence of (i), using the decomposition
of Lemma 6.8.

Part (iii). If $\mathfrak{m}\lhd R$ is the maximal ideal corresponding to
$(0,\mathbf{c})$ then
$\mathbf{a}H^{\Gamma}_{0}\mathbf{a}/\langle\mathfrak{m}\rangle=\mathbf{a}H^{\Gamma}_{0,\mathbf{c}}\mathbf{a}$.
We have a commutative diagram

${Z(\mathbf{a}H^{\Gamma}_{0}\mathbf{a})}$${eH^{\Gamma}_{0}e}$${Z(\mathbf{a}H^{\Gamma}_{0,\mathbf{c}}\mathbf{a})}$${eH^{\Gamma}_{0,\mathbf{c}}e.}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Since $H^{\Gamma}_{0}\to H^{\Gamma}_{0,\mathbf{c}}$ is surjective and applying
invariants, $h\mapsto ehe$, is exact, the right vertical arrow is surjective.
We deduce that the left vertical arrow is also surjective. ∎

If $L$ is a simple $\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a}$-module then
$\mathrm{ann}_{Z}L$ is a maximal ideal of
$Z:=Z(\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a})$ and hence a closed point
of $Y_{\mathbf{c}}$. This is the support of $L$.

######  Lemma 7.11.

Let $\chi$ be a generic character of $eH^{\Gamma}_{\mathbf{c}}e$ and set
$L=\mathbf{a}H^{\Gamma}e\otimes_{eH^{\Gamma}_{\mathbf{c}}e}\chi$. Then
$L|_{\overline{W}}\cong\mathbb{C}\overline{W}$.

###### Proof.

Note that Lemma 6.5 says that the action of $\overline{W}$ on
$V/\\!\\!/\\!\,\Gamma=(\mathfrak{h}\times\mathfrak{h}^{*})/\\!\\!/\\!\,\Gamma$
is generically free. Therefore, the proof of [18, Lemma 2.24] goes through
verbatim. ∎

######  Theorem 7.12.

For all
$\mathbf{c}\in\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})$,

  1. (i)

$H_{\mathbf{c}}^{\Gamma}$ and $\mathbf{a}H_{\mathbf{c}}^{\Gamma}\mathbf{a}$
are Polynomial Identity (PI) algebras and the PI degree of the prime PI-
algebra $\mathbf{a}H_{\mathbf{c}}^{\Gamma}\mathbf{a}$ equals $|\overline{W}|$.

  2. (ii)

Let $L$ be a simple $\mathbf{a}H_{\mathbf{c}}^{\Gamma}\mathbf{a}$-module whose
support is contained in the regular locus of $Y_{\mathbf{c}}$. Then
$L|_{W}\cong\mathbb{C}\overline{W}$.

  3. (iii)

The regular locus of $Y_{\mathbf{c}}$ is contained in the Azumaya locus.

###### Proof.

Part (i). The fact that $H_{\mathbf{c}}^{\Gamma}$ is a PI-algebra follows
directly from Lemma 7.10(i). By Lemma 7.10, we may apply Corollary 7.9 to
$\mathbf{a}H_{\mathbf{c}}^{\Gamma}\mathbf{a}$. In particular, the corollary
implies that the PI-degree of $\mathbf{a}H_{\mathbf{c}}^{\Gamma}\mathbf{a}$ is
independent of $\mathbf{c}$. Taking $\mathbf{c}=0$, it follows from Lemma 6.5
that the PI-degree of
$\mathbf{a}H_{0}^{\Gamma}\mathbf{a}=P^{\Gamma}\rtimes\overline{W}$ equals
$|\overline{W}|$.

Part (ii) and (iii). Our proof follows the outline of the proof of [18,
Theorem 1.7]. As noted in Lemma 6.15, $\mathbf{a}H^{\Gamma}_{\mathbf{c}}e$ is
a Cohen-Macaulay $eH^{\Gamma}_{\mathbf{c}}e$-module. Therefore, if
$\mathfrak{m}\lhd eH^{\Gamma}_{\mathbf{c}}e$ is a maximal ideal contained in
the regular locus of $Y_{\mathbf{c}}$ then
$(\mathbf{a}H^{\Gamma}_{\mathbf{c}}e)_{\mathfrak{m}}$ is a free
$(eH^{\Gamma}_{\mathbf{c}}e)_{{}_{\mathfrak{m}}}$. In other words,
$\mathbf{a}H^{\Gamma}_{\mathbf{c}}e$ localizes to a vector bundle on
$Y_{\mathbf{c}}^{\mathrm{reg}}$. This vector bundle is
$\overline{W}$-equivariant, where $\overline{W}$ acts trivially on
$Y_{\mathbf{c}}^{\mathrm{reg}}$. Since the generic fibre is isomorphic to
$\mathbb{C}\overline{W}$ by Lemma 7.11 and $Y_{\mathbf{c}}^{\mathrm{reg}}$ is
connected, it follows that every fibre is isomorphic to
$\mathbb{C}\overline{W}$.

Since $(\mathbf{a}H^{\Gamma}_{\mathbf{c}}e)_{\mathfrak{m}}$ is a free
$(eH^{\Gamma}_{\mathbf{c}}e)_{{}_{\mathfrak{m}}}$,

$E:=\operatorname{\mathrm{End}}_{(eH^{\Gamma}_{\mathbf{c}}e)_{{}_{\mathfrak{m}}}}((\mathbf{a}H^{\Gamma}_{\mathbf{c}}e)_{\mathfrak{m}})$

is Morita equivalent to $(eH^{\Gamma}_{\mathbf{c}}e)_{\mathfrak{m}}$. In
particular, the fibre

$\mathbf{a}H^{\Gamma}_{\mathbf{c}}e/\mathbf{a}H^{\Gamma}_{\mathbf{c}}e\mathfrak{m}$

of $\mathbf{a}H^{\Gamma}_{\mathbf{c}}e$ at $\mathfrak{m}$ is a simple
$E$-module. But localizing the isomorphism of Theorem 6.16 shows that
$E=(\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a})_{\mathfrak{m}}$. Hence the
fibre of $\mathbf{a}H^{\Gamma}_{\mathbf{c}}e$ at $\mathfrak{m}$ is a simple
$\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a}$ implying (ii). The inclusion of
part (iii) now follows since we have shown that there is a simple
$\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a}$-module supported at
$\mathfrak{m}$ whose dimension equals the PI degree of
$\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a}$. This means that $\mathfrak{m}$
must be in the Azumaya locus by [11, III, Theorem 1.6]. ∎

######  Remark 7.13.

Since $\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a}$ need not have finite
global dimension the usual arguments (see [10, Lemma 3.3]) used to show that
the Azumaya locus of $\mathbf{a}H^{\Gamma}_{\mathbf{c}}\mathbf{a}$ is
contained in the regular locus of $Y_{\mathbf{c}}$ do not apply. We do not
know if the inclusion holds.

##  8\. The algebra of (quantum) Hamiltonian reduction

The action of $\mathrm{T}$ on the Calogero–Moser space
$\operatorname{\mathrm{Spec}}Z(H_{0,\mathbf{c}})$ is Hamiltonian because the
Poisson bracket is homogeneous of degree zero. Since
$\\{\mathrm{eu}_{\mathbf{c}},f\\}=\deg(f)f$ for any homogeneous $f\in
Z(H_{0,\mathbf{c}})$, the (co-)moment map
$\mu^{*}\colon\mathbb{C}[\mathrm{Lie}\,\mathrm{T}]\to Z(H_{0,\mathbf{c}})$ is
given by $\mu^{*}(1)=\mathrm{eu}_{\mathbf{c}}$. Therefore, the Hamiltonian
reduction of $\operatorname{\mathrm{Spec}}Z(H_{0,\mathbf{c}})$ is

$\mu^{-1}(\zeta)/\\!\\!/\\!\,\mathrm{T}=\operatorname{\mathrm{Spec}}\left(Z(H_{0,\mathbf{c}})^{\mathrm{T}}/\langle\mathrm{eu}_{\mathbf{c}}-\zeta\rangle\right),\quad\forall\,\zeta\in\mathbb{C}=(\mathrm{Lie}\,\mathrm{T})^{*}.$

######  Definition 8.1.

For $\zeta\in\mathbb{C}$, the algebra of (quantum) Hamiltonian reduction is

$A_{t,\mathbf{c},\zeta}=A_{t,\mathbf{c},\zeta}(W)=\mathbf{a}H_{t,\mathbf{c}}^{\mathfrak{gl}(n)}\mathbf{a}/\langle\mathbf{a}\mathrm{eu}_{\mathbf{c}}-\zeta\rangle.$

Recall that $P=\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{*}]$. Since
$\mathbf{a}\mathrm{eu}_{\mathbf{c}}$ is central (and hence normal) in
$A_{t,\mathbf{c},\zeta}$ and its symbol $\mathrm{eu}_{0}$ is regular in
$P^{\mathrm{T}}\rtimes W$, Lemma 2.2 immediately implies the following.

######  Lemma 8.2.

We have

$\operatorname{\mathrm{gr}}A_{t,\mathbf{c},\zeta}=(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)\rtimes\overline{W},\quad\operatorname{\mathrm{gr}}(Z(H_{\mathbf{c}})^{\mathrm{T}}/\langle\mathrm{eu}_{\mathbf{c}}\rangle)\cong(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)^{W}.$

The affine variety $(\mathfrak{h}\times\mathfrak{h}^{*})/W$ is a Poisson
variety, where the Poisson bracket on $P^{W}$ comes by restriction from the
standard bracket on $P$. By definition of Hamiltonian reduction,

$\operatorname{\mathrm{Spec}}(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)^{W}$

is again a Poisson scheme.

We recall that Beauville [1] introduced the notion of symplectic
singularities. A normal variety $X$ over $\mathbb{C}$ is said have symplectic
singularities if the smooth locus $U$ of $X$ carries an algebraic symplectic
form $\omega$ such that, for any resolution of singularities $\pi\colon Y\to
X$, $\pi^{*}\omega$ extends to a regular $2$-form on $Y$.

The following is well-known to experts, but we sketch the proof since we do
not know of a suitable reference.

######  Lemma 8.3.

Let $X$ be a symplectic singularity. Then

  1. (i)

$X$ has rational singularities,

  2. (ii)

is Gorenstein, and hence

  3. (iii)

is Cohen-Macaulay.

###### Proof.

Let $n=(1/2)\dim X$. If $\omega\in\Omega^{2}_{X_{\mathrm{reg}}}$ is the
symplectic form then $s=\wedge^{n}\omega$ is a non-vanishing top form. It
trivializes $\Omega_{X_{\mathrm{reg}}}$ and hence implies that the canonical
sheaf $\omega_{X}=j_{*}\Omega_{X_{\mathrm{reg}}}$ is trivial, where $j\colon
X_{\mathrm{reg}}\hookrightarrow X$.

Let $f\colon Y\to X$ be a resolution of singularities which is an isomorphism
over $X_{\mathrm{reg}}$. Since $\omega$ extends to a regular two form on $Y$,
$s$ also extends to a regular form on $Y$. Therefore, in the equality

$K_{Y}=f^{*}(K_{X})+\sum_{i}m_{i}E_{i},$

the coefficients $m_{i}$ are non-negative since they are the order of
vanishing of $f^{*}s$ along $E_{i}$. Here the sum is over all exceptional
divisors of $f$. This means that $X$ has canonical singularities. By [17,
Theoreme 1] this means that $X$ has rational singularities. In particular,
Kempf’s Theorem [28] says that $X$ is Cohen-Macaulay. This implies that the
dualizing complex $\omega_{X}^{{\>\raisebox{2.0pt}{\text{\circle*{1.5}}}}}$ of
$X$ is quasi-isomorphic to $\omega_{X}=\mathcal{O}_{X}$. Thus,
$\mathcal{O}_{X}$ has finite injective dimension i.e. $X$ is Gorenstein. ∎

The variety $X$ is said to be a conic symplectic singularity if it is affine,
$\mathbb{C}[X]$ is $\mathbb{N}$-graded with $\mathbb{C}[X]_{0}=\mathbb{C}$ and
the form $\omega$ is homogeneous of weight $\ell>0$. Examples of conic
symplectic singularities include normalization of a nilpotent orbit closure in
a semi-simple Lie algebra and symplectic quotient singularities.

Let $\mathcal{O}_{\mathrm{min}}\subset\mathfrak{gl}(n)$ denote the minimal
nilpotent orbit (consisting of all nilpotent rank one matrices). Then
$\overline{\mathcal{O}}_{\mathrm{min}}$ is normal and hence is a conic
symplectic singularity; see [1, (2.6)]. If
$W\subset\operatorname{\mathrm{GL}}(n)$ is a finite group then $W$ acts on
$\overline{\mathcal{O}}_{\mathrm{min}}$ preserving the Poisson structure and
hence $\overline{\mathcal{O}}_{\mathrm{min}}/W$ is again a (conic) symplectic
singularity by [1, Proposition 2.4].

######  Lemma 8.4.

There is an isomorphism of Poisson schemes
$\operatorname{\mathrm{Spec}}(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)^{W}\cong\overline{\mathcal{O}}_{\mathrm{min}}/W$.
In particular, $(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)^{W}$ is an
integrally closed domain.

###### Proof.

The map
$\mathfrak{h}\oplus\mathfrak{h}^{*}\to\operatorname{\mathrm{End}}_{\mathbb{C}}(\mathfrak{h})$,
$(v,\lambda)\mapsto v\otimes\lambda$ has image the space of all rank (at most)
one endomorphisms. The equation defined by the polynomial $\mathrm{eu}_{0}$ is
precisely the condition that $\lambda(v)=0$. This means that the image of the
zero set $\mu^{-1}(0)=V(\mathrm{eu}_{0})$ equals the space of rank (at most)
one endomorphisms with zero trace i.e. the rank (at most) one nilpotent
endomorphisms, which is precisely $\overline{\mathcal{O}}_{\mathrm{min}}$. The
image is also invariant under $\mathrm{T}$ giving rise to the (Poisson)
isomorphism of irreducible schemes
$\mu^{-1}(0)/\\!\\!/\\!\,\mathrm{T}\cong\overline{\mathcal{O}}_{\mathrm{min}}$.
Since this isomorphism is $W$-equivariant, it descends to

$\operatorname{\mathrm{Spec}}(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)^{W}=\mu^{-1}(0)/\\!\\!/\\!\,\mathrm{T}\cong\overline{\mathcal{O}}_{\mathrm{min}}/W.$

In particular, the left hand side is a normal domain. ∎

######  Lemma 8.5.

The group $\overline{W}$ acts freely on a dense open subset of
$\overline{\mathcal{O}}_{\mathrm{min}}$ with complement of codimension at
least two.

###### Proof.

Presumably this can be checked directly, however we will deduce it from the
fact that $\overline{W}$ preserves the Poisson structure on
$\overline{\mathcal{O}}_{\mathrm{min}}$.

Since $\overline{\mathcal{O}}_{\mathrm{min}}$ is normal, its smooth locus $U$
has complement of codimension at least two; in fact,
$\overline{\mathcal{O}}_{\mathrm{min}}$ has an isolated singularity. Therefore
it suffices to show that $\overline{W}$ acts freely on an open subset of $U$
with complement of codimension at least two. Let $u\in U$ and
$\overline{W}_{u}$ the stabilizer of $u$ in $\overline{W}$. Then
$\overline{W}_{u}$ acts on the tangent space $T_{u}U$ and it suffices to show
that this action factors through $\mathrm{SL}(T_{u}U)$. But $T_{u}U$ is a
symplectic vector space and the action of $\overline{W}_{u}$ preserves the
symplectic form. Thus, the action of $\overline{W}_{u}$ factors through
$\mathrm{Sp}(T_{u}U)$.

Finally, we need to check that there is at least one point $A$ in
$\overline{\mathcal{O}}_{\mathrm{min}}$ whose stabilizer $\overline{W}_{A}$ is
trivial. Note that each $A\in\mathcal{O}_{\mathrm{min}}$ is characterized by
the fact that there exists $0\neq v\in\mathfrak{h}$ such that
$\mathrm{Im}\,A=\mathbb{C}\cdot v$ and $A(v)=0$. Choose a non-zero
$v\in\mathfrak{h}$ such that $v$ is not a eigenvector of any $w\in W\setminus
Z(W)$ and let $A\in\mathcal{O}_{\mathrm{min}}$ with
$\mathrm{Im}\,A=\mathbb{C}\cdot v$. Then $wAw^{-1}=A$ implies that $v$ is an
eigenvector of $w$. Thus, $\overline{W}_{A}=\\{1\\}$. ∎

######  Proposition 8.6.

The algebra $A_{t,\mathbf{c},\zeta}$ is a prime noetherian
Auslander–Gorenstein ring. It has Gelfand–Kirillov dimension
$2(\dim\mathfrak{h}-1)$.

###### Proof.

As noted above, the variety $\overline{\mathcal{O}}_{\mathrm{min}}$ has
symplectic singularities. By Lemma 8.3, this implies that the ring
$\mathbb{C}[\overline{\mathcal{O}}_{\mathrm{min}}]$ is Gorenstein. Thus,
$P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle$ is a noetherian Gorenstein
domain. Therefore it is Auslander–Gorenstein.

This implies that
$(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)\rtimes\overline{W}$ is a
noetherian Auslander–Gorenstein ring [34]. It is prime by [32, Corollary
12.6]. Hence Lemma 8.2 implies that $A_{t,\mathbf{c},\zeta}(W)$ is also
Auslander–Gorenstein ([5, Theorem 3.9]) and prime. Finally, we note that
$\dim\overline{\mathcal{O}}_{\mathrm{min}}=2(\dim\mathfrak{h}-1)$, which
implies that the Gelfand–Kirillov dimension of $A_{t,\mathbf{c},\zeta}(W)$ is
also $2(\dim\mathfrak{h}-1)$; see [30, Proposition 8.1.14]. ∎

######  Lemma 8.7.

The centre of $(P^{\mathrm{T}}/\langle\mathrm{eu}\rangle)\rtimes\overline{W}$
equals $(P^{\mathrm{T}}/\langle\mathrm{eu}\rangle)^{W}$.

###### Proof.

The proof is similar to that of Proposition 7.1. The key point being that
$P^{\mathrm{T}}/\langle\mathrm{eu}\rangle$ is a domain on which $\overline{W}$
acts faithfully. ∎

######  Remark 8.8.

We note that if $S$ is not a domain then the centre of $S\rtimes W$ can
properly contain $S^{W}$ even if $W$ acts faithfully on $S$.

Let $A$ be a filtered algebra such that
$\operatorname{\mathrm{gr}}_{\mathcal{F}}A$ is a commutative algebra. Then
$\operatorname{\mathrm{gr}}_{\mathcal{F}}A$ is an $\mathbb{N}$-graded Poisson
algebra. If $B$ is an $\mathbb{N}$-graded Poisson algebra, then we say that
$A$ is a filtered quantization of $B$ if there exists a graded Poisson
isomorphism $\operatorname{\mathrm{gr}}_{\mathcal{F}}A\cong B$.

Recall that $e$ is the trivial idempotent in $\mathbb{C}W$.

######  Theorem 8.9.

For $t\neq 0$, the algebra $eA_{t,\mathbf{c},\zeta}e$ is a filtered
quantization of $\overline{\mathcal{O}}_{\mathrm{min}}/W$.

###### Proof.

It is a consequence of the proof of [18, Theorem 1.6] (in particular, Claim
2.25) that $eH_{t,\mathbf{c}}e$ is a filtered quantization of $P^{W}$. Since
the filtration on $eH_{t,\mathbf{c}}^{\mathrm{T}}e$ is by restriction and
$P^{\mathrm{T}\times W}$ is a Poisson subalgebra of $P^{W}$ this implies that
$eH_{t,\mathbf{c}}^{\mathrm{T}}e$ is a filtered quantization of
$P^{\mathrm{T}\times W}$. Finally, since the filtration on
$eA_{t,\mathbf{c},\zeta}(W)e$ is the quotient filtration coming from the
filtration on $eH_{t,\mathbf{c}}^{\mathrm{T}}e$ and the Poisson structure on
$(P^{\mathrm{T}}/\langle\mathrm{eu}_{0}\rangle)^{W}$ comes from the fact that
$\langle\mathrm{eu}_{0}\rangle^{W}$ is a Poisson ideal in $P^{\mathrm{T}\times
W}$, we deduce that $eA_{t,\mathbf{c},\zeta}e$ is a filtered quantization of
$\overline{\mathcal{O}}_{\mathrm{min}}/W$ by applying the (Poisson)
isomorphism of Lemma 8.4. ∎

######  Theorem 8.10.

For Weil generic $(t,\mathbf{c},\zeta)$, $A_{t,\mathbf{c},\zeta}(W)$ has
finite global dimension.

###### Proof.

We assume $t=1$. Then the claim follows from the fact that localization holds
for Weil generic parameters. We use here the notion of sheaves of Cherednik
algebras on $\mathbb{P}(\mathfrak{h})$, as studied in [4]. By [4, Lemma
5.4.1], the sheaf $\mathcal{H}_{\zeta,\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W)$
of Cherednik algebras on the projective space $\mathbb{P}(\mathfrak{h})$ has
global sections

$\Gamma(\mathbb{P}(\mathfrak{h}),\mathcal{H}_{\zeta,\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W))=H_{1,\mathbf{c}}^{\mathrm{T}}/\langle\mathrm{eu}_{\mathbf{c}}-\zeta\rangle.$

This implies that $A_{1,\mathbf{c},\zeta}$ is a direct summand of
$H(\mathbb{P}):=\Gamma(\mathbb{P}(\mathfrak{h}),\mathcal{H}_{\zeta,\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W))$
and hence it suffices to show that $H(\mathbb{P})$ has finite global dimension
for Weil generic $(\mathbf{c},\zeta)$.

The stalks of $\mathcal{H}_{\zeta,\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W)$
have finite global dimension (bounded by $2\dim\mathbb{P}(\mathfrak{h})$).
This implies that the sheaf
$\mathcal{E}xt^{q}_{\mathcal{H}}(\mathcal{M},\mathcal{N})$ is zero for
$q>2\dim\mathbb{P}(\mathfrak{h})$ if $\mathcal{M},\mathcal{N}$ are coherent
over $\mathcal{H}_{\zeta,\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W)$. The
Grothendieck spectral sequence

$E_{2}^{p,q}=H^{p}(\mathbb{P}(\mathfrak{h}),\mathcal{E}xt^{q}_{\mathcal{H}}(\mathcal{M},\mathcal{N}))\Rightarrow\operatorname{\mathrm{Ext}}_{\mathcal{H}}^{p+q}(\mathcal{M},\mathcal{N})$

converges because $H^{p}(\mathbb{P}(\mathfrak{h}),\mathcal{F})=0$ for
$p>2\dim_{\mathbb{C}}\mathbb{P}(\mathfrak{h})$ by Grothendieck’s [26, Theorem
2.7] for any sheaf $\mathcal{F}$ on $\mathbb{P}(\mathfrak{h})$. In particular,
since the second page is bounded we deduce that
$\operatorname{\mathrm{Ext}}_{\mathcal{H}}^{k}(\mathcal{M},\mathcal{N})=0$ for
$k$ sufficiently large (independent of $\mathcal{M},\mathcal{N}$). By [4,
Theorem 1.2.1], the global sections functor
$\Gamma\colon\mathcal{H}_{\zeta,\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W)\text{-}{\mathsf{mod}}\to
H(\mathbb{P})\text{-}{\mathsf{mod}}$ from the category of coherent
$\mathcal{H}_{\zeta,\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W)$-modules to
finitely generated modules over $H(\mathbb{P})$ is an equivalence (i.e.
affinity holds) for Weil generic $(\mathbf{c},\zeta)$. If we assume affinity
then

$\operatorname{\mathrm{Ext}}_{\mathcal{H}}^{i}(\mathcal{M},\mathcal{N})=\operatorname{\mathrm{Ext}}_{H(\mathbb{P})}^{i}(\Gamma(\mathbb{P}(\mathfrak{h}),\mathcal{M}),\Gamma(\mathbb{P}(\mathfrak{h}),\mathcal{N}))$

for any pair of coherent
$\mathcal{H}_{\mathbf{c}}(\mathbb{P}(\mathfrak{h}),W)$-modules
$\mathcal{M},\mathcal{N}$. It follows that
$\operatorname{\mathrm{Ext}}_{H(\mathbb{P})}^{k}(M,N)=0$ (again, independent
of $M,N$) for $k$ sufficiently large and all finitely generated $M,N$. This
implies that $H(\mathbb{P})$ has finite global dimension.

Compare with [7, VI, Theorem 1.10(ii)], which is the proof in the case of
$\mathscr{D}$-modules. ∎

Note that it is not true that $A_{t,\mathbf{c},\zeta}$ always has finite
global dimension. For instance, if $\dim\mathfrak{h}>1$ then take $t=1$ and
$\mathbf{c}=0$. If
$A_{1,0,\zeta}=\mathscr{D}_{\zeta}(\mathbb{P}(\mathfrak{h}))\rtimes\overline{W}$
has finite global dimension then so too would the ring
$\mathscr{D}_{\zeta}(\mathbb{P}(\mathfrak{h}))$ of global $\zeta$-twisted
differential operators on $\mathbb{P}(\mathfrak{h})$. But it is known [33]
that when $\zeta=-1,-2,\dots,-(\dim\mathfrak{h}-1)$ this is not the case.

###  8.1. The double centralizer property

Consider the space $A_{t,\mathbf{c},\zeta}e$ as a left
$A_{t,\mathbf{c},\zeta}$-module and as a right
$eA_{t,\mathbf{c},\zeta}e$-module. The next theorem establishes the double
centralizer property for the bimodule $A_{t,\mathbf{c},\zeta}(W)e$.

######  Theorem 8.11.

$\operatorname{\mathrm{End}}_{eA_{t,\mathbf{c},\zeta}e}(A_{t,\mathbf{c},\zeta}e)\cong
A_{t,\mathbf{c},\zeta}^{\mathrm{op}}$.

###### Proof.

First, we note that
$A_{0,0,0}=\mathbb{C}[\overline{\mathcal{O}}_{\mathrm{min}}]\rtimes\overline{W}$.
Since $\overline{\mathcal{O}}_{\mathrm{min}}$ is a normal variety and the
group $\overline{W}$ acts freely on a dense open subset of
$\overline{\mathcal{O}}_{\mathrm{min}}$ with complement of codimension at
least two (Lemma 8.5), the double centralizer property holds for $A_{0,0,0}$
by [6, Proposition E.2.2(2)]. Therefore the proof of [18, Theorem 1.5(vi)]
shows that it holds for all parameters $(t,\mathbf{c},\zeta)$. ∎

As for the usual rational Cherednik algebra, we say that the parameters
$(t,\mathbf{c},\zeta)$ are aspherical if there exists a nonzero
$A_{t,\mathbf{c},\zeta}$-module $M$ such that $eM=0$. The parameters
$(t,\mathbf{c},\zeta)$ are called spherical if they are not aspherical. The
following is an immediate corollary of Theorem 8.11, as explained in [3,
Corollary 1.6.3].

######  Corollary 8.12.

The algebra $A_{t,\mathbf{c},\zeta}$ is Morita equivalent to
$eA_{t,\mathbf{c},\zeta}e$ if and only if the parameters are spherical.

###  8.2. The centre of $A_{t,\mathbf{c},\zeta}(W)$

Once again, the Satake isomorphism holds in this situation. Recall that
$e\in\mathbb{C}W$ is the trivial idempotent.

######  Corollary 8.13.

The map $z\mapsto ez$ is an isomorphism
$Z(A_{t,\mathbf{c},\zeta}){\;\stackrel{{{}_{\sim}}}{{\to}}\;}Z(eA_{t,\mathbf{c},\zeta}e)$.

We deduce that:

######  Corollary 8.14.

The centre of $A_{0,\mathbf{c},\zeta}$ is isomorphic to
$Z(H_{0,\mathbf{c}})^{\mathrm{T}}/\langle\mathrm{eu}_{\mathbf{c}}-\zeta\rangle$.

As a consequence of the Satake isomorphism, we deduce that:

######  Corollary 8.15.

If $t\neq 0$ then the centre of $A_{t,\mathbf{c},\zeta}$ is $\mathbb{C}$.

###### Proof.

By Corollary 8.13, it suffices to show that the centre of
$eA_{t,\mathbf{c},\zeta}e$ is trivial. By Theorem 8.9, this algebra is a
filtered quantization of
$\mathbb{C}[\overline{\mathcal{O}}_{\mathrm{min}}/W]$. We give the centre of
$eA_{t,\mathbf{c},\zeta}e$ a filtration by restriction. Then
$\operatorname{\mathrm{gr}}Z(eA_{t,\mathbf{c},\zeta}e)$ is contained in
$\mathrm{Cas}(\mathbb{C}[\overline{\mathcal{O}}_{\mathrm{min}}]^{W})$. Since
$\overline{\mathcal{O}}_{\mathrm{min}}/W$ has symplectic singularities, it has
finitely many symplectic leaves. Hence
$\mathrm{Cas}(\mathbb{C}[\overline{\mathcal{O}}_{\mathrm{min}}]^{W})=\mathbb{C}$.
This forces
$\operatorname{\mathrm{gr}}Z(eA_{t,\mathbf{c},\zeta}e)=\mathbb{C}$, and hence
$Z(eA_{t,\mathbf{c},\zeta}e)=\mathbb{C}$ too. ∎

Assume $t=0$. Then
$eH_{0,\mathbf{c}}^{\mathrm{T}}e=eH_{0,\mathbf{c}}^{\mathfrak{gl}(n)}e$ is
commutative. Therefore the quotient $eA_{0,\mathbf{c},\zeta}e$ is commutative
in this case. It follows from Corollary 8.13 that
$Z(A_{0,\mathbf{c},\zeta}){\;\stackrel{{{}_{\sim}}}{{\to}}\;}eA_{0,\mathbf{c},\zeta}e$.

Fixing $t=0$ but considering $\mathbf{c},\zeta$ as formal variables,
$\operatorname{\mathrm{Spec}}Z(A_{0})$ is a flat family over
$\mathfrak{c}:=\operatorname{\mathrm{Hom}}_{W}(\mathcal{S},\mathbb{C})\oplus\mathbb{C}$.
We grade $\mathfrak{c}$ by giving it degree $2$. In this way, $A_{0}$ is a
graded $\mathbb{C}[\mathfrak{c}]$-algebra

######  Theorem 8.16.

$\operatorname{\mathrm{Spec}}Z(A_{0})\to\mathfrak{c}$ is a graded Poisson
deformation of the symplectic singularity $\overline{\mathcal{O}}_{min}/W$.

###### Proof.

We may consider $H_{0}$ as a $\mathbb{C}[\mathfrak{c}]$-algebra, where the
defining relations do not involve $\zeta$. Then $Z(H_{0})$ is a flat graded
Poisson deformation of $P^{W}$ over $\mathfrak{c}$. This implies that
$Z(\mathbf{a}H_{0}^{\Gamma}\mathbf{a})\cong Z(H_{0})^{\Gamma}$ is a graded
Poisson deformation of $P^{W\times\Gamma}$. Finally,
$\mathbf{a}\mathrm{eu}_{\mathbf{c}}-\zeta$ generates a graded Poisson ideal in
$Z(\mathbf{a}H_{0}^{\Gamma}\mathbf{a})$ such that the quotient

$Z(\mathbf{a}H_{0}^{\Gamma}\mathbf{a})/\langle\mathbf{a}\mathrm{eu}_{\mathbf{c}}-\zeta\rangle$

is flat over $\mathbb{C}[\mathfrak{c}]$ by Lemma 8.2. Finally, we note that
the fibre of this map above $0\in\mathfrak{c}$ is the centre of
$A_{0,0,0}=(P^{\mathrm{T}}/\langle\mathrm{eu}\rangle)\rtimes\overline{W}$. By
Lemma 8.7, this equals the ring of functions on the quotient
$\overline{\mathcal{O}}_{min}/W$. ∎

######  Remark 8.17.

As shown by Namikawa [31], the conic symplectic singulartity
$\overline{\mathcal{O}}_{min}/W$ admits a universal graded Poisson deformation
$\mathcal{X}\to\mathfrak{b}$. Therefore, Theorem 8.16 implies that there is a
unique graded map $\mathfrak{c}\to\mathfrak{b}$ such that

$\operatorname{\mathrm{Spec}}Z(A_{0})\cong\mathfrak{c}\times_{\mathfrak{b}}\mathcal{X}$

as Poisson varieties. We expect that one can use arguments as in [2] to
explicitly describe the map $\mathfrak{c}\to\mathfrak{b}$.

## References

  * [1] A. Beauville.  Symplectic singularities.  Invent. Math., 139(3):541–549, 2000.
  * [2] G Bellamy.  Counting resolutions of symplectic quotient singularities.  Compositio Mathematica, 152:99–114, 2016.
  * [3] G. Bellamy.  Symplectic reflection algebras.  In Noncommutative algebraic geometry, volume 64 of Math. Sci. Res. Inst. Publ., pages 167–238. Cambridge Univ. Press, New York, 2016\.
  * [4] G. Bellamy and M. Martino.  Affinity of Cherednik algebras on projective space.  Algebra Number Theory, 8(5):1151–1177, 2014.
  * [5] J.-E. Björk.  The Auslander condition on Noetherian rings.  In Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988), volume 1404 of Lecture Notes in Math., pages 137–173. Springer, Berlin, 1989.
  * [6] C. Bonnafé and R. Rouquier.  Cherednik algebras and Calogero-Moser cells.  arXiv, 1708.09764v3, 2017.
  * [7] A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers.  Algebraic $D$-modules, volume 2 of Perspectives in Mathematics.  Academic Press, Inc., Boston, MA, 1987.
  * [8] J.-F. Boutot.  Singularités rationnelles et quotients par les groupes réductifs.  Invent. Math., 88(1):65–68, 1987.
  * [9] K. A. Brown and K. Changtong.  Symplectic reflection algebras in positive characteristic.  Proc. Edinb. Math. Soc. (2), 53(1):61–81, 2010.
  * [10] K. A. Brown and K. R. Goodearl.  Homological aspects of Noetherian PI Hopf algebras of irreducible modules and maximal dimension.  J. Algebra, 198(1):240–265, 1997.
  * [11] K. A. Brown and K. R. Goodearl.  Lectures on Algebraic Quantum Groups.  Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2002.
  * [12] K. S. Brown.  Cohomology of groups, volume 87 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1994.  Corrected reprint of the 1982 original.
  * [13] W. Bruns and J. Herzog.  Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1993.
  * [14] D. Ciubotaru and M. De Martino.  The Dunkl-Cherednik deformation of a Howe duality.  J. Algebra, 560:914–959, 2020.
  * [15] A. M. Cohen.  Finite complex reflection groups.  Ann. Sci. École Norm. Sup. (4), 9(3):379–436, 1976.
  * [16] C. F. Dunkl and E. M. Opdam.  Dunkl operators for complex reflection groups.  Proc. London Math. Soc. (3), 86(1):70–108, 2003.
  * [17] R. Elkik.  Rationalité des singularités canoniques.  Invent. Math., 64(1):1–6, 1981.
  * [18] P. Etingof and V. Ginzburg.  Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism.  Invent. Math., 147(2):243–348, 2002.
  * [19] M. Feigin and T. Hakobyan.  On Dunkl angular momenta algebra.  J. High Energy Phys., (11):107, front matter+22, 2015.
  * [20] M. Feigin and T. Hakobyan.  Algebra of Dunkl Laplace–Runge–Lenz vector.  Lett. Math. Phys., 112(3):59, 2022.
  * [21] M. Feigin and M. Vrabec.  $q$-Analogue of the degree zero part of a rational Cherednik algebra.  arXiv: 2311.07543.
  * [22] W. Fulton.  Young tableaux, volume 35 of London Mathematical Society Student Texts.  Cambridge University Press, Cambridge, 1997.  With applications to representation theory and geometry.
  * [23] W. Fulton and J. Harris.  Representation theory, volume 129 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1991.  A first course, Readings in Mathematics.
  * [24] V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier.  On the category $\mathcal{O}$ for rational Cherednik algebras.  Invent. Math., 154(3):617–651, 2003.
  * [25] I. G. Gordon.  Symplectic reflection algebras.  In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pages 285–347. Eur. Math. Soc., Zürich, 2008\.
  * [26] R. Hartshorne.  Algebraic Geometry.  Springer-Verlag, New York, 1977.  Graduate Texts in Mathematics, No. 52.
  * [27] G. Heckman.  A remark on the Dunkl differential-difference operators.  In Harmonic analysis on reductive groups (Brunswick, ME, 1989), volume 101 of Progr. Math., pages 181–191. Birkhäuser Boston, Boston, MA, 1991.
  * [28] G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat.  Toroidal embeddings. I.  Lecture Notes in Mathematics, Vol. 339. Springer-Verlag, Berlin-New York, 1973.
  * [29] F. Knop.  Der kanonische Modul eines Invariantenrings.  J. Algebra, 127(1):40–54, 1989.
  * [30] J. C. McConnell and J. C. Robson.  Noncommutative Noetherian Rings, volume 30 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, revised edition, 2001.  With the cooperation of L. W. Small.
  * [31] Y. Namikawa.  Poisson deformations of affine symplectic varieties.  Duke Math. J., 156(1):51–85, 2011.
  * [32] D. S. Passman.  Infinite Crossed Products, volume 135 of Pure and Applied Mathematics.  Academic Press Inc., Boston, MA, 1989.
  * [33] B. Shelton.  The global dimension of rings of differential operators on projective spaces.  Bull. London Math. Soc., 24(2):148–158, 1992.
  * [34] Z. Yi.  Injective homogeneity and the Auslander–Gorenstein property.  Glasgow Math. J., 37(2):191–204, 1995.