# Almost commuting scheme of symplectic matrices and quantum Hamiltonian
reduction

Pallav Goyal

###### Abstract.

Losev introduced the scheme $X$ of almost commuting elements (i.e., elements
commuting upto a rank one element) of $\mathfrak{g}=\mathfrak{sp}(V)$ for a
symplectic vector space $V$ and discussed its algebro-geometric properties. We
construct a Lagrangian subscheme $X^{nil}$ of $X$ and show that it is a
complete intersection of dimension
$\text{dim}(\mathfrak{g})+\frac{1}{2}\text{dim}(V)$ and compute its
irreducible components.

We also study the quantum Hamiltonian reduction of the algebra
$\mathcal{D}(\mathfrak{g})$ of differential operators on the Lie algebra
$\mathfrak{g}$ tensored with the Weyl algebra with respect to the action of
the symplectic group, and show that it is isomorphic to the spherical
subalgebra of a certain rational Cherednik algebra of Type $C$.

##  1\. Introduction

### 1.1.

Let $V:=\mathbb{C}^{2n}$ be a symplectic vector space and let $\mathfrak{g}$
denote the Lie algebra $\mathfrak{sp}(V)=\mathfrak{sp}_{2n}$. The almost
commuting scheme $X$ of $\mathfrak{g}$ was defined by Losev in [Los21] as the
closed subscheme of $\mathfrak{g}\times\mathfrak{g}\times V$ defined by the
ideal $I$ generated by the matrix entries of $[x,y]+i^{2}$, i.e., by all
functions of the form $(x,y,i)\mapsto\lambda([x,y]+i^{2})$ for
$\lambda\in\mathfrak{g}^{*}$. Here, we use the fact that
$\operatorname{Sym}^{2}(V)$ can be identified with $\mathfrak{sp}(V)$ to view
$i^{2}$ as an element of $\mathfrak{sp}(V)$. The geometrical properties of $X$
were studied by Losev who showed that:

######  Theorem 1.1.1 ([Los21]).

The scheme $X$ is reduced, irreducible and a complete intersection of
dimension $2n^{2}+3n=\dim(\mathfrak{g})+\dim(V)$.

In this paper, we consider the reduced subscheme $X^{nil}$ of $X$ defined as:

$X^{nil}:=\\{(x,y,i)\in\mathfrak{g}\times\mathfrak{g}\times
V:[x,y]+i^{2}=0\text{ and }y\text{ is nilpotent}\\}.$

This definition is motivated by the notion of character sheaves, first defined
by Lusztig (see [Lus85, Lus91]). It was shown by Mirković and Vilonen [MV88]
and Ginzburg [Gin89] that over $\mathbb{C}$, a character sheaf on a reductive
algebraic group $K$ can be defined as an $Ad(K)$-equivariant perverse sheaf
$M$ on $K$, such that the corresponding characteristic variety lies in the
nilpotent locus $K\times\mathcal{N}\subseteq K\times\mathfrak{k}^{*}$, where
$\mathfrak{k}=Lie(K)$ and
$\mathcal{N}\subseteq\mathfrak{k}^{*}\simeq\mathfrak{k}$ is the nilpotent
cone. Constructions analogous to $X^{nil}$ were done in [GG06, FG10, FG10a] to
provide ‘mirabolic’ analogs of these character sheaves in Type $A$.

We describe some notation. It is known (for example, see [CM93, Theorem
5.1.3]) that nilpotent conjugacy classes in $\mathfrak{g}$ are parametrized by
the partitions $\lambda$ of $2n$ in which every odd part appears an even
number of times. Let $P_{n}$ be the set of all such partitions and let
$\mathcal{P}_{n}$ denote the subset of those partitions in $P_{n}$ in which
all the parts are even. For each $\lambda\in P_{n}$, let
$\mathcal{N}_{\lambda}$ denote the corresponding nilpotent conjugacy class in
$\mathfrak{g}$. Define for each $\lambda\in P_{n}$:

$X_{\lambda}:=\\{(x,y,i)\in X^{nil}:y\in\mathcal{N}_{\lambda}\\}.$

Let $\overline{X_{\lambda}}$ denote the closure of $X_{\lambda}$ in $X^{nil}$.

Note that we can identify $\mathfrak{g}\times\mathfrak{g}\times V$ with
$T^{*}(\mathfrak{g})\times V$ using the trace form on $\mathfrak{g}$. This
gives $\mathfrak{g}\times\mathfrak{g}\times V$ a natural symplectic structure.
Our first main result reads:

######  Theorem 1.1.2.

  1. (a)

The scheme $X^{nil}$ is a complete intersection in
$\mathfrak{g}\times\mathfrak{g}\times V$ of dimension $2n^{2}+2n$. The
irreducible components of $X^{nil}$ are exactly given by the
$\overline{X_{\lambda}}$ for $\lambda\in\mathcal{P}_{n}$.

  2. (b)

With the standard symplectic structure, $X^{nil}$ is a Lagrangian subscheme of
$\mathfrak{g}\times\mathfrak{g}\times V$.

A similar Lagrangian subscheme was constructed in [GG06] in the context of the
almost commuting scheme of the Lie algebra $\mathfrak{gl}_{n}$. Using Theorem
1.1.2, we provide an independent proof of Theorem 1.1.1 in the style of
[GG06], that eliminates the use of results from [Los06]. Furthermore, the
scheme $X^{nil}$ will be used in $\mathsection\ref{sec:QHRF}$ to provide a
mirabolic analog of Lusztig’s character sheaves in Type $C$.

### 1.2.

In the second half of the paper, we discuss some Hamiltonian reduction
problems arising in the context of the scheme $X$ and some other related
schemes. For this, we define the following subschemes of
$\mathfrak{g}\times\mathfrak{g}=\operatorname{Spec}(\mathbb{C}[\mathfrak{g}\times\mathfrak{g}])=\operatorname{Spec}(\mathbb{C}[x,y])$.
Consider the commuting scheme $C$ which is the (not necessarily reduced)
subscheme of $\mathfrak{g}\times\mathfrak{g}$ defined by the ideal of
$\mathbb{C}[x,y]$ generated by the matrix entries of the commutator $[x,y]$.
Next, define the scheme $A$ to be the (not necessarily reduced) subscheme of
$\mathfrak{g}\times\mathfrak{g}$ defined by the ideal of $\mathbb{C}[x,y]$
generated by all the $2\times 2$ minors of the commutator $[x,y]$.

Note that the set of $\mathbb{C}$-points of the underlying reduced subscheme
of $C$ consists of pairs of elements of $\mathfrak{g}$ that commute with each
other, whereas that of $A$ consists of pairs of elements of $\mathfrak{g}$
whose commutator has rank lesser than or equal to one. The commuting scheme
$C$ is of wide interest, and its geometrical properties (most notably, its
reducedness) are largely unknown. It is known that $C$ is irreducible (see
[Ric79]).

The schemes $X,C$ and $A$ have an action of the group $G=Sp(V)$ obtained by
the adjoint action on $\mathfrak{g}$ and the natural action on $V$. Hence, we
can consider the respective categorical quotients of these schemes by the
action of $G$. While it isn’t known if $C$ is reduced, it was shown in [Los21]
that there’s an isomorphism:

$C//G\longrightarrow X//G,$

which, paired with Theorem 1.1.1, implies that $C//G$ is reduced. (That $C//G$
is reduced was deduced independently in [CN21] slightly earlier, by proving a
version of the Chevalley restriction theorem for the commuting scheme of
$\mathfrak{g}$.) We extend this isomorphism to show that $C//G\simeq
X//G\simeq A//G$. In fact, we prove the following stronger result:

######  Theorem 1.2.1.

We have an isomorphism of schemes:

$X//\\{\pm 1\\}\longrightarrow A//\\{\pm 1\\}=A,$

where $\\{\pm 1\\}\subseteq G$ is the center of the symplectic group. In
particular, the scheme $A$ is reduced.

An analog of the isomorphism $X//G\simeq A//G$ for the Lie algebra
$\mathfrak{gl}_{n}$ was proved in [GG06]. The theorem is deduced from a linear
algebraic lemma (Lemma 3.1.4), which also implies an algebro-geometric analog
of the ‘shifting trick’ in Hamiltonian reduction, that is well-known in the
differential geometric setting (see [GS82], [CS91]).

### 1.3.

The above categorical quotients can (and will) all be viewed as classical
Hamiltonian reductions of certain schemes under the action of the group $G$:

  * •

The scheme $X//G$ is the reduction of the scheme
$\mathfrak{g}\times\mathfrak{g}\times V$ with respect to $G$ at $0$.

  * •

The scheme $C//G$ is the reduction of the scheme
$\mathfrak{g}\times\mathfrak{g}$ with respect to $G$ at $0$.

  * •

The scheme $A//G$ is the reduction of the scheme
$\mathfrak{g}\times\mathfrak{g}$ with respect to $G$ at the closure of the
orbit of rank 1 matrices in $\mathfrak{g}\simeq\mathfrak{g}^{*}$.

So, we can try to study the non-commutative or quantum analogs of these
reduction problems. For this, let $\mathcal{U}\mathfrak{g}$ denote the
universal enveloping algebra of $\mathfrak{g}$, let
$\mathcal{D}(\mathfrak{g})$ denote the algebra of polynomial differential
operators on $\mathfrak{g}$ and let $W_{2n}$ denote the Weyl algebra on $2n$
variables, which is the algebra of polynomial differential operators on the
affine $n$-space. Then, $\mathcal{D}(\mathfrak{g})$ is a quantization of
$\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]\simeq\mathbb{C}[\mathfrak{g}\times\mathfrak{g}^{*}]$,
whereas $W_{2n}$ is a quantization of $\mathbb{C}[V]$. Both the algebras
$\mathcal{D}(\mathfrak{g})$ and $W_{2n}$ have a natural $\mathfrak{g}$-action
(and thus, so does their tensor product.) So, we get quantum co-moment maps:

$\Theta_{0}:\mathcal{U}\mathfrak{g}\longrightarrow\mathcal{D}(\mathfrak{g})$
$\Theta_{2}:\mathcal{U}\mathfrak{g}\longrightarrow\mathcal{D}(\mathfrak{g})\otimes
W_{2n}.$

(These maps are elaborated upon in $\mathsection{\ref{sec:quant}}$.) Then, we
can consider the following non-commutative algebras:

  * •

The reduction $\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\mathfrak{g}}$ of
$\mathcal{D}(\mathfrak{g})\otimes W_{2n}$ at the augmentation ideal of
$\mathcal{U}\mathfrak{g}$.

  * •

The reduction
$\Big{(}\mathcal{D}(\mathfrak{g})/\mathcal{D}(\mathfrak{g})\cdot\Theta_{0}(\mathfrak{g})\Big{)}^{\mathfrak{g}}$
of $\mathcal{D}(\mathfrak{g})$ at the augmentation ideal of
$\mathcal{U}\mathfrak{g}$.

  * •

The reduction
$\Big{(}\mathcal{D}(\mathfrak{g})/\mathcal{D}(\mathfrak{g})\cdot\Theta_{0}(\mathcal{K})\Big{)}^{\mathfrak{g}}$
of $\mathcal{D}(\mathfrak{g})$ at the unique primitive ideal
$\mathcal{K}\subseteq\mathcal{U}\mathfrak{g}$ such that
$\mathrm{gr}(\mathcal{K})\subseteq\mathbb{C}[\mathfrak{g}^{*}]$ is the
defining ideal of the orbit of rank 1 matrices.

(Each of these algebras is discussed in detail in
$\mathsection\ref{sec:quant}.$)

The first of these, the algebra
$\Big{(}\mathcal{D}(\mathfrak{g})/\mathcal{D}(\mathfrak{g})\cdot\Theta_{0}(\mathfrak{g})\Big{)}^{\mathfrak{g}}$,
has been studied classically by Harish-Chandra (see [Har64]) who constructed a
surjective algebra homorphism called the ‘radial parts’ homomorphism:

$\mathcal{D}(\mathfrak{g})^{\mathfrak{g}}\longrightarrow\mathcal{D}(\mathfrak{h})^{W},$

where $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ and $W$ is the
Weyl group. The kernel of this homomorphism was shown to be precisely
$(\mathcal{D}(\mathfrak{g})\cdot\Theta_{0}(\mathfrak{g}))^{\mathfrak{g}}$ in
the works of Wallach [Wal93] and Lavasseur and Stafford [LS95, LS96], implying
that the above map is an isomorphism.

In this paper, we’ll discuss the other two quantum Hamiltonian reduction
problems. For this, we recall the rational Cherednik algebra $H_{c}$ of Type
$C$, first defined in [EG02]. Here, the parameter $c=(c_{long},c_{short})$
lies in $\mathbb{C}^{2}$. Let $e=\frac{1}{|W|}\sum_{w\in W}w$ be the averaging
idempotent of the Weyl group $W$ and consider the spherical subalgebra
$eH_{c}e\subseteq H_{c}$ of the Cherednik algebra. (The notation is elaborated
on in $\mathsection{\ref{sec:RCA}}$.)

We prove the following theorem about these algebras:

######  Theorem 1.3.1.

We have algebra isomorphisms:

$\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\mathfrak{g}}\simeq\Big{(}\mathcal{D}(\mathfrak{g})/\mathcal{D}(\mathfrak{g})\cdot\Theta_{0}(\mathcal{K})\Big{)}^{\mathfrak{g}}\simeq
eH_{c}e,$

for the parameter $c=(-1/4,-1/2)$.

Analogs of this theorem were proved in [EG02] and [GG06] in the
$\mathfrak{gl}_{n}$-setting. This theorem shows that for the very special
choice of parameter $c=(-1/4,-1/2)$, the spherical subalgebra $eH_{c}e$ can be
obtained as a quantum Hamiltonian reduction of the ring of differential
operators. The proof of the first isomorphism in the theorem follows from
Lemma 3.1.4. The proof of the second isomorphism employs a generalization of
the radial parts construction studied by Etingof and Ginzburg in [EG02].

Inspired by the formalism in [GG06, $\mathsection{7}$], we define a certain
category $\mathcal{C}$ of holonomic $(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})$-modules supported on $X^{nil}$. The simple objects of $\mathcal{C}$
will be the mirabolic analogs of Lusztig’s character sheaves in Type $C$. We
construct an exact functor from $\mathcal{C}$ to $\mathcal{O}(eH_{c}e)$, the
category $\mathcal{O}$ of the spherical Cherednik algebra $eH_{c}e$ defined in
[BEG03].

###  1.4. Organization

Here, we give more details about the contents of the paper.

In $\mathsection\ref{sec:subsch},$ we prove Theorem 1.1.2. The proof of the
fact that $X^{nil}$ is Lagrangian is by embedding it into the Lagrangian
subscheme defined in [GG06]. The proof of the rest of the theorem will be seen
to be a consequence of this fact and some elementary
$\mathfrak{sl}_{2}$-theory.

In $\mathsection\ref{sec:class}$, we prove Lemma 3.1.4 and use it to deduce
Theorem 1.2.1. In $\mathsection\ref{sec:RCA}$, we recall some definitions and
results about Cherednik algebras and the generalization of Harish-Chandra’s
radial parts construction in [EG02]. In $\mathsection\ref{sec:quant}$, we
define the algebras alluded to in the statement of Theorem 1.3.1. In
$\mathsection\ref{sec:isoone}$ and $\mathsection\ref{sec:isotwo}$, we
construct maps between these algebras and prove that they are isomorphisms.
Finally, in $\mathsection\ref{sec:QHRF},$ we note some results about the
category $\mathcal{C}$ and provide the construction of the functor from
$\mathcal{C}$ to $\mathcal{O}(eH_{c}e)$.

###  1.5. Acknowledgments

The author would like to thank Victor Ginzburg for motivating this line of
work, for helpful suggestions and for his unparalleled patience, especially
when this paper was being written. Thanks are also due to Ivan Losev for
answering the author’s questions about [Los21]. Finally, discussions with
Santosha Pattanayak led to the discovery of an error in an earlier draft of
this paper, for which the author is extremely grateful.

##  2\. The nilpotent subscheme

In this section, we consider the reduced subscheme of $X$ defined via:

$X^{nil}:=\\{(x,y,i)\in\mathfrak{g}\times\mathfrak{g}\times
V:[x,y]+i^{2}=0\text{ and }y\text{ is nilpotent}\\}.$

In $\mathsection{\ref{sec:Lag}}$, we’ll prove that this is a Lagrangian
subscheme of $\mathfrak{g}\times\mathfrak{g}\times V$. We will then use this
result to provide a new proof of Theorem 1.1.1. Next, in
$\mathsection{\ref{sec:irredcomp}}$, we compute the irreducible components of
$X^{nil}$.

###  2.1. Lagrangian subscheme

To state the precise result, we first describe a symplectic structure on
$\mathcal{X}=\mathfrak{g}\times\mathfrak{g}\times V$, which we define as
$\omega=\omega_{1}+\omega_{2}$. Here, $\omega_{1}$ is the symplectic form on
$\mathfrak{g}\times\mathfrak{g}$ obtained by identifying it with
$T^{*}(\mathfrak{g})$ using the trace form on $\mathfrak{g}$ and $\omega_{2}$
is the form on the symplectic vector space $V$. Next, on the scheme
$\mathcal{M}=\mathfrak{gl}(V)\times\mathfrak{gl}(V)\times V\times V^{*}$, we
have a symplectic form $\omega^{\prime}$ obtained by identifying it with
$T^{*}(\mathfrak{gl}(V)\times V)$ using the trace form on $\mathfrak{gl}(V)$.

######  Theorem 2.1.1.

The scheme $X^{nil}$ is a Lagrangian subscheme of $\mathcal{X}$.

Recall that a (possibly singular) subscheme $Z$ of $\mathcal{X}$ is said to be
Lagrangian if at any smooth point $p$ of $Z$, the tangent space $T_{p}Z$ is a
Lagrangian subspace of $T_{p}\mathcal{X}$.

###### Proof.

The scheme $X^{nil}$ is a closed subscheme of $\mathcal{X}$ defined using the
$\dim(\mathfrak{g})+\frac{1}{2}\dim(V)$ equations coming from the formula
$[x,y]+i^{2}=0$. So,
$\dim(X^{nil})\geq\dim(\mathcal{X})-(\dim(\mathfrak{g})+\frac{1}{2}\dim(V))=\dim(\mathfrak{g})+\frac{1}{2}\dim(V)=\frac{1}{2}\dim(\mathcal{X})$.
Therefore, to prove the theorem, it suffices to show that $X^{nil}$ is
isotropic. For this, consider the embedding:

$\Phi:\mathcal{X}=\mathfrak{g}\times\mathfrak{g}\times
V\longrightarrow\mathfrak{gl}(V)\times\mathfrak{gl}(V)\times V\times
V^{*}=\mathcal{M}$ $(x,y,i)\mapsto(x,y,i_{1},i_{2}),$

where $i_{1}=i/2$ and $i_{2}$ is the symplectic dual of $i$ in $V$. (That is,
$i_{2}$ is the image of $i$ in $V^{*}$ under the identification $V\simeq
V^{*}$ using the symplectic form.)

Recall from [GG06] the scheme of almost commuting matrices
$M\subseteq\mathcal{M}$:

$M:=\\{(x,y,i,j)\in\mathfrak{gl}(V)\times\mathfrak{gl}(V)\times V\times
V^{*}:[x,y]+ij=0\\}.$

Also defined in [GG06] was the closed subscheme $M^{nil}$ of $\mathcal{M}$
obtained by stipulating $y$ to be nilpotent. Then, under the map $\Phi$, we
have $\Phi(X)\subseteq M$ and $\Phi(X^{nil})\subseteq M^{nil}$.

We claim that with the symplectic forms $\omega$ and $\omega^{\prime}$ defined
above, the map $\Phi$ is a symplectic embedding. To see this, we first observe
that we can express the form $\omega^{\prime}$ as a sum
$\omega^{\prime}_{1}+\omega^{\prime}_{2}$, where $\omega^{\prime}_{1}$ is the
symplectic form on $\mathfrak{gl}(V)\times\mathfrak{gl}(V)$ identified with
$T^{*}(\mathfrak{gl}(V))$ and $\omega_{2}^{\prime}$ is the symplectic form on
$V\times V^{*}$ identified with $T^{*}(V).$

Then, it is clear that $\omega_{1}=\Phi^{*}(\omega_{1}^{\prime}).$ Next, if
$i,j$ are two vectors in $V$, then we have:

$\omega_{2}^{\prime}((i_{1},j_{1}),(i_{2},j_{2}))=j_{1}(i_{2})-j_{2}(i_{1})=\dfrac{1}{2}\omega_{2}(i,j)-\dfrac{1}{2}\omega_{2}(j,i)=\omega_{2}(i,j),$

which shows that $\Phi$ preserves the symplectic structure.

By [GG06, Theorem 1.1.4], we know that that $M^{nil}$ is a Lagrangian
subscheme of $\mathcal{M}$. In particular, it is isotropic. Therefore, by
[CG10, Theorem 1.3.30], we get that $\Phi(X^{nil})$ is an isotropic subscheme
of $\mathcal{M}$, proving that $X^{nil}$ is an isotropic subscheme of
$\mathcal{X}$. ∎

######  Corollary 2.1.2.

The scheme $X$ is a complete intersection of dimension
$\dim(\mathfrak{g})+\dim(V)$.

###### Proof.

The scheme $X^{nil}$ is obtained from $X$ by imposing exactly
$n=\frac{1}{2}\dim(V)$ equations, that come from imposing the nilpotence
condition on $y$. Hence, as
$\dim(X^{nil})=\frac{1}{2}\dim(\mathcal{X})=\dim(\mathfrak{g})+\frac{1}{2}\dim(V)$,
we must have that
$\dim(X)\leq\dim(X^{nil})+\frac{1}{2}\dim(V)=\dim(\mathfrak{g})+\dim(V)$. But,
the scheme $X$ is obtained from $\mathcal{X}$ by imposing $\dim(\mathfrak{g})$
equations, and so,
$\dim(X)\geq\dim(\mathcal{X})-\dim(\mathfrak{g})=\dim(\mathfrak{g})+\dim(V).$
Therefore, $X$ is a complete intersection of dimension
$\dim(\mathfrak{g})+\dim(V).$ ∎

In fact, we can generalize Theorem 2.1.1 as follows. Let
$\mathfrak{h}\subseteq\mathfrak{g}$ denote a Cartan subalgebra of
$\mathfrak{g}$ and let $W$ be the Weyl group. Consider the composition map
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}//G\rightarrow\mathfrak{h}//W$, where
the first map is the quotient map and the second one is the Chevalley
restriction isomorphism. Then, we can consider the morphism:

$\pi:X\longrightarrow\mathfrak{h}//W,$

that sends a triple $(x,y,i)$ to $\phi(y)$. It is clear that
$X^{nil}=\pi^{-1}(0).$

######  Proposition 2.1.3.

All the fibers of the map $\pi$ are Lagrangian subschemes of $\mathcal{X}$ and
have dimension $\dim(\mathfrak{g})+\frac{1}{2}\dim(V)$.

###### Proof.

For any $x\in\mathfrak{h}//W$, we have the dimension inequality:

$\dim(\pi^{-1}(x))\geq\dim(X)-\dim(\mathfrak{h}//W)=\dim(\mathfrak{g})+\dfrac{1}{2}\dim(V)=\dfrac{1}{2}\dim(\mathcal{X}).$

Therefore, to show that $\pi^{-1}(x)$ is Lagrangian, it suffices to prove that
it is isotropic.

We consider the symplectic embedding $\Phi:\mathcal{X}\rightarrow\mathcal{M}$
defined in the proof of Theorem 2.1.1. By Corollary 2.3.4 of [GG06], the image
of $\pi^{-1}(x)$ in $\mathcal{M}$ lies inside a Lagrangian subscheme of
$\mathcal{M}$. As a result, we conclude that the image of $\pi^{-1}(x)$ must
be an isotropic subscheme of $\mathcal{M}$, showing that $\pi^{-1}(x)$ must
itself be isotropic. ∎

######  Remark 2.1.4.

In fact, by adapting the proof of Proposition 2.3.2 of [GG06], we can prove
the following: Corresponding to the Hamiltonian $G$-action on the variety
$\mathcal{X}$, we get a moment map
$\mu:\mathcal{X}\rightarrow\mathfrak{g}^{*}\simeq\mathfrak{g}$ given by the
formula $(x,y,i)\mapsto[x,y]+i^{2}$. Consider the map:

$\mu\times\pi:\mathcal{X}=\mathfrak{g}\times\mathfrak{g}\times
V\longrightarrow\mathfrak{g}\times\mathfrak{h}//W,$

that maps a triple $(x,y,i)$ to the pair $([x,y]+i^{2},\phi(y))$. Then, this
map is a flat morphism. As a corollary of this fact, we also get that the
moment map $\mu:\mathcal{X}\rightarrow\mathfrak{g}$ if flat.

With this, we are ready to prove Losev’s theorem. Define the scheme:

$X^{reg}=\\{(x,y,i)\in X:y\text{ is regular, semisimple}\\}.$

In other words, $X^{reg}=\pi^{-1}(\mathfrak{h}^{reg})$, where
$\mathfrak{h}^{reg}$ is the regular semisimple locus of $\mathfrak{h}$. By
Lemma 2.9 of [Los21], the scheme $X^{reg}$ is irreducible.

######  Theorem 2.1.5.

  1. (1)

We have $\overline{X^{reg}}=X$. In particular, the scheme $X$ is irreducible.

  2. (2)

$X$ is a reduced, complete intersection of dimension
$\dim(\mathfrak{g})+\dim(V)$.

###### Proof.

By Corollary 2.1.2, we already know that $X$ is a complete intersection of
$\dim(\mathfrak{g})+\dim(V)$. Consider the big diagonal
$\Delta=(\mathfrak{h}\setminus\mathfrak{h}^{reg})//W$, which is a closed
subscheme of $\mathfrak{h}//W$ of codimension $1$. Then, we have the equality:

$X=\overline{X^{reg}}\cup\pi^{-1}(\Delta).$

Since $\dim(\Delta)=\dim(\mathfrak{h}//W)-1=\frac{1}{2}\dim(V)-1$, by
Proposition 2.1.3, we have
$\dim(\pi^{-1}(\Delta))\leq\dim(\mathfrak{g})+\dim(V)-1$. However, as $X$ is a
complete intersection, any irreducible component must have dimension exactly
$\dim(\mathfrak{g})+\dim(V)$. Therefore, we conclude that $\overline{X^{reg}}$
must be on the only irreducible component, proving that
$X=\overline{X^{reg}}$.

Finally, we note that the action of the group $G$ is free on $X^{reg}$, and
so, $X$ is generically reduced. Therefore, as $X$ is a complete intersection,
it must be Cohen-Macaulay, and thus, we conclude that $X$ is reduced. ∎

###  2.2. Irreducible components of $X^{nil}$

Let $P_{n}$ denote the set of all partitions of $2n$ where every odd part
occurs an even number of times. Let $\mathcal{P}_{n}\subseteq P_{n}$ be the
subset of those partitions where each part is even. For any $\lambda\in
P_{n}$, let $\mathcal{N}_{\lambda}$ denote the corresponding nilpotent
conjugacy class of $\mathfrak{g}$ and define:

$X_{\lambda}:=\\{(x,y,i)\in X^{nil}:y\in\mathcal{N}_{\lambda}\\}.$

Then, it is clear that we have the following disjoint union:

$X^{nil}=\coprod_{\lambda\in P_{n}}X_{\lambda}.$

######  Theorem 2.2.1.

  1. (1)

For each $\lambda\in\mathcal{P}_{n},$ we have
$\dim(X_{\lambda})=\dim(\mathfrak{g})+\frac{1}{2}\dim(V)$.

  2. (2)

For each $\lambda\in P_{n}\setminus\mathcal{P}_{n}$, we have
$\dim(X_{\lambda})<\dim(\mathfrak{g})+\frac{1}{2}\dim(V)$.

As $X^{nil}$ has been shown to be a complete intersection of dimension
$\dim(\mathfrak{g})+\frac{1}{2}\dim(V)$, this theorem implies that the
irreducible components of $X^{nil}$ are given exactly by the closures of those
$X_{\lambda}$ for which $\lambda\in\mathcal{P}_{n}$, completing the proof of
Theorem 1.1.2.

###### Proof.

Let $y$ be any nilpotent element in $\mathfrak{g}$. Let $\mathbb{O}$ denote
the minimal orbit consisting of rank one elements in $\mathfrak{g}$. Define
the reduced schemes: (Also defined in [Los21])

$X_{y}=\\{(x,i)\in\mathfrak{g}\times V:[x,y]+i^{2}=0\\},$
$\underline{X_{y}}=\\{(x,z)\in\mathfrak{g}\times\overline{\mathbb{O}}:[x,y]+z=0\\},$
$Y_{y}=\overline{\mathbb{O}}\cap\\{[x,y]:x\in\mathfrak{g}\\}.$

We have maps:

$\rho_{1}:X_{y}\longrightarrow\underline{X_{y}}$ $(x,i)\mapsto(x,i^{2}),$

and

$\rho_{2}:\underline{X_{y}}\longrightarrow Y_{y}$ $(x,z)\mapsto z.$

The map $\rho_{1}$ is finite, with either one or two points in the fiber at
any point depending on whether $i^{2}$ is zero or not, respectively. The map
$\rho_{2}$ is an affine bundle map which has fibers of dimension equal to
$\dim(\mathfrak{z}_{\mathfrak{g}}(y)).$ Thus, we get that
$\dim(X_{y})=\dim(\mathfrak{z}_{\mathfrak{g}}(y))+\dim(Y_{y})$. Further, if
$y\in\mathcal{N}_{\lambda}$ for some $\lambda\in P_{n}$, we have that
$\dim(X_{\lambda})=\dim(X_{y})+\dim(\mathcal{N}_{\lambda})=\dim(\mathfrak{g})+\dim(Y_{y})$.
So, in order to prove the theorem, we are required to show that
$\dim(Y_{y})=\frac{1}{2}\dim(V)$ for $\lambda\in\mathcal{P}_{n}$ and
$\dim(Y_{y})<\frac{1}{2}\dim(V)$ for $\lambda\not\in\mathcal{P}_{n}$.

For a fixed nilpotent $y$, by Jacobson-Morozov theorem, we can find an
$\mathfrak{sl}_{2}$-triple $(e,f,h)$ in $\mathfrak{g}$ with $e=y$. Identifying
the subspace $\langle e,f,h\rangle\subseteq\mathfrak{g}$ with
$\mathfrak{sl}_{2}$, we get an $\mathfrak{sl}_{2}$-action on the vector space
$V$. By $\mathfrak{sl}_{2}$-theory, the element $h$ acts semisimply on $V$
with integer eigenvalues. Let $V=V_{-}\oplus V_{0}\oplus V_{+}$ be the
decomposition of $V$, such that $V_{-},V_{0}$ and $V_{+}$ denote the span of
negative, zero and positive eigenspaces of $h$ respectively.

Now, we consider the identification of $\mathfrak{sl}_{2}$-representations
$\mathfrak{g}=\mathfrak{sp}(V)=\operatorname{Sym}^{2}(V)$. By Lemma 2.2.2
proved below, for any $v\in V$, we have:

$v\in V_{+}\iff
v^{2}=\operatorname{ad}_{e}(x)=\operatorname{ad}_{y}(x)=[y,x]\text{ for some
}x\in\mathfrak{g}.$

Hence, we conclude that $\dim(Y_{y})=\dim(V_{+})$. As there is a one-to-one
correspondence between positive and negative eigenvectors of $h$, we have that
$\dim(V_{-})=\dim(V_{+})$, and so, $\dim(Y_{y})\leq\frac{1}{2}\dim(V)$. The
equality holds exactly when $V_{0}=0$, that is, when $h$ has no zero
eigenvalues. Zero eigenvalues for $h$ occur only in irreducible
representations having odd dimension. Hence, the dimension inequality becomes
an equality exactly when the space $V$ decomposes into a sum of irreducibles
each having even dimension. However, irreducible components of $V$ correspond
exactly to the Jordan blocks of $e$. Therefore, the dimension equality holds
exactly when each Jordan block of $y$ has even size, that is
$y\in\mathcal{N}_{\lambda}$ for some $\lambda\in\mathcal{P}_{n}$, thus
completing the proof. ∎

######  Lemma 2.2.2.

Let $V$ be a representation of $\mathfrak{sl}_{2}(\mathbb{C})=\langle
e,f,h\rangle$. Let $W=\operatorname{Sym}^{2}(V)$ with the action of
$\mathfrak{sl}_{2}$ induced from the one on $V$. Then, for any $v\in V$, we
have that $v^{2}=e\cdot w$ for some $w\in W$ if and only if $v$ lies in the
span of the positive eigenspaces of $h$ in $V$.

###### Proof.

Suppose $v$ lies in the span of the positive eigenspaces of $h$ in $V$. Then,
we must have that $v^{2}$ lies in the span of the positive eigenspaces of $h$
in $W$, and thus, by $\mathfrak{sl}_{2}$-theory, we must have $v^{2}=e\cdot w$
for some $w\in W$.

Now, we prove the converse. Henceforth, instead of working with
$\operatorname{Sym}^{2}(V)$, we will work with $W=V\otimes V$, for notational
convenience. As $V\otimes V$ contains $\operatorname{Sym}^{2}(V)$ as a proper
subrepresentation, if $v^{2}=e\cdot w$ for some
$w\in\operatorname{Sym}^{2}(V)$, we must have that $v\otimes
v=e\cdot\tilde{w}$ for some $\tilde{w}\in V\otimes V$.

Let $V=\oplus_{i}V_{i}$ be the decomposition of $V$ into a direct sum of
$\mathfrak{sl}_{2}$-representations. Then, we get a corresponding
decomposition $v=\sum_{i}v_{i}$ such that $v_{i}\in V_{i}$ for all $i$. Then,
as $v\otimes v=e\cdot w$, it is clear that there exist $w_{i}\in V_{i}\otimes
V_{i}$ such that $v_{i}\otimes v_{i}=e\cdot w_{i}$ for each $i$. Therefore,
without loss of generality, we can assume that $V$ is irreducible.

Suppose $\dim(V)=n+1$ for some $n\in\mathbb{Z}_{\geq 0}$. Then, there exists
an $h$-eigenbasis $\\{x_{0},x_{1},\dots,x_{n}\\}$ of $V$ such that the actions
of $e$ and $f$ are given by:

$e\cdot x_{i}=x_{i+1},f\cdot x_{j}=x_{j-1},$

for all $0\leq i\leq n-1$ and $1\leq j\leq n$.

Write $v=\sum_{i}c_{i}x_{i}$ for $c_{i}\in\mathbb{C}$. Then, we have $v\otimes
v=\sum_{i,j}c_{i}c_{j}x_{i}\otimes x_{j}$. Pick the smallest $k$ such that
$c_{k}\neq 0$. Then, $c_{k}^{2}x_{k}\otimes x_{k}$ is the summand in $v\otimes
v$ having the strictly smallest eigenvalue for the $h$-action. Therefore, as
we have that $v\otimes v=e\cdot w$ for some $w\in V\otimes V$, there must
exist $w^{\prime}\in V\otimes V$ such that $x_{k}\otimes x_{k}=e\cdot
w^{\prime}$. Hence, without loss of generality, we can assume that $v=x_{k}$
for some $k$.

So, we are reduced to showing that given $x_{k}\otimes x_{k}=e\cdot w$ for
some $w\in V\otimes V$, we must have that $h$ acts on $x_{k}$ with a positive
eigenvalue. For the sake of a contradiction, suppose $h$ acts on $x_{k}$ with
a non-positive eigenvalue. Therefore, we must have that $h$ acts on
$x_{k}\otimes x_{k}$ with a non-positive eigenvalue. As $x_{k}\otimes
x_{k}=e\cdot w$, we must have that $h$ acts on $w$ with a strictly negative
eigenvalue. Let $V^{\prime}\subseteq V$ be the subspace spanned by
$\\{x_{0},x_{1},\dots,x_{n-1}\\}$ (that is, all but the highest weight
vector). Then, we must have that $w\in V^{\prime}\otimes V^{\prime}\subseteq
V\otimes V$.

Consider the linear function defined via:

$f:V\otimes V\longrightarrow\mathbb{C}$ $x_{i}\otimes x_{j}\mapsto(-1)^{i}.$

We claim that $f(e\cdot x)=0$ for all $x\in V^{\prime}\otimes V^{\prime}$. To
see this, we note that $V^{\prime}\otimes V^{\prime}$ is spanned by vectors
for the form $x_{i}\otimes x_{j}$ for $0\leq i,j\leq n-1$. For such $i$ and
$j$, we have:

$e\cdot(x_{i}\otimes x_{j})=x_{i+1}\otimes x_{j}+x_{i}\otimes x_{j+i},$

which makes it clear that $f(e\cdot(x_{i}\otimes x_{j}))=0$.

In particular, since $w\in V^{\prime}\otimes V^{\prime}$, we must have
$f(e\cdot w)=0$. This implies that $f(x_{k}\otimes x_{k})=0$. However, it
follows from the definition of $f$ that $f(x_{k}\otimes x_{k})=(-1)^{k}\neq
0$. This gives a contradiction, and so, the eigenvalue corresponding to
$x_{k}$ must have been positive, completing the proof. ∎

##  3\. Hamiltonian reduction

###  3.1. Classical setting

#### 3.1.1.

In this section, we prove the isomorphism $X//\\{\pm 1\\}\simeq A$ (see
Theorem 3.1.3). We will first define some notation to formulate the precise
statement. All tensor products will be over $\mathbb{C}$, unless specified
otherwise.

Let $\omega$ denote the symplectic form on the vector space $V$. Owing to the
natural action of $G$ on the vector space $V$ that preserves $\omega$, we get
a moment map:

$\mu_{1}:V\longrightarrow\mathfrak{g}^{*},$

that maps the element $v\in V$ to
$v^{2}\in\mathfrak{g}\simeq\mathfrak{g}^{*}$. Here, we use the identification
$\operatorname{Sym}^{2}(V)=\mathfrak{g}$. We can dualize this map to get a co-
moment map:

$\theta_{1}:\mathfrak{g}\longrightarrow\mathbb{C}[V].$

By [CG10, Proposition 1.4.6], we have the following formula for this co-moment
map: Any element $x\in\mathfrak{g}$ maps under $\theta_{1}$ to the following
polynomial function on $V$:

$v\mapsto\dfrac{1}{2}\omega(x\cdot v,v)$

for all $v\in V$. In particular, the image of $\theta_{1}$ in $\mathbb{C}[V]$
is exactly the vector space of polynomial functions on $V$ having degree $2$.
We can extend the above map multiplicatively to get a map
$\mathbb{C}[\mathfrak{g}]\simeq\operatorname{Sym}(\mathfrak{g})\rightarrow\mathbb{C}[V]$,
which we also call $\theta_{1}$, whose image is exactly the subalgebra
$\mathbb{C}[V]_{even}$ of polynomials have even total degree. Define
$K:=\ker(\theta_{1})\subseteq\mathbb{C}[\mathfrak{g}]$. We give a more
explicit description of this ideal $K$ below.

######  Lemma 3.1.1.

Let $\overline{\mathbb{O}}$ denote the closure of the orbit $\mathbb{O}$ of
rank one matrices in $\mathfrak{g}$, such that the scheme structure on
$\overline{\mathbb{O}}$ is given by the reduced structure on it. Then, the
radical ideal in $\mathbb{C}[\mathfrak{g}]$ that defines the scheme
$\overline{\mathbb{O}}$ is generated by the $2\times 2$ minors.

###### Proof.

Consider the map:

$\mu_{1}:V\longrightarrow\mathfrak{g}^{*}\simeq\mathfrak{g}$ $v\mapsto v^{2}.$

The image of $\mu_{1}$ is exactly $\overline{\mathbb{O}}$. Also, the pre-image
of any point in $\mathbb{O}$ consists of exactly two vectors in $V$ that are
negatives of each other, whereas the pre-image of zero is exactly the zero
vector. Therefore, we get an induced map from the categorical quotient:

$\overline{\mu_{1}}:V//\\{\pm 1\\}\longrightarrow\overline{\mathbb{O}}.$

By the above discussion, $\overline{\mu_{1}}$ is a closed embedding that is a
bijection on $\mathbb{C}$-points. Therefore, as the scheme
$\overline{\mathbb{O}}$ is reduced, the map $\overline{\mu_{1}}$ is an
isomorphism.

Hence, the coordinate ring $\mathbb{C}[\overline{\mathbb{O}}]$ of
$\overline{\mathbb{O}}$ is isomorphic to the invariant ring
$\mathbb{C}[V]^{\\{\pm 1\\}}$. Choosing coordinates $p_{1},p_{2},\dots,p_{2n}$
in $V$, this invariant ring is equal to
$\mathbb{C}[p_{1},p_{2},\dots,p_{2n}]^{\\{\pm 1\\}}$. It is clear that this
ring is generated by the polynomials $q_{ij}=p_{i}p_{j}$ for $1\leq i,j\leq
2n$. Also, by the second fundamental theorem ([Wey97, Theorem 2.17A]) for the
group $O(1)=\\{\pm 1\\}$, the relations between these generators are exactly
given by $R_{ijkl}=q_{ij}q_{kl}-q_{il}q_{kj}$ for $1\leq i,j,k,l\leq 2n$.
Since the pullbacks of these relations $R_{ijkl}$’s to the coordinate ring of
$\mathfrak{g}$ are exactly given by the $2\times 2$ minors, this shows that
the defining ideal is exactly generated by these elements. ∎

######  Corollary 3.1.2.

The map $\theta_{1}$ induces an isomorphism of algebras:

$\theta_{1}:\mathbb{C}[\mathfrak{g}]/K\longrightarrow\mathbb{C}[V]^{\\{\pm
1\\}}=\mathbb{C}[V]_{even}.$

The ideal $K$ is the defining ideal of $\overline{\mathbb{O}}$ in
$\mathbb{C}[\mathfrak{g}]$ and is generated by the $2\times 2$ minors.

#### 3.1.2.

The adjoint action of the group $G=Sp(V)$ on the Lie algebra $\mathfrak{g}$
gives rise to a Hamiltonian $G$-action on the symplectic variety
$T^{*}\mathfrak{g}$. Corresponding to this action, we get a moment map:

$\mu_{0}:T^{*}\mathfrak{g}\longrightarrow\mathfrak{g}^{*}.$

If we identify the space $\mathfrak{g}^{*}$ with $\mathfrak{g}$, the map
$\mu_{0}$ is given explicitly by the commutator map on $\mathfrak{g}$. We can
dualize $\mu_{0}$ to get a co-moment map
$\theta_{0}:\operatorname{Sym}(\mathfrak{g})\rightarrow\mathbb{C}[T^{*}\mathfrak{g}]$.

Also, we have a diagonal $G$-action on the space $T^{*}\mathfrak{g}\times V$.
The moment map $\mu_{2}:T^{*}\mathfrak{g}\times V\rightarrow\mathfrak{g}^{*}$
for this action is equal to $\mu_{0}+\mu_{1}$, whereas the co-moment map
$\theta_{2}:\operatorname{Sym}(\mathfrak{g})\rightarrow\mathbb{C}[T^{*}\mathfrak{g}\times
V]\simeq\mathbb{C}[T^{*}\mathfrak{g}]\otimes\mathbb{C}[V]$ is defined via
$\theta_{2}(x)=\theta_{0}(x)\otimes 1+1\otimes\theta_{1}(x)$ for all
$x\in\mathfrak{g}$. Explicitly, the map $\mu_{2}$ is defined via the formula
$(x,y,i)\mapsto[x,y]+i^{2}$.

Now, we can define the schemes that we’ll be dealing with:

  * •

We define the scheme $X\subseteq T^{*}\mathfrak{g}\times V$ as the zero fiber
of the moment map $\mu_{2}$. More specifically, the defining ideal $I$ of $X$
in $\mathbb{C}[T^{*}\mathfrak{g}\times V]$ is the one generated by
$\theta_{2}(\mathfrak{g})$ in $\mathbb{C}[T^{*}\mathfrak{g}\times V]$. Using
the formula for $\theta_{2}$ above, this ideal is equal to the one generated
by the matrix entries of the expression $[x,y]+i^{2}$.

  * •

We define the scheme $A\subseteq T^{*}\mathfrak{g}$ as the pre-image of
$\overline{\mathbb{O}}$ under the moment map $\mu_{0}$. That is, the defining
ideal $J$ of $A$ in $\mathbb{C}[T^{*}\mathfrak{g}]$ is the one generated by
$\theta_{0}(K)$. By Corollary 3.1.2, this ideal is generated by all $2\times
2$ minors of the commutator $[x,y]$ for
$(x,y)\in\mathfrak{g}\times\mathfrak{g}$.

There is a projection morphism $\phi:X\rightarrow A$ that maps a triple
$(x,y,i)\in X$ to the pair $(x,y)\in A$. We will show that:

######  Theorem 3.1.3.

The induced morphism on the categorical quotients:

$\phi:X//\\{\pm 1\\}\longrightarrow A//\\{\pm 1\\}=A,$

is an isomorphism.

The proof of this theorem will be a consequence of the following linear
algebraic lemma, which will also be useful for proving the quantum Hamiltonian
reduction statement in $\mathsection{\ref{sec:isotwo}}$. The algebras in the
following lemma aren’t necessarily assumed to be commutative.

######  Lemma 3.1.4.

Let $\mathcal{A}$ be an associative $\mathbb{C}$-algebra generated by a vector
space $\mathcal{V}\subseteq\mathcal{A}$. Suppose the linear map
$S:\mathcal{V}\rightarrow\mathcal{V}$ that maps $v\mapsto-v$ for
$v\in\mathcal{V}$ extends to an algebra anti-automorphism
$S:\mathcal{A}\rightarrow\mathcal{A}$. Let $\mathcal{B}_{0},\mathcal{B}_{1}$
be $\mathbb{C}$-algebras and consider algebra homomorphisms
$f_{i}:\mathcal{A}\rightarrow\mathcal{B}_{i}$ for $i=0,1$. Suppose $f_{1}$ is
surjective and let $\mathcal{I}=\ker(f_{1})$. Let
$f_{2}:\mathcal{V}\rightarrow\mathcal{B}_{0}\otimes\mathcal{B}_{1}$ be the
linear map defined via the formula $v\mapsto f_{0}(v)\otimes 1+1\otimes
f_{1}(v)$. Then, there exists a vector space isomorphism:

$\mathcal{B}_{0}/(\mathcal{B}_{0}\cdot
f_{0}(S(\mathcal{I})))\simeq((\mathcal{B}_{0}\otimes\mathcal{B}_{1})/(\mathcal{B}_{0}\otimes\mathcal{B}_{1})\cdot
f_{2}(\mathcal{V})),$

induced by the linear map
$\mathcal{B}_{0}\rightarrow\mathcal{B}_{0}\otimes\mathcal{B}_{1}$ that sends
$b\mapsto b\otimes 1$ for $b\in\mathcal{B}_{0}$.

###### Proof.

We start by observing that:

$\mathcal{B}_{0}/(\mathcal{B}_{0}\cdot
f_{0}(S(\mathcal{I})))\simeq\mathcal{B}_{0}\otimes_{\mathcal{A}}(\mathcal{A}/S(\mathcal{I})).$

By definition, the space
$\mathcal{B}_{0}\otimes_{\mathcal{A}}(\mathcal{A}/S(\mathcal{I}))$ is equal to
the quotient of the algebra
$\mathcal{B}_{0}^{op}\otimes(\mathcal{A}/S(\mathcal{I}))$ by the left ideal
generated by elements of the form $f_{0}(v)\otimes 1-1\otimes f_{1}(v)$ for
$v\in\mathcal{V}$. Here, we have identified the underlying vector space of the
opposite algebra $\mathcal{B}_{0}^{op}$ with that of the algebra
$\mathcal{B}_{0}$.

Next, we note that the map $S$ induces an algebra anti-isomorphism:

$S:\mathcal{A}/S(\mathcal{I})\rightarrow\mathcal{A}/S(S(\mathcal{I}))=\mathcal{A}/\mathcal{I}=\mathcal{B}_{1}.$

Hence, we can define a map:

$\mathcal{B}_{0}\otimes_{\mathcal{A}}(\mathcal{A}/S(\mathcal{I}))\rightarrow((\mathcal{B}_{0}\otimes\mathcal{B}_{1})/(\mathcal{B}_{0}\otimes\mathcal{B}_{1})\cdot
f_{2}(\mathcal{V})),$

via the formula $b_{0}\otimes b_{1}\mapsto b_{0}\otimes S(b_{1})$. The fact
that $S$ is an anti-isomorphism ensures that the map is well-defined and gives
an isomorphism between the two spaces, thus completing the proof. ∎

######  Remark 3.1.5.

One can weaken the hypothesis of the lemma and assume that $\mathcal{B}_{0}$
and $\mathcal{B}_{1}$ are $\mathcal{A}$-modules (rather than algebras) and
prove a slightly modified statement, but we don’t need that generality here.

###### Proof of Theorem 3.1.3.

To prove the isomorphsim of schemes, we need to prove that their respective
coordinate rings are isomorphic. That is, we need to show that there is an
isomorphism:

$\phi^{*}:\mathbb{C}[T^{*}\mathfrak{g}]/J=\mathbb{C}[A]\rightarrow\mathbb{C}[X]^{\\{\pm
1\\}}=\Big{(}\mathbb{C}[T^{*}\mathfrak{g}]\otimes\mathbb{C}[V])/I\Big{)}^{\\{\pm
1\\}},$

induced by the map $p\mapsto p\otimes 1$ for
$p\in\mathbb{C}[T^{*}\mathfrak{g}]$. As the group $\\{\pm 1\\}$ acts trivially
on $\mathbb{C}[T^{*}\mathfrak{g}],$ we have the equality of algebras:

$\Big{(}\mathbb{C}[T^{*}\mathfrak{g}]\otimes\mathbb{C}[V])/I\Big{)}^{\\{\pm
1\\}}=(\mathbb{C}[T^{*}\mathfrak{g}]\otimes\mathbb{C}[V]_{even})/(I^{\\{\pm
1\\}}).$

We apply Lemma 3.1.4 by taking $\mathcal{A}=\operatorname{Sym}(\mathfrak{g})$,
$\mathcal{V}=\mathfrak{g}$, $\mathcal{B}_{0}=\mathbb{C}[T^{*}\mathfrak{g}]$,
$\mathcal{B}_{1}=\mathbb{C}[V]_{even}$ and $f_{i}=\theta_{i}$ for $i=0,1,2$.
The ideal $\mathcal{I}=\ker(f_{1})=\ker(\theta_{1})$, in this case, is equal
to $K$, which is $S$-invariant as it is generated by homogeneous polynomials
of degree 2 by Corollary 3.1.2. Then, the conclusion of Lemma 3.1.4 gives us
the required isomorphism. ∎

######  Remark 3.1.6.

Theorem 3.1.3 is a special case of the following ‘shifting trick’, which is
also an immediate consequence of Lemma 3.1.4:

Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $G$ be the corresponding
adjoint group and fix $\mathbb{O}$ to be a $G$-orbit in
$\mathfrak{g}\cong\mathfrak{g}^{*}$ under the adjoint action. Let $Y$ be an
affine variety with a Hamiltonian $G$-action. Then, the Hamiltonian reduction
of $Y$ at the negative orbit closure $-\overline{\mathbb{O}}$ is isomorphic to
the Hamiltonian reduction of the variety $Y\times\overline{\mathbb{O}}$ at
$0\in\mathfrak{g}$.

#### 3.1.3.

As a consequence of Theorem 3.1.3, we have an isomorphism of schemes
$X//G\simeq A//G$. Therefore, we have morphisms:

$C//G\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}A//G\stackrel{{\scriptstyle\phi}}{{\longleftarrow}}X//G.$

Set-theoretically, these morphisms are as follows: The map $\psi$ sends a pair
$(x,y)$ of commuting matrices to the almost commuting pair $(x,y)$. The map
$\phi$ sends a triple $(x,y,i)$ to the pair $(x,y)$. We have shown that $\phi$
is an isomorphism. The morphism $\psi$ is an isomorphism too, because it is
proven in [Los21] that we have an isomorphism $C//G\rightarrow X//G$, and that
morphism composed with $\phi$ gives $\psi$.

######  Remark 3.1.7.

The fact that $\psi$ is an isomorphism can also be proven independently by
mimicking the proof of Theorem 12.1 of [EG02], making use of Weyl’s
fundamental theorem of invariant theory for $\mathfrak{g}=\mathfrak{sp}(V)$.

Combining Theorem 1.2.1 with Theorem 1.3 of [Los21], we have the following
corollary:

######  Corollary 3.1.8.

We have an algebra isomorphism:

$\mathbb{C}[\mathfrak{h}\times\mathfrak{h}]^{W}=\mathbb{C}[(\mathfrak{h}\times\mathfrak{h})//W]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{C}[A//G]=\Big{(}\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]/J\Big{)}^{\mathfrak{g}}.$

###  3.2. Reminder on rational Cherednik algebras and the Harish-Chandra
homomorphism

#### 3.2.1.

Fix a Cartan sublagebra $\mathfrak{h}\subseteq\mathfrak{g}=\mathfrak{sp}(V)$
and a root system $R\subseteq\mathfrak{h}^{*}$. The space $\mathfrak{h}$ has
an action of the Weyl group $W=(\mathbb{Z}/(2))^{n}\rtimes S_{n}$. For each
$\alpha\in R$, let $s_{\alpha}\in W$ denote the reflection of $\mathfrak{h}$
relative to the root $\alpha$. Fix a $W$-invariant function
$c:R\rightarrow\mathbb{C}$. For root systems of Type $C$, it turns out that
there are exactly two $W$-orbits in $R$ given by the set of all long roots and
the set of all short roots. Hence, such a $W$-invariant function $c$ can be
viewed as a pair of complex numbers $c=(c_{long},c_{short})\in\mathbb{C}^{2}$.

We recall from [EG02] the definition of the rational Cherednik algebra $H_{c}$
of Type $C$, which is the quotient of the algebra
$\operatorname{Sym}(\mathfrak{h})\otimes\operatorname{Sym}(\mathfrak{h}^{*})\otimes\mathbb{C}[W]$
by the ideal generated by the following relations:

$wxw^{-1}=w(x),wyw^{-1}=w(y),$ $[y,x]=\langle
x,y\rangle-\dfrac{1}{2}\sum_{\alpha\in
R}c(\alpha)\langle\alpha,y\rangle\langle x,\alpha^{\vee}\rangle,$

for all $w\in W$, $x\in\mathfrak{h}^{*}$ and $y\in\mathfrak{h}$. Let
$e=\frac{1}{|W|}\sum_{w\in W}w$ be the averaging idempotent in
$\mathbb{C}[W]$. Then, we can construct the spherical subalgebra
$eH_{c}e\subseteq H_{c}$, which will be our main object of interest. It is
known that the subalgebras $\mathbb{C}[\mathfrak{h}]^{W}$ and
$\operatorname{Sym}(\mathfrak{h})^{W}$ generate the algebra $eH_{c}e$.

Another description of the algebra $eH_{c}e$ given in [EG02] will be useful
for us, and we recall that here. Define the rational Calogero-Moser operator
$L_{c}$, which is a differential operator on the space $\mathfrak{h}^{reg}$,
as follows (see [OP83]):

$L_{c}:=\Delta_{\mathfrak{h}}-\dfrac{1}{2}\sum_{\alpha\in
R}\dfrac{c(\alpha)(c(\alpha)+1)}{\alpha^{2}}\cdot(\alpha,\alpha),$

where $\Delta_{h}$ is the Laplacian operator on the Cartan subalgebra
$\mathfrak{h}$. Let $\mathcal{D}(\mathfrak{h}^{reg})_{-}$ denote the
subalgebra of $\mathcal{D}(\mathfrak{h}^{reg})$ spanned by differential
operators $D\in\mathcal{D}(\mathfrak{h}^{reg})$ such that
$degree(D)+order(D)\leq 0$. Let $\mathcal{C}_{c}$ denote the centralizer of
the operator $L_{c}$ in $\mathcal{D}(\mathfrak{h}^{reg})_{-}$. Finally,
consider the subalgebra $\mathcal{B}_{c}$ of $\mathcal{D}(\mathfrak{h}^{reg})$
generated by $\mathcal{C}_{c}$ and $\mathbb{C}[\mathfrak{h}]^{W}$, the algebra
of $W$-invariant polynomial functions on $\mathfrak{h}$.

######  Theorem 3.2.1.

[EG02, Theorem 4.9] We have an embedding (known as the Dunkl embedding)
$\Theta:eH_{c}e\hookrightarrow{}\mathcal{D}(\mathfrak{h}^{reg})^{W}$ such that
$\Theta(\mathbb{C}[\mathfrak{h}]^{W})=\mathbb{C}[\mathfrak{h}]^{W}$ and
$\Theta(\operatorname{Sym}(\mathfrak{h})^{W})=\mathcal{C}_{c}$. Moreover,
$\Theta(\Delta_{\mathfrak{h}})=L_{c}$. Furthermore, $\Theta$ induces an
isomorphism of algebras $eH_{c}e\simeq\mathcal{B}_{c}$.

The algebra $H_{c}$ has a filtration such that all the elements of $W$ and the
generators of $\operatorname{Sym}(\mathfrak{h}^{*})$ have degree zero, and the
generators of $\operatorname{Sym}(\mathfrak{h})$ have degree one. This induces
a filtration on the spherical subalgebra $eH_{c}e$. The algebra
$\mathcal{D}(\mathfrak{h}^{reg})$ has a filtration given by the degree of
differential operators. Then, we have the following PBW theorem for Cherednik
algebras:

######  Proposition 3.2.2.

[EG02, Corollary 4.4] With the above filtrations, we have an isomorphism:

$\mathrm{gr}(eH_{c}e)\simeq\mathbb{C}[\mathfrak{h}\times\mathfrak{h}]^{W}.$

Furthermore, under this isomorphism, the associated graded map
$\mathrm{gr}(\Theta):\mathbb{C}[\mathfrak{h}\times\mathfrak{h}]^{W}\simeq\mathrm{gr}(eH_{c}e)\rightarrow\mathrm{gr}(\mathcal{D}(\mathfrak{h}^{reg}))=\mathbb{C}[\mathfrak{h}^{reg}\times\mathfrak{h}^{*}]^{W}$
is exactly the algebra homomorhpism induced by the natural embedding
$\mathfrak{h}^{reg}\hookrightarrow{}\mathfrak{h}$.

#### 3.2.2.

We recall the construction of the ‘univeral’ Harish-Chandra homomorphism from
[EG02]. Let $\mathfrak{g}^{rs}$ denote the subset of regular semisimple
elements of $\mathfrak{g}$. Then, we have that
$\mathfrak{h}^{reg}=\mathfrak{h}\cap\mathfrak{g}^{rs}$. Next, inside the
universal enveloping algebra $\mathcal{U}\mathfrak{g}$, consider the space
$(\mathcal{U}\mathfrak{g})^{\operatorname{ad}(\mathfrak{h})}$ of
$\operatorname{ad}(\mathfrak{h})$-invariants. Then,
$(\mathcal{U}\mathfrak{g})^{\operatorname{ad}(\mathfrak{h})}\cdot\mathfrak{h}$
is a two-sided ideal of the algebra
$(\mathcal{U}\mathfrak{g})^{\operatorname{ad}(\mathfrak{h})}$, and so, we can
define the quotient algebra
$(\mathcal{U}\mathfrak{g})_{\mathfrak{h}}:=(\mathcal{U}\mathfrak{g})^{\operatorname{ad}(\mathfrak{h})}/((\mathcal{U}\mathfrak{g})^{\operatorname{ad}(\mathfrak{h})}\cdot\mathfrak{h}).$

By Proposition 6.1 of [EG02], and the paragraph following its proof, there
exists a canonical algebra isomorphism:

$\Psi:\mathcal{D}(\mathfrak{g}^{reg})^{\mathfrak{g}}\longrightarrow(\mathcal{D}(\mathfrak{h}^{reg})\otimes(\mathcal{U}\mathfrak{g})_{\mathfrak{h}})^{W}.$

Next, fix a $\mathcal{U}\mathfrak{g}$-module $\mathfrak{V}$ and let
$\mathfrak{V}\langle 0\rangle$ denote its zero weight space. Then, by
definition, this space $\mathfrak{V}\langle 0\rangle$ is acted upon by
$\mathfrak{h}$ by zero, and is thus stable under the action of
$(\mathcal{U}\mathfrak{g})^{\operatorname{ad}(\mathfrak{h})}.$ Hence, we have
an action of $(\mathcal{U}\mathfrak{g})_{\mathfrak{h}}$ on
$\mathfrak{V}\langle 0\rangle$, giving an algebra homomorphism
$\chi:(\mathcal{U}\mathfrak{g})_{\mathfrak{h}}\rightarrow\operatorname{End}_{\mathbb{C}}(\mathfrak{V}\langle
0\rangle)$. In particular, if $V\langle 0\rangle$ is one-dimensional, we get a
homomorphism
$\chi:(\mathcal{U}\mathfrak{g})_{\mathfrak{h}}\rightarrow\mathbb{C}$, and so,
we can compose it with the above algebra isomorphism to get:

$\Psi_{\mathfrak{V}}:=\chi\circ\Psi:\mathcal{D}(\mathfrak{g}^{reg})^{\mathfrak{g}}\longrightarrow\mathcal{D}(\mathfrak{h}^{reg})^{W}.$

This is the Harish-Chandra homomorphism associated with the representation
$\mathfrak{V}$.

Let $\mathrm{Ann}(\mathfrak{V})\cdot\mathcal{U}\mathfrak{g}$ be the
annihilator of $\mathfrak{V}$ and consider the ideal
$(\mathcal{D}(\mathfrak{g})\cdot\operatorname{ad}(\mathrm{Ann}(\mathfrak{V})))^{\mathfrak{g}},$
which is a two-sided ideal inside $\mathcal{D}(\mathfrak{g})^{\mathfrak{g}}$.
Then, it follows from definitions and the proof of [EG02, Proposition 6.1]
that the kernel of the homomorphism $\Psi_{\mathfrak{V}}$ contains the ideal
$(\mathcal{D}(\mathfrak{g})\cdot\operatorname{ad}(\mathrm{Ann}(\mathfrak{V})))^{\mathfrak{g}}.$

#### 3.2.3.

For explicit computations in subsequent sections, we fix some coordinates. We
identify the symplectic vector space $V$ with the space $\mathbb{C}^{2n}$,
which has the standard symplectic form given by the following $2n\times 2n$
skew-symmetric matrix:

$\begin{pmatrix}0&I\\\ -I&0\end{pmatrix}.$

The subspace $L\subseteq V=\mathbb{C}^{2n}$, consisting of vectors whose last
$n$ coordinates are zero, is a Lagrangian subspace of $V$ under the above
symplectic form and the dual vector space $L^{*}$ can be identified with the
space of vectors whose first $n$ coordinates are zero.

The Lie algebra $\mathfrak{sp}(V)=\mathfrak{sp}_{2n}$ is the space of
endomorphisms preserving this form. Then, $\mathfrak{sp}(V)$ can be viewed as
the subspace of $\mathfrak{gl}(V)=\mathfrak{gl}_{2n}$ consisting of block
matrices of the form:

$\begin{pmatrix}A&B\\\ C&D\end{pmatrix},$

where $A=-D^{T},B=B^{T}$ and $C=C^{T}$. The subspace $\mathfrak{h}$ consisting
of diagonal matrices is a Cartan subalgera of $\mathfrak{sp}_{2n}$.
Identifying $\mathfrak{h}\simeq\mathfrak{h}^{*}$ using the trace form, we fix
an orthonormal basis $r_{1},r_{2},\dots,r_{n}$ of $\mathfrak{h}^{*}$ given by
$r_{i}=(E_{i,i}-E_{i+n,i+n})/\sqrt{2}$, where $E_{i,j}\in\mathfrak{gl}_{2n}$
is the elementary matrix whose only non-zero entry is in the $i^{th}$ row of
the $j^{th}$ column and is equal to 1.

Then, we have a choice of a root system $R$ given by:

$R:=\Bigg{\\{}\pm\dfrac{(r_{i}+r_{j})}{\sqrt{2}},\pm\dfrac{(r_{i}-r_{j})}{\sqrt{2}},\pm\sqrt{2}r_{i}:1\leq
i<j\leq n\Bigg{\\}}.$

Here, the long roots are given the vectors $\pm\sqrt{2}r_{i}$, whereas the
rest are all short roots. The set of positive roots $R^{+}\subseteq R$ is
obtained by replacing all ‘$\pm$’-signs by ‘$+$’ in the above defintion. Let
$\\{e_{\alpha}\\}_{\alpha\in R}$ denote the set of root vectors in
$\mathfrak{h}$ that form a Cartan-Weyl basis of the Lie algebra $\mathfrak{g}$
chosen so that $(e_{\alpha},e_{-\alpha})=1$, where $(.,.)$ is the trace form.
Then, we can express the $e_{\alpha}$’s explicitly in terms of elementary
matrices $E_{i,j}$ as follows:

$\alpha=\dfrac{(r_{i}+r_{j})}{\sqrt{2}}\implies
e_{\alpha}=e_{-\alpha}^{T}=\dfrac{E_{i,j+n}+E_{j,i+n}}{\sqrt{2}}$
$\alpha=\dfrac{(r_{i}-r_{j})}{\sqrt{2}}\implies
e_{\alpha}=e_{-\alpha}^{T}=\dfrac{E_{i,j}-E_{j+n,i+n}}{\sqrt{2}}$
$\alpha=\sqrt{2}r_{i}\implies e_{\alpha}=e_{-\alpha}^{T}=E_{i,i+n}.$

###  3.3. Definitions and statement of the main theorem

#### 3.3.1.

We set up some notation. Let $\omega$ denote the symplectic form on the vector
space $V$. Let $L$ be a fixed Lagrangian subspace of $V$. Then, we define the
Weyl algebra $W_{2n}$ as the algebra of polynomial differential operators on
$L$, also denoted by $\mathcal{D}(L)$. Explicitly, the Weyl algebra is an
associative $\mathbb{C}$-algebra generated by the variables
$x_{1},x_{2},\dots,x_{n},y_{1},y_{2},\dots,y_{n}$ that satisfy the following
relations:

$[x_{i},x_{j}]=0,[y_{i},y_{j}]=0,[y_{i},x_{j}]=\delta_{i,j},1\leq i,j\leq n.$

Here, the $x_{i}$’s correspond to a choice of coordinates on the vector space
$L^{*}$ and and the $y_{i}$’s represent the partial derivatives
$\partial_{x_{i}}$’s with respect to $x_{i}$’s. We have a direct sum
decomposition $W_{2n}=W_{2n,even}\oplus W_{2n,odd}$ as vector spaces, where
$W_{2n,even}$ is the space spanned by all the monomials in the $x_{i}$’s and
$y_{i}$’s having even total degree and $W_{2n,odd}$ is the space spanned by
the monomials having odd total degree. It is clear that $W_{2n,even}$ is a
subalgebra of $W_{2n}$ which is generated by all monomials of the form
$x_{i}x_{j},y_{i}y_{j}$ and $x_{i}y_{j}$ for $1\leq i,j\leq n$.

Recall from $\mathsection{\ref{sec:class}}$ that corresponding to the $G$
action on $V$, we have a co-moment map:

$\theta_{1}:\mathfrak{g}\longrightarrow\mathbb{C}[V],$

such that the image of $\phi$ in $\mathbb{C}[V]$ is exactly the vector space
of polynomial functions on $V$ having degree $2$.

Next, we note that the algebra $W_{2n}$ is a quantization of $\mathbb{C}[V]$.
To make this statement precise, we define the symmetrization map, which is the
following map of vector spaces:

$\operatorname{Sym}:\mathbb{C}[V]\longrightarrow W_{2n}$
$\lambda_{1}\lambda_{2}\cdots\lambda_{k}\mapsto\dfrac{1}{k!}\sum_{\sigma\in
S_{k}}\lambda_{\sigma(1)}\lambda_{\sigma(2)}\cdots\lambda_{\sigma(k)},$

where each $\lambda_{i}$ is a linear function on $V$. (Here, we have used that
there is a canonical isomorphism of vector spaces $V^{*}\cong L\oplus L^{*}$
using the symplectic form). The map Sym is a vector space isomorphism. Note
that both of these spaces have Lie algebra structures, where the Lie bracket
on $\mathbb{C}[V]$ is given by the Poisson bracket and the bracket on $W_{2n}$
is the one induced by the commutator of the associative product. The following
lemma follows by a straightforward computation:

######  Lemma 3.3.1.

The restriction of the map $\operatorname{Sym}$ to the subspace
$\mathbb{C}[V]_{\leq 2}\subseteq\mathbb{C}[V]$ of polynomials of degree lesser
than or equal to two is a Lie algebra homomorphism .

Now, we define the composition:

$\Theta_{1}:=\operatorname{Sym}\circ\text{
}\theta_{1}:\mathfrak{g}\longrightarrow W_{2n}.$

Then, $\Theta_{1}$ is a Lie algebra homomorphism, and so it induces an algebra
homomorphism $\mathcal{U}\mathfrak{g}\rightarrow W_{2n}$, which we also denote
by $\Theta_{1}$. Note that the image of $\Theta_{1}$ lies in the even
subalgebra $W_{2n,even}$. Define
$\mathcal{K}:=\ker(\Theta_{1})\subseteq\mathcal{U}\mathfrak{g}$.

Viewing the Lie algebra $\mathfrak{sp}(V)=\mathfrak{sp}_{2n}$ as a subspace of
$\mathfrak{gl}_{2n}$, using the embedding defined in the previous section, the
map $\Theta_{1}$ can be written in terms of coordinates via the formula:

$\begin{pmatrix}A&B\\\
C&-A^{T}\end{pmatrix}\mapsto\dfrac{1}{2}\Bigg{(}\sum_{i,j=1}^{n}(2a_{i,j}x_{i}y_{j}+b_{i,j}x_{i}x_{j}-c_{i,j}y_{i}y_{j})+\operatorname{Tr}(A)\Bigg{)},$

where $A=(a_{i,j}),B=(b_{i,j})$ and $C=(c_{i,j})$.

Next, we define the action of $\mathfrak{g}$ on the space
$\mathcal{D}(\mathfrak{g})$ of polynomial differential operators on
$\mathfrak{g}$. For this, fix $x\in\mathfrak{g}$ and consider the linear map:

$\operatorname{ad}_{x}:\mathfrak{g}\longrightarrow\mathfrak{g}$
$y\mapsto[x,y].$

Using the identification $\mathfrak{g}\simeq\mathfrak{g}^{*}$, the assignment
$x\mapsto\operatorname{ad}_{x}$ gives a Lie algebra homomorphism
$\mathfrak{g}\rightarrow\operatorname{End}(\mathfrak{g}^{*})$, where
$\operatorname{End}(\mathfrak{g}^{*})$ denotes the space of vector space
endomorphisms of $\mathfrak{g}^{*}$. Any element in
$\operatorname{End}(\mathfrak{g}^{*})$ can be uniquely extended to a
derivation on
$\operatorname{Sym}(\mathfrak{g}^{*})\simeq\mathbb{C}[\mathfrak{g}]$ via
Leibniz rule. Since
$\operatorname{Der}(\mathbb{C}[\mathfrak{g}])\subseteq\mathcal{D}(\mathfrak{g})$,
we have constructed a map:

$\Theta_{0}:\mathfrak{g}\longrightarrow\mathcal{D}(\mathfrak{g}),$

which is a Lie algebra homomorphism. Hence, this induces an algebra
homomorphism $\mathcal{U}\mathfrak{g}\rightarrow\mathcal{D}(\mathfrak{g})$,
which we also denote by $\Theta_{0}$. Then, given $x\in\mathfrak{g}$ and
$d\in\mathcal{D}(\mathfrak{g})$, the action of $\mathfrak{g}$ on
$\mathcal{D}(\mathfrak{g})$ is defined via:

$x\cdot d:=[\Theta_{0}(x),d].$

We now define the algebras we will be working with:

  * •

Consider the algebra $\mathcal{D}(\mathfrak{g})\otimes W_{2n}$. We have the
diagonal $\mathfrak{g}$-action on this algebra: Given $d\otimes
w\in\mathcal{D}(\mathfrak{g})\otimes W_{2n}$ and $x\in\mathfrak{g}$, we define
the action via:

$x\cdot(d\otimes w):=[\Theta_{0}(x),d]\otimes w+d\otimes[\Theta_{1}(x),w].$

Let $\Theta_{2}:\mathfrak{g}\rightarrow\mathcal{D}(\mathfrak{g})\otimes
W_{2n}$ be the Lie algebra homomorphism defined via
$\Theta_{2}=\Theta_{0}\otimes 1+1\otimes\Theta_{1}$. This can be extended to
get an algebra homomorphism
$\Theta_{2}:\mathcal{U}\mathfrak{g}\rightarrow\mathcal{D}(\mathfrak{g})\otimes
W_{2n}$, which is the co-moment map for the above action. Then, we have the
quantum Hamiltonian reduction $\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\mathfrak{g}}$ of
$\mathcal{D}(\mathfrak{g})\otimes W_{2n}$ at the augmentation ideal
$\mathcal{U}\mathfrak{g}^{+}$ of $\mathcal{U}\mathfrak{g}$.

  * •

Define $\mathfrak{J}\subseteq\mathcal{D}(\mathfrak{g})$ to be the left ideal
generated by the image of $\mathcal{K}\subseteq\mathcal{U}\mathfrak{g}$ under
the co-moment map $\Theta_{0}$, that is,
$\mathfrak{J}:=\mathcal{D}(\mathfrak{g})\cdot(\Theta_{0}(\ker(\Theta_{1})))$.
Then, we have the algebra
$\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}$ which is
the quantum Hamiltonian reduction of the algebra $\mathcal{D}(\mathfrak{g})$
at the ideal $\mathcal{K}\subseteq\mathcal{U}\mathfrak{g}$.

  * •

Finally, we consider the spherical subalgebra $eH_{c}e$ of the rational
Cherednik algebra $H_{c}$ with the paramater $c$ given by
$c=(c_{long},c_{short})=(-1/4,-1/2).$

Our goal is to prove Theorem 1.3.1 by constructing algebra isomorphisms:

$\Psi:\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\longrightarrow
eH_{c}e,$
$\Phi:\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\longrightarrow\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\mathfrak{g}}.$

These isomorphisms are established in Theorems 3.4.3 and 3.5.1 respectively.

#### 3.3.2.

We define some filtrations on the algebras we will be working with. On the
algebra $\mathcal{D}(\mathfrak{g})$, we have an increasing filtration given by
the degree of the differential operators. The associated graded with respect
to this filtration is given by
$\mathrm{gr}(\mathcal{D}(\mathfrak{g}))\simeq\mathbb{C}[T^{*}\mathfrak{g}]\simeq\mathbb{C}[\mathfrak{g}\times\mathfrak{g}],$
identifying $\mathfrak{g}^{*}$ with $\mathfrak{g}$ using the trace pairing.

On the even part of the Weyl algebra $W_{2n,even}$, we define a version of the
Bernstein filtration. Note that the algebra $W_{2n,even}$ is generated by
elements of the form $x_{i}x_{j},y_{i}y_{j}$ and $x_{i}y_{j}+y_{j}x_{i}$. We
define the filtration on $W_{2n,even}$ by placing each of these elements in
degree 1. Thus, any element which is a product of $2m$ of the $x_{i}$’s and
$y_{i}$’s lies in the $m^{th}$ filtered piece. Under this filtration, we have
the associated graded $\mathrm{gr}(W_{2n,even})=\mathbb{C}[V]_{even}$.

Using the two filtrations above, we also get a tensor product filtration on
$\mathcal{D}(\mathfrak{g})\otimes W_{2n,even}$ such that the associated graded
is $\mathrm{gr}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n,even})=\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]\otimes\mathbb{C}[V]_{even}$.
Next, on the algebra $\mathcal{U}\mathfrak{g}$, we have the PBW filtration,
and so by PBW theorem, we have the associated graded with respect to this
filtration
$\mathrm{gr}(\mathcal{U}\mathfrak{g})\simeq\operatorname{Sym}\mathfrak{g}$.

######  Remark 3.3.2.

Note that the $\mathfrak{g}$-action on each of these algebras is filtration
preserving, and so, as $\mathfrak{g}$ is reductive, taking
$\mathfrak{g}$-invariants commutes with computing associated graded.

Finally, the Cherednik algebra $H_{c}$ has a filtration such that all the
elements of $W$ and the generators of $\operatorname{Sym}(\mathfrak{h}^{*})$
have degree zero, and the generators of $\operatorname{Sym}(\mathfrak{h})$
have degree one. This also induces a filtration on the spherical subalgebra
$eH_{c}e$.

Before moving on to the construction of the maps $\Phi$ and $\Psi$, we prove a
lemma that relates the ideals in the non-commutative setting with the ideals
in the commutative setting:

######  Lemma 3.3.3.

We have an inclusion of ideals $J\subseteq\mathrm{gr}(\mathfrak{J})$, where
$J\subseteq\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]$ is the defining ideal
of pairs of matrices whose commutator has rank lesser than or equal to 1.

Recall from the definitions given in $\mathsection\ref{sec:class}$ that the
ideal $J$ is generated by all $2\times 2$ minors of the commutator $[x,y]$ for
$(x,y)\in\mathfrak{g}\times\mathfrak{g}$.

###### Proof.

First, we describe the associated graded versions of the maps $\Theta_{1}$ and
$\Theta_{0}$, in order to compute $\mathrm{gr}(\mathfrak{J})$. The map
$\Theta_{1}:\mathcal{U}\mathfrak{g}\rightarrow W_{2n,even}\subseteq W_{2n}$
was defined on $\mathfrak{g}\subseteq\mathcal{U}\mathfrak{g}$ as the
composition $\operatorname{Sym}\circ\text{ }\theta_{1}$. Then, this is a
filtration-preserving map, and we have:

$\theta_{1}=\mathrm{gr}(\Theta_{1}):\operatorname{Sym}\mathfrak{g}=\mathrm{gr}(\mathcal{U}\mathfrak{g})\longrightarrow\mathrm{gr}(W_{2n,even})=\mathbb{C}[V]_{even}\subseteq\mathbb{C}[V].$

This induces a map
$V=\operatorname{Spec}(\mathbb{C}[V])\rightarrow\operatorname{Spec}(\operatorname{Sym}\mathfrak{g}))=\mathfrak{g}^{*}\simeq\mathfrak{g}$
that sends a vector $v$ to the rank one endomorphism $v^{2}\in\mathfrak{g}$.
By Lemma 3.1.1, we know that the ideal $\ker(\theta_{1})$ is generated by all
$2\times 2$ minors in the entries of $\mathfrak{g}$.

Next, we consider the map
$\Theta_{0}:\mathcal{U}\mathfrak{g}\rightarrow\mathcal{D}(\mathfrak{g})$. In
this case, we have the associated graded map:

$\theta_{0}=\mathrm{gr}(\Theta_{0}):\operatorname{Sym}\mathfrak{g}=\mathrm{gr}(\mathcal{U}\mathfrak{g})\longrightarrow\mathrm{gr}(\mathcal{D}(\mathfrak{g}))=\mathbb{C}[\mathfrak{g}\times\mathfrak{g}].$

This is the co-moment map for the diagonal adjoint action of $G$ on
$\mathfrak{g}\times\mathfrak{g}$. This morphism $\theta_{0}$ induces a map
$\mathfrak{g}\times\mathfrak{g}=\operatorname{Spec}(\mathbb{C}[\mathfrak{g}\times\mathfrak{g}])\rightarrow\operatorname{Spec}(\operatorname{Sym}\mathfrak{g})=\mathfrak{g}^{*}\simeq\mathfrak{g}$
sending a pair $(x,y)$ to the commutator $[x,y]$.

Hence, if we consider the ideal generated by the image of $\ker(\theta_{1})$
under the map $\theta_{0}$, then this is exactly generated by the $2\times 2$
minors of the commutator $[x,y]$ for $(x,y)\in\mathfrak{g}\times\mathfrak{g}$.
But these are exactly the generators of the ideal $J$, and so
$J=\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]\cdot\theta_{0}(\ker(\theta_{1}))$.
Also, by definition,
$\mathfrak{J}=\mathcal{D}(\mathfrak{g})\cdot\Theta_{0}(\ker(\Theta_{1}))$.
Hence, to prove that $J\subseteq\mathrm{gr}(\mathfrak{J})$, it suffices to
show that $\mathrm{gr}(\ker(\Theta_{1}))=\ker(\theta_{1})$. It is clear that
$\mathrm{gr}(\ker(\Theta_{1}))\subseteq\ker(\theta_{1})$.

To see that the inclusion is an equality, we first describe the images of the
maps $\theta_{1}$ and $\Theta_{1}$. We have that $\theta_{1}(\mathfrak{g})$ is
exactly the space of degree $2$ polynomials in $\mathbb{C}[V]$, and so,
$Im(\theta_{1})$ is equal to the subalgebra $\mathbb{C}[V]_{even}$ of all
polynomials having even total degree. Applying the symmetrization map, we see
that $Im(\Theta_{1})$ is exactly $W_{2n,even}$. Thus, we have a short exact
sequence of filtered $\mathcal{U}\mathfrak{g}$-modules:

${0}$${\ker(\Theta_{1})}$${\mathcal{U}\mathfrak{g}}$${W_{2n,even}}$${0.}$

We claim that the filtrations on $\ker(\Theta_{1})$ and $W_{2n,even}$ are
exactly the ones induced on them by $\mathcal{U}\mathfrak{g}$. For
$\ker(\Theta_{1})$, this is true by definition. For $W_{2n,even}$, this
follows by observing that the function $\Theta_{1}$ maps
$\mathfrak{g}\subseteq\mathcal{U}\mathfrak{g}$ to exactly the space spanned by
the elements $x_{i}x_{j},y_{i}y_{j}$ and $x_{i}y_{j}+y_{j}x_{i}$. Hence, this
is a strict exact sequence of filtered modules, and so, by [Sjö73, Lemma 1],
we can take the associated graded to get an exact sequence of
$\mathrm{gr}(\mathcal{U}\mathfrak{g})=\operatorname{Sym}\mathfrak{g}$-modules:

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This implies that $\mathrm{gr}(\ker(\Theta_{1}))=\ker(\theta_{1})$, completing
the proof. ∎

######  Remark 3.3.4.

The ideal $\ker(\Theta_{1})$ is a primitive ideal of $\mathcal{U}\mathfrak{g}$
and the equality $\mathrm{gr}(\ker(\Theta_{1}))=\ker(\theta_{1})$ can also be
seen in a more general setting in the works of Joseph, where he constructs
minimal realizations of simple Lie algebras in the Weyl algebra (see [Jos74],
[Jos76]). This ideal is often referred to as the Joseph ideal in the
literature.

###  3.4. Construction of the isomorphism $\Psi$

In this section, we construct the isomorphism:

$\Psi:\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\longrightarrow
eH_{c}e.$

For this, we recall the map $\Theta_{1}:\mathfrak{g}\rightarrow W_{2n}$
defined in the previous section. The Weyl algebra $W_{2n}$ is the space of the
polynomial differential operators on the vector space $L$, and we have fixed
coordinates $x_{1},x_{2},\dots,x_{n}$ on $L^{*}$. Let $\mathfrak{V}$ be the
vector space spanned by all expressions of the form $(x_{1}x_{2}\cdots
x_{n})^{-1/2}\cdot P$, where $P$ is a Laurent polynomial in
$x_{1},x_{2},\dots,x_{n}$, i.e $P\in\mathbb{C}[x_{1}^{\pm 1},x_{2}^{\pm
1},\cdots,x_{n}^{\pm 1}]$. Then, the map $\Theta_{1}$ gives an action of
$\mathfrak{g}$ on $\mathfrak{V}$, where any element $f\in\mathfrak{g}$ acts on
$\mathfrak{V}$ by $\Theta_{1}(f)\subseteq W_{2n}$ via formal differentiation
of Laurent polynomials.

Recall that the Cartan $\mathfrak{h}\subseteq\mathfrak{g}$ is spanned by
elements of the form $E_{i,i}-E_{n+i,n+i}$ for $1\leq i\leq n$. Then, under
$\Theta_{1}$, the image of this element is the differential operator
$(x_{i}y_{i}+y_{i}x_{i})/2=x_{i}y_{i}+\frac{1}{2}=x_{i}\partial_{x_{i}}+\frac{1}{2}$.
Hence, the $\mathfrak{g}$-action on $\mathfrak{V}$ has a one-dimensional zero
weight space $\mathfrak{V}\langle 0\rangle$ spanned by the element
$(x_{1}x_{2}\cdots x_{n})^{-1/2}\cdot 1$. Then, using the construction of the
radial parts homomorphism in $\mathsection\ref{sec:RCA}$, we get an algebra
homomorphism:

$\Psi:\mathcal{D}(\mathfrak{g})^{\mathfrak{g}}\longrightarrow\mathcal{D}(\mathfrak{h}^{reg})^{W}.$

Let $\Delta_{\mathfrak{g}}$ and $\Delta_{\mathfrak{h}}$ be the Laplacian
operators on $\mathfrak{g}$ and $\mathfrak{h}$ respectively. Recall from
$\mathsection\ref{sec:RCA}$ the Caloger-Moser differential operator $L_{c}$
for $c=(c_{long},c_{short})=(-1/4,-1/2)$ and let $\mathcal{C}_{c}$ denote the
centralizer of $L_{c}$ in $\mathcal{D}(\mathfrak{h}^{reg})$. Also, let
$\operatorname{Sym}(\mathfrak{g})\subseteq\mathcal{D}(\mathfrak{g})$ denote
the subalgebra of differential operators on $\mathfrak{g}$ with constant
coefficients. The computations in the following lemma are the origin of the
precise value of our parameter $c$.

######  Lemma 3.4.1.

  1. (a)

We have the equality:

$\Psi(\Delta_{\mathfrak{g}})=L_{c}.$

  2. (b)

The map $\Psi$ induces an isomorphism:

$\operatorname{Sym}(\mathfrak{g})^{\mathfrak{g}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}C_{c}.$

###### Proof.

  1. (a)

By Proposition 6.2 of [EG02], we have:

$\Psi(\Delta_{\mathfrak{g}})=\Delta_{\mathfrak{h}}-\sum_{\alpha\in
R}\dfrac{e_{\alpha}\cdot e_{-\alpha}}{\alpha^{2}}.$

We will evaluate $e_{\alpha}\cdot e_{-\alpha}|_{\mathfrak{V}\langle 0\rangle}$
for the long roots and the short roots separately:

    1. (i)

Suppose $\alpha$ is a long root. (For such roots, $(\alpha,\alpha)=2$.) Hence,
$\alpha=\sqrt{2}r_{i}$ for some $i$, where
$r_{i}=(E_{i,i}-E_{i+n,i+n})/\sqrt{2}$. Then,
$e_{\alpha}=e_{-\alpha}^{T}=E_{i,i+n}$, which implies that:

$\Theta_{1}(e_{\alpha}\cdot
e_{-\alpha})=\dfrac{({x_{i}}^{2})\cdot(-\partial_{x_{i}}^{2})}{4}.$

It is straightforward to see that this operator acts by
$\frac{-3}{16}\operatorname{Id}_{\mathfrak{V}\langle
0\rangle}=c_{long}(c_{long}+1)\operatorname{Id}_{\mathfrak{V}\langle
0\rangle}$ on the space spanned by $(x_{1}x_{2}\cdots x_{n})^{-1/2}\cdot 1$.

    2. (ii)

Suppose $\alpha$ is a short root. (For such roots, $(\alpha,\alpha)=1$.) Then,
$\alpha=(r_{i}+r_{j})/\sqrt{2}$ or $\alpha=(r_{i}-r_{j})/\sqrt{2}$ for some
$i\neq j$. In the first case,
$e_{\alpha}=e_{-\alpha}^{T}=(E_{i,j+n}+E_{j,i+n})/\sqrt{2}$, and so:

$\Theta_{1}(e_{\alpha}\cdot
e_{-\alpha})=\dfrac{({x_{i}}{x_{j}})\cdot(-\partial_{x_{i}}\partial_{x_{j}})}{2}.$

In the second case,
$e_{\alpha}=e_{-\alpha}^{T}=(E_{i,j}+E_{j+n,i+n})/\sqrt{2}$, giving that:

$\Theta_{1}(e_{\alpha}\cdot
e_{-\alpha})=\dfrac{(x_{i}\partial_{x_{j}})\cdot(x_{j}\partial_{x_{i}})}{2}.$

Then, in either case, the differential operator acts by
$\frac{-1}{8}\operatorname{Id}_{\mathfrak{V}\langle
0\rangle}=\frac{1}{2}c_{short}(c_{short}+1)\operatorname{Id}_{\mathfrak{V}\langle
0\rangle}$ on the space spanned by $(x_{1}x_{2}\cdots x_{n})^{-1/2}\cdot 1$.

Hence, we conclude that:

$\Psi(\Delta_{\mathfrak{g}})=\Delta_{\mathfrak{h}}-\dfrac{1}{2}\sum_{\alpha\in
R}\dfrac{c(\alpha)(c(\alpha)+1)}{\alpha^{2}}\cdot(\alpha,\alpha),$

which is exactly the Calogero-Moser operator $L_{c}$ of Type $C$ for the
parameter $c$.

  2. (b)

This follows from the proof of Proposition 7.2 of [EG02], which works for any
reductive Lie algebra $\mathfrak{g}$.

∎

By the above lemma, we see that
$\mathcal{C}_{c}\subseteq\operatorname{Im}(\Psi)$. Next, the restriction of
$\Psi$ to
$\mathbb{C}[\mathfrak{g}]^{\mathfrak{g}}\subseteq\mathcal{D}(\mathfrak{g})^{\mathfrak{g}}$
is exactly the Chevalley restriction map
$\mathbb{C}[\mathfrak{g}]^{\mathfrak{g}}\rightarrow\mathbb{C}[\mathfrak{h}]^{W}\subseteq\mathcal{D}(\mathfrak{h}^{reg})$,
and so, $\mathbb{C}[\mathfrak{h}]^{W}\subseteq\operatorname{Im}(\Psi)$. Now,
by Theorem 3.2.1, the image of the spherical subalgebra $eH_{c}e$ under the
Dunkl homomorphism
$\Theta:eH_{c}e\rightarrow\mathcal{D}(\mathfrak{h}^{reg})^{W}$ is exactly the
subalgebra of $\mathcal{D}(\mathfrak{h}^{reg})^{W}$ generated by
$\mathcal{C}_{c}$ and $\mathbb{C}[\mathfrak{h}]^{W}$. Hence, we have the
following corollary:

######  Corollary 3.4.2.

We have the inclusion of algebras
$\Theta(eH_{c}e)\subseteq\operatorname{Im}(\Psi)$.

Next, let $\mathrm{Ann}(\mathfrak{V})\subseteq\mathcal{U}\mathfrak{g}$ denote
the annihilator of the representation $\mathfrak{V}$. Then, as remarked in
$\mathsection\ref{sec:RCA}$, we have that
$(\mathcal{D}(\mathfrak{g})\cdot\operatorname{ad}(\mathrm{Ann}(\mathfrak{V})))^{\mathfrak{g}}\subseteq\ker(\Psi)$.
Finally, we note that the action of $\mathfrak{g}$ on $\mathfrak{V}$ was
defined via the map $\Theta_{1}$, and so,
$\ker(\Theta_{1})\subseteq\mathrm{Ann}(\mathfrak{V})$, which implies that
$\mathfrak{J}^{\mathfrak{g}}=(\mathcal{D}(\mathfrak{g})\cdot\Theta_{0}(\ker(\Theta_{1})))^{\mathfrak{g}}\subseteq(\mathcal{D}(\mathfrak{g})\cdot\operatorname{ad}(\mathrm{Ann}(\mathfrak{V})))^{\mathfrak{g}}$.

######  Theorem 3.4.3.

We have an isomorphism of algebras:

$\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\simeq
eH_{c}e.$

###### Proof.

The proof of this theorem is based on a commutative diagram that is very
similar to the one present in the proof of Theorem 1.3.1 of [GG06].

As noted above, we have
$\mathfrak{J}^{\mathfrak{g}}\subseteq(\mathcal{D}(\mathfrak{g})\cdot\operatorname{ad}(\mathrm{Ann}(\mathfrak{V})))^{\mathfrak{g}}\subseteq\ker(\Psi)$,
and so, the map $\Psi$ can be factored to induce an algebra hommomorphism
(which we also denote by $\Psi$):

$\Psi:\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\longrightarrow\mathcal{D}(\mathfrak{h}^{reg}).$

Identifying $\mathfrak{h}$ with $\mathfrak{h}^{*}$ using the trace form, we
get an algebra isomorphism
$\phi:\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{*}]\rightarrow\mathbb{C}[\mathfrak{h}\times\mathfrak{h}]$.
Then, we have the following diagram:

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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{49.12213pt}{-48.57364pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\text{gr}\Psi}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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This diagram commutes, and so, we get that all the arrows must be bijections.
In particular, the image of $\mathrm{gr}(\Psi)$ in
$\mathrm{gr}(\mathcal{D}(\mathfrak{h}^{reg}))=\mathbb{C}[\mathfrak{h}\times\mathfrak{h}^{reg}]$
can be identified with the image of $\mathrm{gr}(\Theta)$. This identification
can also be obtained as the associated graded of the embedding
$\Theta(eH_{c}e)\subseteq\operatorname{Im}(\Psi)$ from Corollary 3.4.2, and
so, this embedding must itself be an equality. Hence, we can compose with
$\Theta^{-1}$ (as $\Theta$ is injective) to get an algebra homomorphism:

$\Theta^{-1}\circ\Psi:\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\longrightarrow
eH_{c}e.$

It follows from the commutative diagram that the associated graded version of
this map gives the bijection between
$\mathrm{gr}\Big{(}(\mathcal{D}(\mathfrak{g})/\mathfrak{J})^{\mathfrak{g}}\Big{)}$
and $\mathrm{gr}(eH_{c}e)$. Hence, the map $\Theta^{-1}\circ\Psi$ itself must
be a bijection, which is exactly the claim of the theorem. ∎

######  Corollary 3.4.4.

  1. (1)

We have an isomorphism of commutative algebras:

$\mathrm{gr}\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{gr}(eH_{c}e).$

  2. (2)

All the maps in the above commutative diagram are isomorphisms. In particular,
we get that:

    1. (a)

We have an isomorphism:

$\mathbb{C}[A//G]=\Big{(}\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]/J\Big{)}^{\mathfrak{g}}\simeq\mathrm{gr}\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}.$

    2. (b)

We have the equality of ideals
$\mathrm{gr}(\mathfrak{J})^{\mathfrak{g}}=J^{\mathfrak{g}}$ in the ring
$\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]^{\mathfrak{g}}$ (cf. Lemma 3.3.3).

###  3.5. Construction of the isomorphism $\Phi$

Now, we construct an isomorphism of non-commutative algebras:

$\Phi:\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\longrightarrow\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\mathfrak{g}}.$

In fact, we prove the following stronger result:

######  Theorem 3.5.1.

There is an isomorphism of vector spaces:

$\Phi:\mathcal{D}(\mathfrak{g})/\mathfrak{J}\longrightarrow\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\\{\pm 1\\}}.$

Furthermore, the map $\Phi$ restricts to an algebra isomorphism between the
respective spaces of $\mathfrak{g}$-invariants.

###### Proof.

As the group $\\{\pm 1\\}$ acts trivially on the algebra
$\mathcal{D}(\mathfrak{g})$, we have an equality of vector spaces:

$\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\\{\pm
1\\}}=(\mathcal{D}(\mathfrak{g})\otimes
W_{2n,even})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n,even})\cdot\Theta_{2}(\mathfrak{g}).$

We apply Lemma 3.1.4 taking $\mathcal{A}=\mathcal{U}\mathfrak{g}$,
$\mathcal{V}=\mathfrak{g}$, $\mathcal{B}_{0}=\mathcal{D}(\mathfrak{g})$,
$\mathcal{B}_{1}=W_{2n,even}$ and $f_{i}=\Theta_{i}$ for $i=0,1,2$. Here, the
ideal $\mathcal{I}=\ker(f_{1})=\ker(\Theta_{1})$ is equal to $\mathcal{K}$,
which is $S$-invariant because we have a commutative diagram:

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{\Theta_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where the right vertical map $S^{\prime}:W_{2n}\rightarrow W_{2n}$ is an
algebra anti-homomorphism defined by sending the generators
$x_{1},\dots,x_{n},y_{1},\dots,y_{n}$ of $W_{2n}$ to $ix_{1},\dots
ix_{n},iy_{1},\dots,iy_{n}$ respectively, where $i=\sqrt{-1}$. Then, the
conclusion from Lemma 3.1.4 gives us the required vector space isomorphism.

Furthermore, as this isomorphism is induced by the map
$\mathcal{D}(\mathfrak{g})\rightarrow\mathcal{D}(\mathfrak{g})\otimes W_{2n}$
given by $D\mapsto D\otimes 1$ for $D\in\mathcal{D}(\mathfrak{g})$, which is
an algebra homomorphism, the restriction to $\mathfrak{g}$-invariants gives an
algebra isomorphism. ∎

######  Corollary 3.5.2.

We have isomorphisms of commutative algebras:

$\mathrm{gr}\Big{(}\mathcal{D}(\mathfrak{g})/\mathfrak{J}\Big{)}^{\mathfrak{g}}\simeq\mathrm{gr}\Big{(}(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})/(\mathcal{D}(\mathfrak{g})\otimes
W_{2n})\cdot\Theta_{2}(\mathfrak{g})\Big{)}^{\mathfrak{g}}\simeq\Big{(}\mathbb{C}[\mathfrak{g}\times\mathfrak{g}\times
V]/I\Big{)}^{\mathfrak{g}}=\mathbb{C}[X//G].$

###### Proof.

Recall from $\mathsection{\ref{sec:class}}$ that we have an isomorphism of
schemes $\phi:X//G\rightarrow A//G$. This gives an isomorphism between the
coordinate rings:

$\phi^{*}:\Big{(}\mathbb{C}[\mathfrak{g}\times\mathfrak{g}]/J\Big{)}^{\mathfrak{g}}=\mathbb{C}[A//G]\rightarrow\mathbb{C}[X//G]=\Big{(}\mathbb{C}[\mathfrak{g}\times\mathfrak{g}\times
V]/I\Big{)}^{\mathfrak{g}}.$

Then, we can consider the commutative diagram:

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