# One dimensional representations of finite $W$-algebras,
Dirac reduction and the orbit method

Lewis Topley

###### Abstract.

In this paper we study the variety of one dimensional representations of the
finite $W$-algebra for classical Lie algebras, giving a precise description of
the dimensions of the irreducible components. We apply this to prove a
conjecture of Losev describing the image of his orbit method map. In order to
do so we first establish new Yangian-type presentations of semiclassical
limits of the $W$-algebras attached to distinguished nilpotent elements in
classical Lie algebras, using Dirac reduction.

Dedicated to my teacher Sasha Premet, with admiration and gratitude.

###  1\. Introduction

1.1. Let $G$ be a complex connected reductive algebraic group with Lie algebra
$\mathfrak{g}$, nilpotent cone $\mathcal{N}(\mathfrak{g})$. Identify
$\mathfrak{g}$ with $\mathfrak{g}^{*}$ by a choice of non-degenerate
$G$-invariant trace form on $\mathfrak{g}$. The primitive spectrum
$\operatorname{Prim}U(\mathfrak{g})$ is the set of all primitive ideals of the
enveloping algebra, equipped with the Jacobson topology. These ideals are
classically studied via their invariants, and the most important of these are
the associated variety and Goldie rank. The associated variety
$\mathcal{V\\!A}(I)$ is defined to be the vanishing locus in $\mathfrak{g}$ of
the associated graded ideal $\operatorname{gr}I\subseteq
S(\mathfrak{g})=\mathbb{C}[\mathfrak{g}]$ with respect to the PBW filtration.
A celebrated theorem of Kostant states that $\mathcal{N}(\mathfrak{g})$ is the
vanishing locus of the positive degree invariant polynomials
$S(\mathfrak{g})^{G}_{+}$ whilst Joseph’s irreducibility theorem states that
$\mathcal{V\\!A}(I)$ is irreducible (see [Ja04] for a detailed survey).
Together with Dixmier’s lemma, these results show that the associated variety
is the closure of a nilpotent orbit. The Goldie rank is defined to be the
uniform dimension of the primitive quotient $U(\mathfrak{g})/I$.

It is natural to consider the decomposition
$\operatorname{Prim}U(\mathfrak{g})=\bigcup_{\mathcal{O}}\operatorname{Prim}_{\mathcal{O}}U(\mathfrak{g})$,
were the union is taken over all nilpotent $G$-orbits $\mathcal{O}$ and
$\operatorname{Prim}_{\mathcal{O}}U(\mathfrak{g})=\\{I\in\operatorname{Prim}U(\mathfrak{g})\mid\mathcal{V\\!A}(I)=\overline{\mathcal{O}}\\}.$
Now fix an orbit $\mathcal{O}\subseteq\mathcal{N}(\mathfrak{g})$ and
$e\in\mathcal{O}$, and let $U(\mathfrak{g},e)$ denote the finite $W$-algebra,
first associated to $(\mathfrak{g},e)$ by Premet [Pr02]. The reductive part of
the centraliser $G^{e}(0)$ acts naturally on $U(\mathfrak{g},e)$ by algebra
automorphisms, and this induces an action of the component group
$\Gamma=G^{e}(0)/G^{e}(0)^{\circ}$ on the category of finite dimensional
modules. Losev famously gave a new construction of $U(\mathfrak{g},e)$ via
deformation quantization [Lo10a] and used this to show that
$\operatorname{Prim}_{\mathcal{O}}U(\mathfrak{g})$ is in bijection with
$U(\mathfrak{g},e)\operatorname{-mod}_{\operatorname{f.d.}}/\Gamma$ [Lo11].
The one dimensional representations of $U(\mathfrak{g},e)$ play an especially
important role here for two reasons: on one hand the images under Skryabin’s
equivalence are all completely prime, and therefore play a key role in
Joseph’s theory of Goldie rank polynomials [Lo15], and on the other hand they
classify quantizations of $G$-equivariant coverings of $\mathcal{O}$ [Lo10b].

1.2. The above narrative leads us to consider the affine scheme
$\mathcal{E}(\mathfrak{g},e):=\operatorname{Spec}U(\mathfrak{g},e)^{\operatorname{ab}}$
associated to the maximal abelian quotient. By Hilbert’s nullstellensatz the
closed points classify the one dimensional representations of
$U(\mathfrak{g},e)$. The work of Losev and Premet [Lo10a, Pr14] shows that
$\mathcal{E}(\mathfrak{g},e)$ is nonempty and in [Pr10, PT14] the first steps
were made towards a full description of $\mathcal{E}(\mathfrak{g},e)$.

Recall that the sheets of $\mathfrak{g}$ are the maximal irreducible subsets
consisting of orbits of constant dimension. They are classified via the theory
of decomposition classes which, in turn, are classified by the
Lusztig–Spaltenstein induction data. One of the main themes of [PT14], which
we build upon in this paper, is the interplay between the sheets of
$\mathfrak{g}$ and the structure of $\mathcal{E}(\mathfrak{g},e)$. In the case
where $\mathfrak{g}$ is classical (i.e. a simple Lie algebra of type A, B, C
or D) we described a combinatorial procedure for describing the sheets of
$\mathfrak{g}$ containing a given orbit $\mathcal{O}$, and we named it the
Kempken–Spaltenstein (KS) algorithm. This procedure was one of the key tools
for [PT14, Theorem 1], which states that for $\mathfrak{g}$ classical the
variety $\mathcal{E}(\mathfrak{g},e)$ is an affine space if and only if $e$
lies in a unique sheet of $\mathfrak{g}$. The first goal of this paper is to
elucidate the structure of $\mathcal{E}(\mathfrak{g},e)$ when $\mathfrak{g}$
is classical and $e$ is singular, i.e. lies in multiple sheets.

Once again let $G$ be connected and reductive and let
$\mathcal{S}_{1},...,\mathcal{S}_{l}$ be the set of all sheets containing
$\mathcal{O}\in\mathcal{N}(\mathfrak{g})/G$. If $e+\mathfrak{g}^{f}$ denotes
the Slodowy slice to $\mathcal{O}$ at $e$ then we define the Katsylo variety

(1.1) $\displaystyle
e+X:=(e+\mathfrak{g}^{f})\cap\bigcup_{i=1}^{l}\mathcal{S}_{i}.$

In [Ka82] Katsylo used this variety to construct a geometric quotient of the
variety $\bigcup_{i=1}^{l}\mathcal{S}_{i}$. Perhaps the first indication that
$e+X$ should influence the representation theory of $U(\mathfrak{g},e)$
appeared in [Pr10]. Premet used reduction modulo $p$ to show that there is a
surjective map on the sets of irreducible components

(1.2)
$\displaystyle\operatorname{Comp}\mathcal{E}(\mathfrak{g},e)\relbar\joinrel\twoheadrightarrow\operatorname{Comp}(e+X)$

which restricts to a dimension preserving bijection on some subset of
$\operatorname{Comp}\mathcal{E}(\mathfrak{g},e)$. This is our first main
result.

######  Theorem 1.1.

When $\mathfrak{g}$ is a simple Lie algebra of classical type, the map (1.2)
is a dimension preserving bijection.

The dimensions of the irreducible components of $e+X$ can be calculated from
the KS algorithm, which depends only on the partition associated to $e$; see
Proposition 6.2(2) and Proposition 8.2. Thus Theorem 1.1 provides an effective
method for computing dimensions of all components of
$\mathcal{E}(\mathfrak{g},e)$. We note that these dimensions were calculated
in low ranks in [BG18].

1.3. In [Lo16] Losev demonstrated that for every conic symplectic singularity
there is an initial object in the category of filtered quantizations of
Poisson deformations (see also [ACET20]). We call this the universal
quantization. Using this result, he then showed that every coadjoint orbit
uniquely gives rise to a quantization of the affinization of a certain cover
of a nilpotent orbit, and that each such quantization give rise to a
completely prime primitive ideal. Thus we have a map
$\mathcal{J}:\mathfrak{g}^{*}/G\to\operatorname{Prim}U(\mathfrak{g})$, which
is known to be an embedding whenever $\mathfrak{g}$ is classical [Lo16,
Theorem 5.3]. The search for such a map is motivated by the orbit method of
Kostant and Kirillov, and we will refer to the map as Losev’s orbit method
map. An introduction to the classical theory of the orbit method can be found
in [Vo94].

It is important to understand and characterise the primitive ideals appearing
in the image of the orbit method map for $\mathfrak{g}$. Losev has conjectured
that they are precisely the ideals one obtains by applying Skryabin’s
equivalence to the one dimensional representations of the $W$-algebras
associated to $\mathfrak{g}$, and taking their annihilators. In the final
Section of this paper we deduce his conjecture from Theorem 1.1.

######  Theorem 1.2.

For $\mathfrak{g}$ classical, the image of $\mathcal{J}$ consists of primitive
ideals obtained from one dimensional representations of $W$-algebras under
Skryabin’s equivalence.

The main technique appearing in the proof is the application of quantum
Hamiltonian reduction to the universal quantizations of affinizations of
nilpotent orbit covers.

1.4. There are several new tools involved in the proof of Theorem 1.1, and we
now briefly describe the most important ones. One of the basic ideas comes
from deformation theory. The finite $W$-algebra is a filtered quantization of
the transverse Poisson structure on the Slodowy slice
$\mathbb{C}[e+\mathfrak{g}^{f}]$ and so there are two natural degenerations
associated to $\mathcal{E}(\mathfrak{g},e)$. On the one hand, we may
degenerate $U(\mathfrak{g},e)$ to the classical finite $W$-algebra
$S(\mathfrak{g},e)\cong\mathbb{C}[e+\mathfrak{g}^{f}]$ and then abelianise,
which leads to the spectrum of the maximal Poisson abelian quotient
$\mathcal{F}(\mathfrak{g},e):=\operatorname{Spec}S(\mathfrak{g},e)^{\operatorname{ab}}$.
On the other hand we may abelianise and then degenerate, which leads us to the
asymptotic cone of $\mathcal{E}(\mathfrak{g},e)$, denoted
$\mathbb{C}\mathcal{E}(\mathfrak{g},e):=\operatorname{Spec}(\operatorname{gr}U(\mathfrak{g},e)^{\operatorname{ab}})$.

One of the general results of this paper states that there is a closed
immersion of schemes inducing a bijection on closed points, which we prove
using reduction to prime characteristic

(1.3)
$\displaystyle\mathbb{C}\mathcal{E}(\mathfrak{g},e)\lhook\joinrel\longrightarrow\mathcal{F}(\mathfrak{g},e).$

Furthermore by considering the rank strata and symplectic leaves of the
Poisson structure of $e+\mathfrak{g}^{f}$ we see that the reduced subscheme
associated to $\mathcal{F}(\mathfrak{g},e)$ is $e+X$. Therefore the main
theorem will follow if we can show that
$\operatorname{Comp}\mathcal{E}(\mathfrak{g},e)$ is no larger than
$\operatorname{Comp}\mathbb{C}\mathcal{E}(\mathfrak{g},e)$. It is an
elementary fact from commutative algebra that
$\operatorname{Comp}\mathcal{E}(\mathfrak{g},e)\leq\operatorname{Comp}\mathbb{C}\mathcal{E}(\mathfrak{g},e)$
provided $\mathbb{C}\mathcal{E}(\mathfrak{g},e)$ is reduced, and so our
approach is to show that
$\operatorname{gr}U(\mathfrak{g},e)^{\operatorname{ab}}$ has no nilpotent
elements. By (1.3) it suffices to show that
$S(\mathfrak{g},e)^{\operatorname{ab}}$ is reduced.

In this paragraph we take $\mathfrak{g}$ classical. By passing to the
completion of $S(\mathfrak{g},e)^{\operatorname{ab}}$ at the maximal graded
ideal and using the fact that the Slodowy slice is transverse to every point
of $e+X$ we are able to reduce the problem of showing that
$S(\mathfrak{g},e)^{\operatorname{ab}}$ is reduced to the case where $e$ is
distinguished. To be more precise, $S(\mathfrak{g},e)^{\operatorname{ab}}$ is
reduced if and only if the completion at the maximal graded ideal is so, and
we show that this completion is reduced if and only if
$(S(\tilde{\mathfrak{g}},\tilde{e})_{x}^{\\!\wedge})^{\operatorname{ab}}$ is
reduced, where $x$ is a point on the Slodowy slice attached to a distinguished
element in a larger classical Lie algebra. For the proof we make use of the
fact that transverse Poisson manifolds are locally diffeomorphic, which
follows from Weinstein’s splitting theorem [We83, Theorem 2.1].

1.5. Now Theorem 1.1 will follow if we can show that
$S(\mathfrak{g},e)^{\operatorname{ab}}$ is reduced for distinguished elements
$e$. Again we introduce some new ideas to attack this problem. If $X$ is a
complex Poisson scheme of finite type and $H$ is a reductive group acting
rationally by Poisson automorphisms then the invariant subscheme $X^{H}$ can
be equipped with a Poisson structure via Dirac reduction. The reduced Poisson
algebra will be denoted $R(\mathbbm{k}[X],H)$. Now if
$\mathfrak{g}=\operatorname{Lie}(G)$ for $G$ reductive and
$H\subseteq\operatorname{Aut}(\mathfrak{g})$ is a reductive group fixing an
$\mathfrak{sl}_{2}$-triple $\\{e,h,f\\}\subseteq\mathfrak{g}$ then we prove
the following isomorphism of Poisson algebras

(1.4) $\displaystyle
R(S(\mathfrak{g},e),H)\overset{\sim}{\longrightarrow}S(\mathfrak{g}^{H},e).$

Some special cases of (1.4) were discovered by Ragoucy [Ra01].

We apply this isomorphism in the case where $\mathfrak{g}=\mathfrak{gl}_{n}$
and $H=\mathbb{Z}/2\mathbb{Z}$ is generated by some involution $\tau$. Then
$\mathfrak{g}^{\tau}=\mathfrak{so}_{n}$ or $\mathfrak{g}=\mathfrak{sp}_{n}$.
It follows from the work of Brundan and Kleshchev that $S(\mathfrak{g},e)$ is
a quotient of a (semiclassical) shifted Yangian $y_{n}(\sigma)$ depending on
$e$. If $e$ is distinguished then we can identify an involution $\tau$ on
$y_{n}(\sigma)$ defined so that the Poisson homomorphism
$y_{n}(\sigma)\twoheadrightarrow S(\mathfrak{g},e)$ is $\tau$-equivariant. As
a consequence we can apply the Dirac reduction procedure to the shifted
Yangian $y_{n}(\sigma)$ and thus obtain a presentation of the Poisson
structure on $S(\mathfrak{g},e)$. We mention that obtaining presentations of
finite $W$-algebras outside of type A is one of the key open problems in the
field, and so we hope this result will present fertile ground for further
discoveries.

Let $\mathfrak{m}_{0}$ be the maximal graded ideal. Finally we use the
presentation of $S(\mathfrak{g},e)^{\operatorname{ab}}$ to calculate
generators and certain relations of
$\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g},e)^{\operatorname{ab}}$.
Considering the relationship between $\mathcal{F}(\mathfrak{g},e)$ and $e+X$
mentioned earlier we see that the reduced algebra of
$\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g},e)^{\operatorname{ab}}$ is
naturally identified with the coordinate ring on the tangent cone
$\mathbb{C}[\operatorname{TC}_{e}(e+X)]$. Michaël Bulois has recently
demonstrated that the sheets of $\mathfrak{g}$ containing $e$ are transversal
at $e$ (work in preparation [Bu]), which allows us to calculate the dimensions
of the irreducible components of $\operatorname{TC}_{e}(e+X)$, once again they
are determined by the KS algorithm. Using a combinatorial argument we then
show that the relations mentioned above give a full presentation for
$\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g},e)^{\operatorname{ab}}$.
It follows quickly that both
$\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g},e)^{\operatorname{ab}}$
and $S(\mathfrak{g},e)^{\operatorname{ab}}$ are reduced, which allows us to
conclude the proof of Theorem 1.1.

1.6. Since the presentation of the distinguished semiclassical finite
$W$-algebras in types B, C, D is an important result in its own right we
formulate it straight away. For the proof combine Theorem 3.7 and Theorem 4.9.

######  Theorem 1.3.

Let $\mathfrak{g}=\mathfrak{so}_{n}$ or $\mathfrak{sp}_{n}$ and let $e$ be a
distinguished nilpotent element with partition
$\lambda=(\lambda_{1},...,\lambda_{n})$. Then $S(\mathfrak{g},e)$ is generated
as a Poisson algebra by elements

(1.5) $\displaystyle\\{\eta_{i}^{(2r)}\mid 1\leq i\leq n,\
r\in\mathbb{Z}_{>0}\\}\cup\\{\theta_{i}^{(r)}\mid 1\leq i<n,\
r-\frac{\lambda_{i+1}-\lambda_{i}}{2}\in\mathbb{Z}_{>0}\\}$

together with the following relations

(1.6) $\displaystyle\\{\eta_{i}^{(2r)},\eta_{j}^{(2s)}\\}=0$ (1.7)
$\displaystyle\big{\\{}\eta_{i}^{(2r)},\theta_{j}^{(s)}\big{\\}}=(\delta_{i,j}-\delta_{i,j+1})\sum_{t=0}^{r-1}\eta_{i}^{(2t)}\theta_{j}^{(2r+s-1-2t)}$
(1.8) $\displaystyle\ \ \ \
\big{\\{}\theta_{i}^{(r)},\theta_{i}^{(s)}\big{\\}}=0$ $\displaystyle\text{
for }r+s-1\text{ odd}$ (1.9) $\displaystyle\ \ \ \
\big{\\{}\theta_{i}^{(r)},\theta_{i}^{(s)}\big{\\}}=\sum_{t=r}^{s-1}\frac{1}{2}\theta_{i}^{(t)}\theta_{i}^{(2m-t)}+(-1)^{s+s_{i,i+1}}\sum_{t=0}^{m}2\eta_{i+1}^{(2m-2t)}\widetilde{\eta}_{i}^{(2t)}$
$\displaystyle\text{ for }r<s,r+s-1=2m$ (1.10)
$\displaystyle\big{\\{}\theta_{i}^{(r+1)},\theta_{i+1}^{(s)}\big{\\}}-\big{\\{}\theta_{i}^{(r)},\theta_{i+1}^{(s+1)}\big{\\}}=\frac{1}{2}\theta_{i}^{(r)}\theta_{i+1}^{(s)}$
(1.11) $\displaystyle\big{\\{}\theta_{i}^{(r)},\theta_{j}^{(s)}\big{\\}}=0$
$\displaystyle\text{ for }|i-j|>1$ (1.12)
$\displaystyle\Big{\\{}\theta_{i}^{(r)},\big{\\{}\theta_{i}^{(s)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}+\Big{\\{}\theta_{i}^{(s)},\big{\\{}\theta_{i}^{(r)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}=0$
$\displaystyle\text{ for }|i-j|=1\text{ and }r+s\text{ odd}$ (1.13)
$\displaystyle\Big{\\{}\theta_{i}^{(r)},\big{\\{}\theta_{i}^{(s)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}+\Big{\\{}\theta_{i}^{(s)},\big{\\{}\theta_{i}^{(r)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}=$
$\displaystyle\ \ \ \ \ \ \ \ \ \
2(-1)^{s+s_{i,i+1}-1}\delta_{i,j+1}\sum_{m_{1}=0}^{m-1}\sum_{m_{2}=0}^{m_{1}}\eta_{i+1}^{(2(m-m_{1}-1))}\widetilde{\eta}_{i}^{(2m_{2})}\theta_{j}^{(2(m_{1}-m_{2})+t)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+2(-1)^{s+s_{i,i+1}-1}\delta_{i+1,j}\sum_{m_{1}=0}^{m-1}\sum_{m_{2}=0}^{m-m_{1}-1}\widetilde{\eta}_{i}^{(2m_{1})}\eta_{i+1}^{(2m_{2})}\theta_{j}^{(2(m-m_{1}-m_{2}-1)+t)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{ for }|i-j|=1\text{ and }r+s=2m\text{ even}$ (1.14)
$\displaystyle\eta_{i}^{(2r)}=0$ for $\displaystyle 2r>\lambda_{1}$ (1.15)
$\displaystyle\sum_{t=0}^{\lambda_{i}/2}\eta_{i}^{(2t)}\theta_{i}^{(\lambda_{i}-2t+s_{1,2}+1)}=0$
for $\displaystyle\ i=1,...,n-1\text{ when }\mathfrak{g}=\mathfrak{sp}_{n}.$

where we adopt the convention $\eta_{i}^{(0)}=\widetilde{\eta}_{i}^{(0)}=1$
and the elements $\\{\widetilde{\eta}_{i}^{(r)}\mid r\in 2\mathbb{Z}_{\geq
0}\\}$ are defined via the recursion

(1.16)
$\displaystyle\widetilde{\eta}_{i}^{(2r)}:=-\sum_{t=1}^{r}\eta_{i}^{(2t)}\widetilde{\eta}_{i}^{(2r-2t)}.$

1.7. To conclude the introduction we describe the structure of the paper,
which is divided into two parts. The first part is very algebraic, dealing
with Dirac reduction and the presentation of semiclassical $W$-algebras in the
distinguished case. The second part is more geometric, studying the
degenerations of $\mathcal{E}(\mathfrak{g},e)$ using Lusztig–Spaltenstein
induction and the tangent cone of the Katsylo variety.

Part I: We begin Section 2 by giving an elementary introduction to the version
of Dirac reduction used in this paper. In Subsection 2.3 we prove the
isomorphism (1.4), which should have independent interest. In Section 3 we
describe the semiclassical shifted Yangian $y_{n}(\sigma)$ by generators and
relations. The main results on the Dirac reduction of $y_{n}(\sigma)$ are
presented in Subsection 3.4, including the canonical grading, the loop
filtration, the PBW theorem and the presentation by generators and relations.
All of the results about $R(y_{n}(\sigma),\tau)$ are ultimately deduced from
similar results on $y_{n}(\sigma)$. In Section 4 we recall the definition of
Brundan–Kleshchev’s isomorphism $y_{n}(\sigma)\twoheadrightarrow
S(\mathfrak{g},e)$ and show by an explicit calculation that this is
$\tau$-equivariant for a suitable choice of involution on $y_{n}(\sigma)$. By
(1.4) this leads to a surjection $R(y_{n}(\sigma),\tau)\twoheadrightarrow
S(\mathfrak{g}^{\tau},e)$ and in Subsection 4.5 we complete the picture by
describing a full set of Poisson generators for the kernel.

Part II: In Section 5 we gather together some important general facts about
degenerations and completions of schemes, as well as reviewing the theory of
rank stratification and symplectic leaves of a Poisson scheme. In Section 6 we
explain how $S(\mathfrak{g},e)^{\operatorname{ab}}$ is related to Katsylo
variety $e+X$, and recall the aforementioned results of Bulois allowing us to
enumerate the irreducible components of $\operatorname{TC}_{e}(e+X)$ and
calculate their dimensions. In Section 7 we introduce quantum finite
$W$-algebras and prove the existence of the closed immersion (1.3). The proof
of the latter uses a reduction modulo $p$ argument similar to Premet’s
construction of the component map (1.2) in [Pr10, Theorem 1.2], along with the
identification of reduced schemes
$\mathcal{F}(\mathfrak{g},e)_{\operatorname{red}}=\mathbb{C}[e+X]$ from
Section 6. Finally in Section 8 we describe the Kempken–Spaltenstein
algorithm, as well as its relationship with sheets, and then use this to
construct an algebraic variety $X_{\lambda}$ associated to a distinguished
nilpotent orbit $\mathcal{O}$ with partition $\lambda$, which we call the
combinatorial Katsylo variety. In Theorem 8.7 we use the presentation of
$S(\mathfrak{g},e)^{\operatorname{ab}}$ obtained in part I of the paper to
demonstrate that $\mathbb{C}[X_{\lambda}]\twoheadrightarrow
S(\mathfrak{g},e)^{\operatorname{ab}}\twoheadrightarrow\mathbb{C}[\operatorname{TC}_{e}(e+X)]$,
and we show that these are isomorphisms by comparing the dimensions of the
irreducible components. In particular this implies that
$S(\mathfrak{g},e)^{\operatorname{ab}}$ is reduced for $e$ distinguished.
Theorem 8.8 reduces the general case to the distinguished case. Finally we
conclude the proof of the main theorem in Subsection 8.7, making use of
deformation techniques gathered in Section 5.

In Subsection 9.1 we recall the definition of Losev’s orbit method map and we
prove a slight refinement of Theorem 1.2. In Subsection 9.2, 9.3 we recall the
role of the Namikawa–Weyl group in the classification of quantizations of
conic symplectic singularities, and study the relationship with primitive
ideals. Finally in Subsection 9.4 we prove the main theorem by applying
quantum Hamiltonian reduction to the quantizations of orbit covers arising
from generalised Springer maps.

###### Contents

  1. 1 Introduction
  2. I  Presentations of classical $W$-algebras
    1. 2 Dirac reduction for classical finite $W$-algebras
      1. 2.1 Invariants via Dirac reduction
      2. 2.2 Classical finite $W$-algebras
      3. 2.3 Dirac reduction applied to $W$-algebras
    2. 3 Dirac reduction for shifted Yangians
      1. 3.1 Poisson algebras by generators and relations
      2. 3.2 Chevalley–Serre presentations for shifted current Lie–Poisson algebras
      3. 3.3 The semiclassical shifted Yangian
      4. 3.4 The Dirac reduction of the shifted Yangian
    3. 4 Finite $W$-algebras for classical Lie algebras
      1. 4.1 Partitions for nilpotent orbits in classical Lie algebras
      2. 4.2 The Dynkin pyramid and the centraliser
      3. 4.3 Symplectic and orthogonal subalgebras
      4. 4.4 Generators of the $W$-algebra
      5. 4.5 The semiclassical Brundan–Kleshchev homomorphism
  3. II One dimensional representations of $W$-algebras
    1. 5 Generalities on Poisson schemes
      1. 5.1 Conic degenerations of affine schemes
      2. 5.2 Completions and nilpotent elements of graded algebras
      3. 5.3 Poisson schemes
    2. 6 Sheets and induction
      1. 6.1 Lusztig–Spaltenstein induction, sheets and slices
      2. 6.2 Classical finite $W$-algebras and Slodowy slices
      3. 6.3 The Katsylo variety and the tangent cone
    3. 7 Abelian quotients of finite $W$-algebras
      1. 7.1 The finite $W$-algebra
      2. 7.2 Bounding the asymptotic cone
    4. 8 The abelian quotient of the classical $W$-algebra
      1. 8.1 Rigid, singular and distinguished partitions
      2. 8.2 The Kempken–Spaltenstein algorithm
      3. 8.3 Distinguished elements and induction
      4. 8.4 The combinatorial Katsylo variety
      5. 8.5 The semiclassical abelianisation I: the distinguished case
      6. 8.6 The semiclassical abelianisation II: the general case
      7. 8.7 The abelianisation of the finite $W$-algebra via deformation theory
    5. 9 The orbit method
      1. 9.1 Losev’s orbit method map
      2. 9.2 The Namikawa–Weyl group and the Poisson automorphism group
      3. 9.3 Primitive ideals arising from generalised Springer maps
      4. 9.4 Quantum Hamiltonian reduction of universal quantizations

#### Notation and conventions

The following notation will be used throughout the paper. For $i\in\mathbb{Z}$
we write $\mathbb{Z}_{>i}=\\{i+1,i+2,...\\}$. All algebras and vector spaces
defined over $\mathbb{C}$, except in Section 7 where we use reduction modulo a
large prime. We use capital letters $G,H,...$ for algebraic groups and gothic
script $\mathfrak{g},\mathfrak{h},...$ for their Lie algebras. All schemes
appearing in this paper will be affine schemes of finite type over
$\mathbb{C}$, thus the reader may almost always think of complex affine
varieties, except that the coordinate rings will often be non-reduced (the
consideration of nilpotent elements will be vital to our main results).
Occasionally we write $\operatorname{m-Spec}(A)$ for the variety of closed
points in the prime spectrum of a commutative algebra $A$. If $X$ is a
noetherian scheme then we write $\operatorname{Comp}(X)$ for the set of
irreducible components of $X$.

The associated graded algebra of an almost commutative, filtered associative
algebra is equipped with a Poisson structure in the usual manner. If $A$ is a
commutative algebra and $I\subseteq A$ an ideal, then we write
$\operatorname{gr}_{I}A$ for the graded algebra with respect to the $I$-adic
filtration.

#### Acknowledgements

I would like to offer thanks to Simon Goodwin, Ivan Losev, Sasha Premet and
Dmytro Matvieievskyi for useful comments on an early draft of this paper. I’m
especially grateful to Ivan for suggesting some of the constructions used in
the proof of Theorem 1.2, and to Sasha to whom this paper is dedicated - his
insightful teaching first introduced to these fascinating problems. I have
also benefited from many interesting conversations and email correspondence
with Jon Brundan, Michaël Bulois, Paul Levy, Anne Moreau, and Daniele Valeri,
and am grateful for all their help. Some of these results were announced at
the conference “Geometric and automorphic aspects of $W$ -algebras”, Lille
2019. This research is supported by the UKRI Future Leaders Fellowship project
“Geometric representation theory and $W$-algebras”, grant number MR/S032657/1.

##  Part I  Presentations of classical $W$-algebras

###  2\. Dirac reduction for classical finite $W$-algebras

####  2.1. Invariants via Dirac reduction

Let $X$ be an complex affine Poisson variety and suppose
$z_{1},...,z_{n}\in\mathbb{C}[X]$ such that the determinant of the matrix
$(\\{z_{i},z_{j}\\})_{1\leq i,j\leq n}$ is a unit. In his seminal paper
[Dir50] Dirac defined a new Poisson bracket on $\mathbb{C}[X]$, such that the
$z_{i}$ are Casimirs, thus equipping $\mathbb{C}[X]/(z_{1},...,z_{n})$ with a
Poisson structure. In fact this is a special case of the following procedure:
say that $I\subseteq\mathbb{C}[X]$ is a Dirac ideal if
$N_{\mathbb{C}[X]}(I)\twoheadrightarrow\mathbb{C}[X]/I$ surjects where
$N_{\mathbb{C}[X]}(I):=\\{f\in\mathbb{C}[X]\mid\\{f,I\\}\subseteq I\\}$
denotes the Poisson idealiser. Then the subscheme associated to $I$ inherits a
Poisson structure from $X$, and we call this induced structure the Dirac
reduction [LPV13, 5.4.3].

Now let $H$ be a linearly reductive group acting on $X$ by Poisson
automorphisms. We regard the set of invariants $X^{H}$ as a (not necessarily
reduced) affine scheme with global sections
$\mathbb{C}[X^{H}]:=\mathbb{C}[X]_{H}$, where
$\mathbb{C}[X]_{H}=\mathbb{C}[X]/I_{H}$ and $I_{H}:=(h\cdot f-f\mid h\in
H,f\in\mathbb{C}[X])$. Although $I_{H}$ is usually not a Poisson ideal, it is
always a Dirac ideal (this is a corollary of Lemma 2.1) so that $X^{H}$
acquires the structure of a Poisson scheme.

The following suggests an alternative approach of the Poisson structure on
$X^{H}$, better-suited to calculations.

######  Lemma 2.1.

The map $\mathbb{C}[X]^{H}\to\mathbb{C}[X^{H}]$ is surjective and

(2.1) $\displaystyle\mathbb{C}[X]^{H}/\mathbb{C}[X]^{H}\cap
I_{H}\overset{\sim}{\longrightarrow}\mathbb{C}[X^{H}]$

is a Poisson isomorphism.

###### Proof.

If $f\in\mathbb{C}[X]^{H}$, $g\in\mathbb{C}[X]$ and $h\in H$ then $\\{f,h\cdot
g-g\\}=h\cdot\\{f,g\\}-\\{f,g\\}$ and so $\mathbb{C}[X]^{H}\subseteq
N_{\mathbb{C}[X]}(I_{H})$ is a Poisson subalgebra. Since $H$ is linearly
reductive and acts locally finitely we can decompose
$\mathbb{C}[X]=\mathbb{C}[X]^{H}\oplus\mathbb{C}[X]_{1}$ where
$\mathbb{C}[X]_{1}$ is the kernel of the $H$-equivariant projection
$\mathbb{C}[X]\to\mathbb{C}[X]^{H}$. If $V\subseteq\mathbb{C}[X]_{1}$ is an
irreducible $H$-submodule then $\\{h\cdot v-v\mid h\in H,v\in V\\}\subseteq V$
is $H$-stable and nonzero hence equal to $V$, and it follows that
$I_{H}=(\mathbb{C}[X]_{1})$. Therefore the composition
$\mathbb{C}[X]^{H}\hookrightarrow
N_{\mathbb{C}[X]}(I_{H})\to\mathbb{C}[X^{H}]=\mathbb{C}[X]_{H}$ is surjective.
∎

Now let $A$ be any Poisson algebra with a locally finite action of $H$ by
Poisson automorphisms. Write $I_{H}:=(h\cdot a-a\mid h\in H,a\in A)$ and
define the Dirac reduction of $A$ by $H$ by

(2.2) $\displaystyle R(A,H):=A^{H}/A^{H}\cap I_{H}$

If $\tau\in\operatorname{Aut}(A)$ is a semisimple Poisson automorphism of
finite order, then we often abuse notation writing $R(A,\tau)$ for $R(A,H)$
where $H$ is the group generated by $\tau$.

Since $H$ is linearly reductive the functor of $H$-invariants is exact on the
category of locally finite $H$-modules. This implies that $R(-,H)$ is a right
exact functor from the category of $H$-locally finite Poisson algebras, to the
category of Poisson algebras.

######  Lemma 2.2.

Suppose the following:

  1. (1)

$V$ is a direct sum of locally finite $H$-modules with $H$-stable
decomposition $V=V^{H}\oplus V_{1}$.

  2. (2)

$S(V)$ is a Poisson algebra with $H$ acting by Poisson automorphisms.

Then the natural map $S(V^{H})\to R(S(V),H)$ is an isomorphism of commutative
algebras.

###### Proof.

Since $S(V)=S(V^{H})\oplus(V_{1})$ and $S(V^{H})$ is $H$-fixed we must have
$S(V)_{1}\subseteq(V_{1})$. Combining with $V_{1}\subseteq S(V)_{1}$ we deduce
that $I_{H}=(V_{1})$ which proves the map $S(V^{H})\to R(S(V),H)$ is an
isomorphism of commutative algebras. ∎

######  Remark 2.3.

Let $\mathfrak{g}$ be a Lie algebra and $H$ a linearly reductive group of
automorphisms of $\mathfrak{g}$ acting locally finitely. In this case the
composition $S(\mathfrak{g}^{H})\to S(\mathfrak{g})^{H}\to
R(S(\mathfrak{g}),H)$ is a Poisson homomorphism and so Lemma 2.2 shows that
$S(\mathfrak{g}^{H})\overset{\sim}{\longrightarrow}R(S(\mathfrak{g}),H)$ as
Poisson algebras.

####  2.2. Classical finite $W$-algebras

For the rest of the Section we fix a connected reductive algebraic group $G$
such that the derived subgroup is simply connected, and write
$\mathfrak{g}=\operatorname{Lie}(G)$. Let $\kappa$ be a choice of non-
degenerate trace form on $\mathfrak{g}$ which is preserved by
$\operatorname{Aut}(\mathfrak{g})$. Pick a nilpotent element
$e\in\mathfrak{g}$ and write $\chi:=\kappa(e,\cdot)\in\mathfrak{g}^{*}$. Pick
an $\mathfrak{sl}_{2}$-triple $\\{e,h,f\\}$ and write
$\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}}\mathfrak{g}(i)$ for the grading by
$\operatorname{ad}(h)$-eigenspaces. Throughout the paper we use the notation
$\mathfrak{g}(\leq\\!i)=\bigoplus_{j\leq i}\mathfrak{g}(j)$ and similar for
$\mathfrak{g}(\\!<\\!i)$. Since $e\in\mathfrak{g}(2)$ we see that $\chi$
restricts to a character on $\mathfrak{g}(\operatorname{\\!<\\!--1})$. Make
the following notation $\mathfrak{g}(\operatorname{\\!<\\!--
1}):=\\{x-\chi(x)\mid x\in\mathfrak{g}(\operatorname{\\!<\\!--1})\\}\subseteq
S(\mathfrak{g})$. The nilpotent Lie algebra
$\mathfrak{g}(\operatorname{\\!<\\!0})$ is algebraic and we write
$\mathfrak{g}(\operatorname{\\!<\\!0})=\operatorname{Lie}G(\operatorname{\\!<\\!0})$.

The (classical) finite $W$-algebra associated to $(\mathfrak{g},e)$ is a
Poisson reduction of $S(\mathfrak{g})$

(2.3) $\displaystyle
S(\mathfrak{g},e):=(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!
--1})_{\chi})^{G(<0)}.$

In more detail, the Poisson normaliser $N=\\{f\in
S(\mathfrak{g})\mid\\{f,\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}\\}\subseteq S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi})$ is a Poisson subalgebra of $S(\mathfrak{g})$ with $N\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$ embedded as a
Poisson ideal, and $S(\mathfrak{g},e)$ is equipped with a Poisson structure
via the isomorphism $N/N\cap S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!
--1})_{\chi}\overset{\sim}{\longrightarrow}S(\mathfrak{g},e)$.

The Kazhdan grading is defined on $S(\mathfrak{g})$ by placing
$\mathfrak{g}(i)$ in Kazhdan degree $i+2$. Notice that
$\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$ generates a homogeneous ideal
and that $S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}$ inherits a connected grading in non-negative degrees, with
$S(\mathfrak{g},e)$ embedded as a graded subalgebra, with Poisson bracket in
degree $-2$.

Since the grading is good for $e$ the map $\mathfrak{g}^{e}\to
S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$
is injective.

######  Theorem 2.4.

Let $\mathfrak{m}_{0}\subseteq
S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$
denote the unique maximal graded ideal.

  1. (1)

There exists a Kazhdan graded map $\theta:\mathfrak{g}^{e}\to
S(\mathfrak{g},e)$ such that

(2.4) $\displaystyle\theta(x)-x\in\mathfrak{m}_{0}^{2}.$

Furthermore $\theta$ can be chosen to be equivariant with respect to any
reductive group of Poisson automorphisms acting rationally on
$S(\mathfrak{g},e)$ by graded automorphisms.

  2. (2)

If $\theta$ is any map satisfying (2.4) then $S(\mathfrak{g}^{e})\to
S(\mathfrak{g},e)$ is an isomorphism of commutative algebras.

###### Proof.

It follows from [GG02, Lemma 2.1] that the restriction homomorphism
$S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}\cong\mathbb{C}[\chi+\mathfrak{g}(\operatorname{\\!<\\!--
1})^{\perp}]\to\mathbb{C}[\chi+\mathfrak{g}^{f}]\cong S(\mathfrak{g}^{e})$
gives a $G^{e}(0)$-equivariant isomorphism $S(\mathfrak{g},e)\to
S(\mathfrak{g}^{e})$ of commutative algebras, where
$\mathfrak{g}(\operatorname{\\!<\\!--
1})^{\perp}:=\\{\eta\in\mathfrak{g}^{*}\mid\eta(\mathfrak{g}(\operatorname{\\!<\\!
--1}))=0\\}$. Taking the inverse isomorphism restricted to
$\mathfrak{g}^{e}\subseteq S(\mathfrak{g}^{e})$ gives the desired map
$\theta$. If $H$ is any reductive group of Poisson isomorphisms then $\theta$
can be replaced with an $H$-equivariant map using the standard trick of
projecting onto isotypic components of $S(\mathfrak{g},e)$ for the $H$-action.

Now if $\theta$ is any map satisfying (2.4) then $S(\mathfrak{g}^{e})\to
S(\mathfrak{g},e)$ and it suffices to show that this map is surjective. If
$x_{1},...,x_{r}\in\mathfrak{g}^{e}$ is a homogenous basis then [Ja04, Lemma
7.1] shows that $\theta(x_{1}),...,\theta(x_{r})$ generate a graded radical
ideal of finite codimension. The only such ideal is the maximal graded ideal
of $S(\mathfrak{g},e)$ and this implies that $\theta(x_{1}),...,\theta(x_{r})$
generate $S(\mathfrak{g},e)$. ∎

####  2.3. Dirac reduction applied to $W$-algebras

Now fix a reductive subgroup $H\subseteq\operatorname{Aut}(\mathfrak{g})$
fixing our choice of $\mathfrak{sl}_{2}$-triple. Make the notation
$\mathfrak{g}_{0}:=\mathfrak{g}^{H}$ and let $\mathfrak{g}_{1}$ denote an
$H$-invariant complement to $\mathfrak{g}_{0}$ in $\mathfrak{g}$. Also write
$G_{0}\subseteq G$ for the connected component of the subgroup consisting of
$g\in G$ such that $[\operatorname{Ad}(g),H]=1$.

######  Lemma 2.5.

  1. (1)

The restriction of $\kappa$ to $\mathfrak{g}_{0}$ is non-degenerate.

  2. (2)

$G_{0}$ is reductive and $\mathfrak{g}_{0}=\operatorname{Lie}(G_{0})$.

###### Proof.

Since $\mathfrak{g}_{1}$ is spanned by elements $\\{h\cdot x-x\mid h\in
H,x\in\mathfrak{g}\\}$ it follows that
$\kappa(\mathfrak{g}_{0},\mathfrak{g}_{1})=0$. Since $\kappa$ is non-
degenerate we deduce part (1).

Now suppose that $\kappa$ is defined from a representation
$\rho:\mathfrak{g}\to\mathfrak{gl}(V)$. If
$\mathfrak{n}\subseteq\mathfrak{g}_{0}$ is a nilpotent ideal then by Engel
there exists $k>0$ such that $\rho(\mathfrak{n})^{k}=0$. Hence
$(\rho(x)\rho(n))^{k}=0$ for all $x\in\mathfrak{g}_{0}$ and
$n\in\mathfrak{n}$, so $\kappa(x,n)=0$. Since $\kappa$ is non-degenerate the
nilradical of $\mathfrak{g}_{0}$ is trivial.

If $\operatorname{Ad}_{G}:G\to\operatorname{GL}(\mathfrak{g})$ is the adjoint
representation then we consider
$\rho_{0}:=\operatorname{Ad}_{\operatorname{GL}(\mathfrak{g})}\circ\operatorname{Ad}_{G}:G\to\operatorname{GL}(\operatorname{End}(\mathfrak{g}))$
and let $\rho:G\to\operatorname{GL}(W)$ be any faithful representation
admitting $\rho_{0}$ as a direct summand. If we identify $G$ (resp.
$\mathfrak{g}$) with its image in $\operatorname{GL}(W)$ (resp.
$\mathfrak{gl}(W)$) via $\rho$ (resp. $d_{0}\rho$), and identify $H$ with a
subset of $\operatorname{End}(\mathfrak{g})\subseteq W$, then $G_{0}$ is
precisely the subgroup of $\operatorname{GL}(W)$ fixing $H$, and similar for
$\mathfrak{g}_{0}$. Now apply [Hum75, Theorem 13.2] to see that
$\mathfrak{g}_{0}=\operatorname{Lie}(G_{0})$. Since the nilradical of
$\mathfrak{g}_{0}$ is trivial, $G_{0}$ is reductive thanks to [Hum75, Theorem
13.5]. ∎

Since $\mathfrak{g}_{0}$ is a graded subalgebra of $\mathfrak{g}$ containing
$e$ we can consider $S(\mathfrak{g}_{0},e)$. Furthermore $H$ preserves
$\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$ and the induced action on
$S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}$ stabilises the $G(\operatorname{\\!<\\!0})$-invariants, so that
$H$ acts by Poisson automorphisms on $S(\mathfrak{g},e)$.

We are now ready to formulate one of our first main theorems, stated in (1.4).

######  Theorem 2.6.

There is a natural Poisson isomorphism

(2.5) $\displaystyle
R(S(\mathfrak{g},e),H)\overset{\sim}{\longrightarrow}S(\mathfrak{g}_{0},e).$

Since $\kappa$ is non-degenerate on $\mathfrak{g}$ and $\mathfrak{g}_{0}$ and
$\operatorname{Aut}(\mathfrak{g})$-invariant, it follows that
$\mathfrak{g}_{0}^{\perp}=\\{x\in\mathfrak{g}\mid\kappa(x,\mathfrak{g}_{0})=0\\}$
is an $H$-stable complement to $\mathfrak{g}_{0}$ in $\mathfrak{g}$. Hence
$\mathfrak{g}_{1}=\mathfrak{g}_{0}^{\perp}$. This implies

(2.6) $\chi(\mathfrak{g}_{1})=0.$

######  Lemma 2.7.

  1. (1)

$S(\mathfrak{g}_{0})\cap S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}=S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})_{\chi}$;

  2. (2)

$S(\mathfrak{g})\mathfrak{g}_{1}\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}=S(\mathfrak{g})(\mathfrak{g}_{1}(\operatorname{\\!<\\!--
1})+\mathfrak{g}_{1}\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi})$.

###### Proof.

Observe that $\mathfrak{g}_{0}$ and $\mathfrak{g}_{1}$ are graded subspaces of
$\mathfrak{g}$. It suffices to show that the left hand side is contained in
the right, for both (1) and (2).

Let $x\in\mathfrak{g}(\operatorname{\\!<\\!--1})$ and $f\in S(\mathfrak{g})$,
and let $f=f_{0}+f_{1}$ and $x=x_{0}+x_{1}$ be the decompositions over
$S(\mathfrak{g})=S(\mathfrak{g}_{0})\oplus S(\mathfrak{g})\mathfrak{g}_{1}$
and $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ respectively. By
(2.6) the projections of $f(x-\chi(x))$ to $S(\mathfrak{g}_{0})$ and
$S(\mathfrak{g})\mathfrak{g}_{1}$ are $f_{0}(x_{0}-\chi(x_{0}))$ and
$f_{0}x_{1}+f_{1}(x-\chi(x))$ respectively. Therefore if $f(x-\chi(x))$ lies
in $S(\mathfrak{g}_{0})$ we must have $f_{0}x_{1}+f_{1}(x-\chi(x))=0$ and
$f(x-\chi(x))=f_{0}(x_{0}-\chi(x_{0}))$, proving (1).

Now suppose that $f(x-\chi(x))\in S(\mathfrak{g})\mathfrak{g}_{1}\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$ and that
$x=x_{0}+x_{1}$ where $x_{i}\in\mathfrak{g}_{i}(\operatorname{\\!<\\!--1})$.
Then $f(x_{1}-\chi(x_{1}))\in
S(\mathfrak{g})\mathfrak{g}_{1}(\operatorname{\\!<\\!--1})$ by (2.6), whilst
$f(x_{0}-\chi(x_{0}))\in S(\mathfrak{g})\mathfrak{g}_{1}$ implies that $f\in
S(\mathfrak{g})\mathfrak{g}_{1}$ so that $f(x_{0}-\chi(x_{0}))\in
S(\mathfrak{g})(\mathfrak{g}_{1}\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})_{\chi})$. This proves (2). ∎

######  Lemma 2.8.

The ideal $S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$ is
the direct sum of its intersections with $S(\mathfrak{g}_{0})$ and
$S(\mathfrak{g})\mathfrak{g}_{1}$. Therefore
$S(\mathfrak{g})=S(\mathfrak{g}_{0})\oplus S(\mathfrak{g})\mathfrak{g}_{1}$
gives a $G_{0}(\operatorname{\\!<\\!0})$-module decomposition

(2.7) $\displaystyle
S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}=S(\mathfrak{g}_{0})/S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!
--1})_{\chi}\oplus
S(\mathfrak{g})\mathfrak{g}_{1}/S(\mathfrak{g})(\mathfrak{g}_{1}(\operatorname{\\!<\\!
--1})+\mathfrak{g}_{1}\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}).$

###### Proof.

We begin by proving the claim

(2.8) $\displaystyle S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}=S(\mathfrak{g}_{0})\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}\oplus
S(\mathfrak{g})\mathfrak{g}_{1}\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}.$

It suffices to show that the right hand side contains the left. Let $f\in
S(\mathfrak{g})$ and $x-\chi(x)\in\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}$. Since $\mathfrak{g}(\operatorname{\\!<\\!--
1})=\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})\oplus\mathfrak{g}_{1}(\operatorname{\\!<\\!--1})$ we can consider two
cases: (i) if $x\in\mathfrak{g}_{1}$ then by (2.6) we have $f(x-\chi(x))\in
S(\mathfrak{g})\mathfrak{g}_{1}\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$; (ii) if
$x\in\mathfrak{g}_{0}$ then we can write $f=f_{0}+f_{1}\in
S(\mathfrak{g}_{0})\oplus S(\mathfrak{g})\mathfrak{g}_{1}$, in which case
$f_{0}(x-\chi(x))\in S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!
--1})_{\chi}\subseteq S(\mathfrak{g}_{0})\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$ and
$f_{1}(x-\chi(x))\in S(\mathfrak{g})\mathfrak{g}_{1}\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$. Now the Lemma
follows from Lemma 2.7. ∎

We write $S(\mathfrak{g})_{0}$ for the $H$-invariants and
$S(\mathfrak{g})_{1}$ for the unique $H$-stable complement. Use similar
notation for $S(\mathfrak{g},e)$. Consider the two sets

(2.9) $\displaystyle N$ $\displaystyle:=$ $\displaystyle\\{f\in
S(\mathfrak{g})\mid g\cdot f-f\in
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}\text{ for all
}g\in G(\operatorname{\\!<\\!0})\\};$ (2.10) $\displaystyle N_{0}$
$\displaystyle:=$ $\displaystyle\\{f\in S(\mathfrak{g}_{0})\mid g\cdot f-f\in
S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}\text{
for all }g\in G_{0}(\operatorname{\\!<\\!0})\\}.$

By differentiating the locally finite actions on $G(\operatorname{\\!<\\!0})$
on $S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}$ and of $G_{0}(\operatorname{\\!<\\!0})$ on
$S(\mathfrak{g}_{0})/S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!
--1})_{\chi}$ we see that

(2.13) $\displaystyle\begin{array}[]{c}\\{\mathfrak{g}(\operatorname{\\!<\\!--
1}),N\cap S(\mathfrak{g})_{0}\\}\subseteq N\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi};\vspace{4pt}\\\
\\{\mathfrak{g}_{0}(\operatorname{\\!<\\!--1}),N_{0}\\}\subseteq N_{0}\cap
S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})_{\chi}.\end{array}$

We consider the projection

(2.14) $\displaystyle\rho:S(\mathfrak{g})\to S(\mathfrak{g}_{0})$

across the decomposition $S(\mathfrak{g})=S(\mathfrak{g}_{0})\oplus
S(\mathfrak{g})\mathfrak{g}_{1}$.

######  Proposition 2.9.

  1. (i)

$N\cap S(\mathfrak{g})_{0}\subseteq S(\mathfrak{g})$ and $N_{0}\subseteq
S(\mathfrak{g}_{0})$ are Poisson subalgebras;

  2. (ii)

$\rho(N\cap S(\mathfrak{g})_{0})\subseteq N_{0}$ and the map $\rho:N\cap
S(\mathfrak{g})_{0}\to N_{0}$ is a Poisson homomorphism;

  3. (iii)

The map $\pi:S(\mathfrak{g})\to
S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$
restricts to a surjective Poisson homomorphism

$N\cap S(\mathfrak{g})_{0}\twoheadrightarrow S(\mathfrak{g},e)_{0};$

  4. (iv)

The map $\pi_{0}:S(\mathfrak{g}_{0})\to
S(\mathfrak{g}_{0})/S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!
--1})_{\chi}$ restricts to a surjective Poisson homomorphism

$N_{0}\twoheadrightarrow S(\mathfrak{g}_{0},e);$

  5. (v)

The kernel of the map $N\cap S(\mathfrak{g})_{0}\twoheadrightarrow
S(\mathfrak{g},e)_{0}$ is contained in the kernel of the map
$\pi_{0}\circ\rho$. Thus $\rho$ induces a Poisson homomorphism
$S(\mathfrak{g},e)_{0}\twoheadrightarrow S(\mathfrak{g}_{0},e)$;

  6. (vi)

$S(\mathfrak{g},e)_{0}\cap S(\mathfrak{g},e)S(\mathfrak{g},e)_{1}$ is
contained in the kernel of $S(\mathfrak{g},e)_{0}\twoheadrightarrow
S(\mathfrak{g}_{0},e)$, inducing a surjective Poisson homomorphism

(2.15) $\displaystyle R(S(\mathfrak{g},e),H)\twoheadrightarrow
S(\mathfrak{g}_{0},e).$

###### Proof.

Since $H$ acts on $S(\mathfrak{g})$ by Poisson automorphisms the invariant
subspace $S(\mathfrak{g})_{0}$ is a Poisson subalgebra. Therefore the proofs
of the two claims in (i) are identical, and we will only prove that
$N_{0}\subseteq S(\mathfrak{g}_{0})$ is a Poisson subalgebra. It is evidently
closed under multiplication so we only need to show that it is closed under
the bracket. Suppose that $f_{1},f_{2}\in N_{0}$, that $g\in
G_{0}(\operatorname{\\!<\\!0})$ and that $g\cdot f_{i}-f_{i}=h_{i}\in
S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}$. We
have $g\cdot\\{f_{1},f_{2}\\}=\\{g\cdot f_{1},g\cdot
f_{2}\\}=\\{f_{1}+h_{1},f_{2}+h_{2}\\}$ and so (i) will follow if we can show
that $\\{f_{1},h_{2}\\},\\{h_{1},f_{2}\\},\\{h_{1},h_{2}\\}\in
S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}$. Since
$\chi$ vanishes on $\mathfrak{g}(\\!<\\!\\!-2)$ we have
$[\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})_{\chi},\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})_{\chi}]\subseteq\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}$,
therefore $\\{h_{1},h_{2}\\}\in
S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}$ by the
Leibniz rule. To complete the proof of (i) we observe that $N_{0}\cap
S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}$ is a
Poisson ideal of $N_{0}$, which follows quickly from (2.13).

We now address (ii). Let $f\in N\cap S(\mathfrak{g})_{0}$ and write
$f=f_{0}+f_{1}$ according to the decomposition
$S(\mathfrak{g})=S(\mathfrak{g}_{0})\oplus S(\mathfrak{g})\mathfrak{g}_{1}$.
If $g\in G_{0}(\operatorname{\\!<\\!0})$ then $g\cdot f_{0}-f_{0}\in
S(\mathfrak{g}_{0})\cap S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}=S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})_{\chi}$ by Lemma 2.7 and Lemma 2.8. This shows that $\rho(f)=f_{0}\in N$.

Finally, to see that $\rho:N\cap S(\mathfrak{g})_{0}\to N_{0}$ is a Poisson
homomorphism it suffices to show that $N\cap S(\mathfrak{g})_{0}\cap
S(\mathfrak{g})\mathfrak{g}_{1}$ is a Poisson ideal of $N\cap
S(\mathfrak{g})_{0}$. Recall that $\mathfrak{g}_{1}=\\{h\cdot x-x\mid h\in
H,x\in\mathfrak{g}\\}$ and $S(\mathfrak{g})_{1}=\\{h\cdot x-x\mid h\in H,x\in
S(\mathfrak{g})\\}$. If $f\in S(\mathfrak{g})_{0}$ then $\\{f,h\cdot
x-x\\}=h\cdot\\{f,x\\}-\\{f,x\\}\in S(\mathfrak{g})_{1}$ for any $x\in
S(\mathfrak{g})_{1}$. Lemma 2.2 shows that the ideals generated by
$\mathfrak{g}_{1}$ and $S(\mathfrak{g})_{1}$ coincide, hence
$\\{S(\mathfrak{g})_{0},\mathfrak{g}_{1}\\}\subseteq
S(\mathfrak{g})\mathfrak{g}_{1}$, which shows that $S(\mathfrak{g})_{0}\cap
S(\mathfrak{g})\mathfrak{g}_{1}$ is a Poisson ideal of $S(\mathfrak{g})_{0}$.
This completes the proof of (ii).

The map $\pi$ restricts to a surjection $N\twoheadrightarrow
S(\mathfrak{g},e)$ by definition. Since the latter map is $H$-equivariant we
get $N\cap S(\mathfrak{g})_{0}\to S(\mathfrak{g},e)_{0}$, which proves (iii),
whilst (iv) is proven similarly.

We move on to (v). The kernel of $N\cap S(\mathfrak{g})_{0}\to
S(\mathfrak{g},e)_{0}$ is $N\cap S(\mathfrak{g})_{0}\cap
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}$. This is mapped
to $S(\mathfrak{g}_{0})\cap S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!
--1})_{\chi}=S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--
1})_{\chi}$ by $\rho$, thanks to Lemma 2.7 and Lemma 2.8. Finally
$S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!--1})_{\chi}$ lies
in the kernel of $\pi_{0}$, which proves (v). Now we take $f\in N\cap
S(\mathfrak{g})_{0}$ such that $\pi(f)\in
S(\mathfrak{g},e)S(\mathfrak{g},e)_{1}$. Using Lemma 2.8 again we see that
$S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--
1})_{\chi}$ decomposes as the direct sum of the image of $S(\mathfrak{g}_{0})$
and the image of $S(\mathfrak{g})\mathfrak{g}_{1}$. Therefore
$S(\mathfrak{g},e)_{1}$, and the ideal which it generates, are contained in
the image of $S(\mathfrak{g})\mathfrak{g}_{1}$. It follows immediately that

$\displaystyle f\in
S(\mathfrak{g})S(\mathfrak{g})_{1}+S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!
--1})_{\chi}$ $\displaystyle\rho(f)\in
S(\mathfrak{g})\mathfrak{g}(\operatorname{\\!<\\!--1})_{\chi}\cap
S(\mathfrak{g}_{0})=S(\mathfrak{g}_{0})\mathfrak{g}_{0}(\operatorname{\\!<\\!
--1})_{\chi}$

Hence $\rho(f)$ is in the kernel of $\pi_{0}$, completing (vi). This concludes
the proof. ∎

######  Proof of Theorem 2.6.

Thanks to Proposition 2.9 we have
$\phi:R(S(\mathfrak{g},e),H)\twoheadrightarrow S(\mathfrak{g}_{0},e)$.

Pick an $H$-equivariant map $\theta:\mathfrak{g}^{e}\to S(\mathfrak{g},e)$
satisfying the properties of Theorem 2.4, and define
$\theta_{0}:\mathfrak{g}_{0}^{e}\to S(\mathfrak{g}_{0},e)$ via

$\theta_{0}(x):=\phi(\theta(x)+S(\mathfrak{g},e)_{0}\cap
S(\mathfrak{g},e)S(\mathfrak{g},e)_{1})$

Property (2.4) for $\theta$ implies (2.4) for $\theta_{0}$. By Theorem 2.4(2)
this implies that $\phi$ is surjective. Applying Proposition 2.2 we see that
$R(S(\mathfrak{g},e),H)$ is a polynomial algebra generated by the image of
$\theta(\mathfrak{g}^{e}_{0})$ under the map $S(\mathfrak{g},e)_{0}\to
R(S(\mathfrak{g},e),H)$. It follows that $\phi$ maps a basis of
$R(S(\mathfrak{g},e),H)$ to a basis of $S(\mathfrak{g}_{0},e)$, hence it is an
isomorphism. $\hfill\qed$

###  3\. Dirac reduction for shifted Yangians

####  3.1. Poisson algebras by generators and relations

Let $X$ be a set. The free Lie algebra $L_{X}$ on $X$ is the initial object in
the category of (complex) Lie algebras generated by $X$ and can be constructed
as the Lie subalgebra of the free algebra $\mathbb{C}\langle X\rangle$
generated by the vector space spanned by $X$. When $L$ is a Lie algebra
generated by $X$ we say that $L$ has relations $Y\subseteq L_{X}$ if $Y$
generates the kernel of $L_{X}\twoheadrightarrow L$.

The free Poisson algebra generated by $X$ is the initial object in the
category of (complex) Poisson algebras generated by $X$. It can be constructed
as the symmetric algebra $S(L_{X})$ together with its Poisson structure. If
there is a Poisson surjection $S(L_{X})\twoheadrightarrow A$ then we say that
$A$ is Poisson generated by $X$. It is important to distinguish this from $A$
being generated by $X$ as a commutative algebra, as both notions will occur
frequently.

We say that a (complex) Poisson algebra $A$ has Poisson generators $X$ and
relations $Y\subseteq S(L_{X})$ if there is a surjective Poisson homomorphism
$S(L_{X})\twoheadrightarrow A$ and the kernel is the Poisson ideal generated
by $Y$.

Let $X$ be a set and $Y\subseteq L_{X}\subseteq S(L_{X})$. Write $I$ (resp.
$J$) for the ideal of $S(L_{X})$ (resp. $L_{X}$) generated by $Y$ . It is easy
to see that the natural map $S(L_{X})\to S(L_{X})/I$ induces an isomorphism

(3.1) $\displaystyle S(L_{X}/J)\overset{\sim}{\longrightarrow}S(L_{X})/I.$

####  3.2. Chevalley–Serre presentations for shifted current Lie–Poisson
algebras

Throughout this Section we fix $n\in\mathbb{Z}_{>0}$. Following [BK06] a shift
matrix is an $n\times n$ array $\sigma=(s_{i,j})_{1\leq i,j\leq n}$ of non-
negative integers with zero on the diagonal, satisfying

(3.2) $\displaystyle s_{i,k}=s_{i,j}+s_{j,k}$

whenever $i\leq j\leq k$ or $k\leq j\leq i$. A shift matrix is said to be
symmetric if it is equal to its transpose and is said to be even if the
entries are integers. This nomenclature arises from the fact that shift
matrices classify certain gradings [BK06, (7.6)] and an even shift matrix
corresponds to a grading supported on the even integers. Symmetric shift
matrices correspond to Dynkin gradings, i.e. those which arise from
$\mathfrak{sl}_{2}$-triples.

The current algebra is the Lie algebra
$\mathfrak{c}_{n}:=\mathfrak{gl}_{n}\otimes\mathbb{C}[t]$. It has a basis
consisting of elements $\\{e_{i,j}t^{r}\mid 1\leq i,j\leq n,\
r\in\mathbb{Z}_{\geq 0}\\}$ where we write $x\otimes t^{r}=xt^{r}$ for
$x\in\mathfrak{gl}_{n}$ and $r\geq 0$. For any shift matrix
$\sigma=(s_{i,j})_{1\leq i,j\leq n}$ we define the shifted current algebra
$\mathfrak{c}_{n}(\sigma)$ to be the subalgebra spanned by

(3.3) $\displaystyle\\{e_{i,j}t^{r}\mid 1\leq i,j\leq n,\
r-s_{i,j}\in\mathbb{Z}_{\geq 0}\\}.$

######  Lemma 3.1.

The Lie subalgebra $\mathfrak{u}_{n}(\sigma)\subseteq\mathfrak{c}_{n}(\sigma)$
spanned by elements (3.3) with $i<j$ is generated as a complex Lie algebra by

(3.4) $\displaystyle\\{e_{i;r}\mid 1\leq i<n,\ r-s_{i,i+1}\in\mathbb{Z}_{\geq
0}\\}$

subject to the relations

(3.5) $\displaystyle\big{[}e_{i;r},e_{j;s}\big{]}$ $\displaystyle=$
$\displaystyle 0\text{ \ \ \ for \ \ \ }|i-j|\neq 1,$ (3.6)
$\displaystyle{}\big{[}e_{i;r+1},e_{i+1;s}\big{]}-\big{[}e_{i;r},e_{i+1;s+1}\big{]}$
$\displaystyle=$ $\displaystyle 0,$ (3.7)
$\displaystyle{}\Big{[}e_{i;r_{1}}\big{[}e_{i;r_{2}},e_{j;r_{3}}\big{]}\Big{]}+\Big{[}e_{i;r_{2}}\big{[}e_{i;r_{1}},e_{j;r_{3}}\big{]}\Big{]}$
$\displaystyle=$ $\displaystyle 0\text{ \ \ \ for all }|i-j|=1.$

###### Proof.

Write $\underline{0}$ for the $n\times n$ zero matrix. It follows from (3.2)
that the linear map
$\mathfrak{u}_{n}(\sigma)\to\mathfrak{u}_{n}(\underline{0})$ defined by
$e_{i,j}t^{r}\mapsto e_{i,j}t^{r-s_{i,j}}$ is a Lie algebra isomorphism and so
it suffices to prove the Lemma when $\sigma=\underline{0}$.

Let $\widehat{\mathfrak{u}}_{n}$ be the Lie algebra with generators (3.4) and
relations (3.5)–(3.7), with $\sigma=\underline{0}$. We inductively define
elements $e_{i,j;r}\in\widehat{\mathfrak{u}}_{n}$ by setting
$e_{i,i+1;r}:=e_{i;r}$ and $e_{i,j;r}:=[e_{i,j-1;r},e_{j-1,j;0}]$ for $1\leq
i<j\leq n$. There is a homomorphism
$\widehat{\mathfrak{u}}_{n}\twoheadrightarrow\mathfrak{u}_{n}(\underline{0})$
given by $e_{i;r}\mapsto e_{i,i+1}t^{r}$ and, in order to show that it is an
isomorphism, we show that $\widehat{\mathfrak{u}}_{n}$ is spanned by the
elements $\\{e_{i,j;r}\mid 1\leq i<j\leq n,\ r\in\mathbb{Z}_{\geq 0}\\}$.

Following (1)–(7) in the proof of [BK05, Lemma 5.8] verbatim we have for all
$i,j,k,l,r,s$

(3.8)
$\displaystyle[e_{i,j;r},e_{k,l;s}]=\delta_{j,k}e_{i,l;r+s}-\delta_{i,l}e_{k,j;r+s}.$

Define an ascending filtration on
$\widehat{\mathfrak{u}}_{n}=\bigcup_{d>0}\operatorname{F}_{d}\widehat{\mathfrak{u}}_{n}$
satisfying
$\operatorname{F}_{d}\widehat{\mathfrak{u}}_{n}=\sum_{\begin{subarray}{c}d_{1}+d_{2}=d\end{subarray}}[\operatorname{F}_{d_{1}}\widehat{\mathfrak{u}}_{n},\operatorname{F}_{d_{2}}\widehat{\mathfrak{u}}_{n}]$
by placing $e_{i;r}$ in degree 1. We prove by induction that
$\operatorname{F}_{d}\widehat{\mathfrak{u}}_{n}$ is spanned by elements
$e_{i,j;r}$ with $j-i\leq d$. The base case $d=1$ holds by definition. For
$d_{1}+d_{2}=d>1$ we know by the inductive hypothesis that
$\operatorname{F}_{d_{1}}\widehat{\mathfrak{u}}_{n}$ and
$\operatorname{F}_{d_{2}}\widehat{\mathfrak{u}}_{n}$ are spanned by elements
$e_{i,j;r}$. Using (3.8) we complete the induction, which finishes the proof.
∎

######  Theorem 3.2.

$S(\mathfrak{c}_{n}(\sigma))$ is Poisson generated by

(3.11) $\displaystyle\begin{array}[]{c}\\{d_{i;r}\mid 1\leq i\leq n,\
r\in\mathbb{Z}_{\geq 0}\\}\cup\\{e_{i;r}\mid 1\leq i<n,\
r-s_{i,i+1}\in\mathbb{Z}_{\geq 0}\\}\\\ \cup\\{f_{i;r}\mid 1\leq i<n,\
r-s_{i+1,i}\in\mathbb{Z}_{\geq 0}\\}\end{array}$

subject to relations

(3.18) $\displaystyle\big{\\{}d_{i;r},d_{j;s}\big{\\}}=0,$
$\displaystyle\big{\\{}d_{i;r},e_{j;s}\big{\\}}=(\delta_{i,j}-\delta_{i,j+1})e_{j;r+s},$
$\displaystyle\big{\\{}d_{i;r},f_{j;s}\big{\\}}=(\delta_{i,j+1}-\delta_{i,j})f_{j;r+s},$
$\displaystyle\big{\\{}e_{i;r_{1}},e_{i+1;r_{2}+1}\big{\\}}-\big{\\{}e_{i;r_{1}+1},e_{i+1;r_{2}}\big{\\}}=0,$
$\displaystyle\big{\\{}f_{i;r_{1}},f_{i+1;r_{2}+1}\big{\\}}-\big{\\{}f_{i;r_{1}+1},e_{i+1;r_{2}}\big{\\}}=0,$
$\displaystyle\big{\\{}e_{i;r},e_{j;s}\big{\\}}=0\text{ for }|i-j|\neq 1,$
$\displaystyle\big{\\{}f_{i;r},f_{j;s}\big{\\}}=0\text{ for }|i-j|\neq 1,$
(3.20)
$\displaystyle\Big{\\{}e_{i;r_{1}},\big{\\{}e_{i;r_{2}},e_{j;r_{3}}\big{\\}}\Big{\\}}+\Big{\\{}e_{i;r_{2}},\big{\\{}e_{i;r_{1}},e_{j;r_{3}}\big{\\}}\Big{\\}}=0\text{
for }|i-j|=1,$
$\displaystyle\Big{\\{}f_{i;r_{1}},\big{\\{}f_{i;r_{2}},f_{j;r_{3}}\big{\\}}\Big{\\}}+\Big{\\{}e_{i;r_{2}},\big{\\{}e_{i;r_{1}},e_{j;r_{3}}\big{\\}}\Big{\\}}=0\text{
for }|i-j|=1.$

###### Proof.

By (3.1) it suffices to show that $\mathfrak{c}_{n}(\sigma)$ is generated as a
Lie algebra by (3.11) subject to relations
(LABEL:e:shiftedcurrentfreerels1)–(3.18). Let
$(\widehat{\mathfrak{c}}_{n}(\sigma),\\{\cdot,\cdot\\})$ denote the Lie
algebra with these generators and relations. Define a map from the set (3.11)
to $\mathfrak{c}_{n}(\sigma)$ by $d_{i;r}\mapsto e_{i,i}t^{r},\ e_{i;r}\mapsto
e_{i,i+1}t^{r},\ f_{i;r}\mapsto e_{i+1,i}t^{r}$. One can easily verify using
(3.8) that this extends to a surjective Lie algebra homomorphism
$\widehat{\mathfrak{c}}_{n}(\sigma)\twoheadrightarrow\mathfrak{c}_{n}(\sigma)$.
To show that this is an isomorphism it suffices to show that the elements

(3.21) $\displaystyle\\{e_{i,j;r}\mid 1\leq i,j\leq n,\
r-s_{i,j}\in\mathbb{Z}_{\geq 0}\\}\subseteq\widehat{\mathfrak{c}}_{n}(\sigma)$

defined inductively by setting $e_{i,i+1;r}:=e_{i,r}$, $e_{i+1,i;r}:=f_{i;r}$
and

(3.22) $\displaystyle e_{i,j;r}:=\\{e_{i;s_{i,i+1}},e_{i+1,j;r-s_{i,i+1}}\\}$
$\displaystyle\text{ for }i<j;$ (3.23) $\displaystyle
e_{i,j;r}:=\\{f_{i-1;s_{i,i-1}},e_{i-1,j;r-s_{i,i-1}}\\}$ $\displaystyle\text{
for }i>j.$

form a spanning set. Using (LABEL:e:shiftedcurrentfreerels1),(3.18), (3.18)
and a simple inductive argument one can see that
$\widehat{\mathfrak{c}}_{n}(\sigma)$ is a direct sum of three subalgebras: the
diagonal subalgebra, spanned by the elements $d_{i;r}$, and the upper and
lower triangular subalgebras $\mathfrak{u}_{n}^{+}(\sigma)$ and
$\mathfrak{u}_{n}^{-}(\sigma)$ generated by the elements $e_{i;r}$,
respectively by the elements $f_{i;r}$.

In order to complete the proof it suffices to show that
$\mathfrak{u}_{n}^{+}(\sigma)$ and $\mathfrak{u}_{n}^{-}(\sigma)$ are spanned
by the elements defined in (3.22) and (3.23) respectively. Since the argument
is identical for $\mathfrak{u}_{n}^{+}(\sigma)$ and
$\mathfrak{u}_{n}^{-}(\sigma)$ we only need to consider the former, where the
claim follows from Lemma 3.1 ∎

The following theorem is one of the key stepping stones for understanding the
Dirac reduction of the shifted Yangian. The algebra described here is the
twisted shifted current Lie–Poisson algebra.

######  Theorem 3.3.

If $\sigma$ is even and symmetric then $S(\mathfrak{c}_{n}(\sigma))$ admits a
Poisson automorphism

(3.24) $\displaystyle\tau:e_{i,j}t^{r}\mapsto(-1)^{r-1+s_{i,j}}e_{j,i}t^{r}.$

The Dirac reduction
$R(S(\mathfrak{c}_{n}(\sigma),\tau)=S(\mathfrak{c}_{n}(\sigma)^{\tau})$ is
Poisson generated by

(3.26) $\displaystyle\begin{array}[]{l}\\{\eta_{i;2r-1}\mid 1\leq i\leq n,\
r\in\mathbb{Z}_{>0}\\}\cup\\{\theta_{i;r}\mid 1\leq i\leq n,\
r\in\mathbb{Z}_{\geq s_{i,i+1}}\\}\end{array}$

subject to the relations

(3.27) $\displaystyle\big{\\{}\eta_{i;2r-1},\eta_{j;2s-1}\big{\\}}$
$\displaystyle=$ $\displaystyle 0,\vspace{4pt}$ (3.28)
$\displaystyle\big{\\{}\eta_{i;2r-1},\theta_{j;s}\big{\\}}$ $\displaystyle=$
$\displaystyle(\delta_{i,j}-\delta_{i,j+1})\theta_{j;2r-1+s},\vspace{4pt}$
(3.29) $\displaystyle\big{\\{}\theta_{i;r},\theta_{j;s}\big{\\}}$
$\displaystyle=$ $\displaystyle 0\text{ if }|i-j|>1,\vspace{4pt}$ (3.30)
$\displaystyle\big{\\{}\theta_{i;r},\theta_{i+1;s+1}\big{\\}}$
$\displaystyle=$
$\displaystyle\big{\\{}\theta_{i;r+1},\theta_{i+1;s}\big{\\}},$ (3.33)
$\displaystyle\big{\\{}\theta_{i;r},\theta_{i;s}\big{\\}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}2(-1)^{s-1+s_{i,i+1}}(\eta_{i;r+s}-\eta_{i+1;r+s})&\text{
if }r+s\text{ is odd}\\\ 0&\text{ if }r+s\text{ is even},\end{array}\right.$
(3.34)
$\displaystyle\Big{\\{}\theta_{i;r_{1}},\big{\\{}\theta_{i;r_{2}},\theta_{j;r_{3}}\big{\\}}\Big{\\}}+\Big{\\{}\theta_{i;r_{2}},\big{\\{}\theta_{i;r_{1}},\theta_{j;r_{3}}\big{\\}}\Big{\\}}=0\text{
for }|i-j|=1,r_{1}+r_{2}\text{ odd}.$
$\displaystyle\Big{\\{}\theta_{i;r_{1}},\big{\\{}\theta_{i;r_{2}},\theta_{j;r_{3}}\big{\\}}\Big{\\}}+\Big{\\{}\theta_{i;r_{2}},\big{\\{}\theta_{i;r_{1}},\theta_{j;r_{3}}\big{\\}}\Big{\\}}=\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (3.35)
$\displaystyle
2(-1)^{r_{2}-1+s_{i,i+1}}(\delta_{i+1,j}+\delta_{i,j+1})\theta_{j;r_{1}+r_{2}+r_{3}}\
\ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle\text{ for
}|i-j|=1,r_{1}+r_{2}\text{ even}.$

###### Proof.

Relation (3.8) implies that $\tau$ gives a Poisson automorphism. By Remark 2.3
we can identify $R(S(\mathfrak{c}_{n}(\sigma),\tau)$ with
$S(\mathfrak{c}_{n}(\sigma)^{\tau})$, so it suffices to check that
$S(\mathfrak{c}_{n}(\sigma)^{\tau})$ has the stated Poisson presentation. Use
(3.1) to reduce the claim to a statement about Lie algebras.

Consider the Lie algebra $\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$ which is
generated by the set (3.26) subject to relations (3.27)–(3.35). We define a
map from the set (3.26) to $\mathfrak{c}_{n}(\sigma)^{\tau}$ by sending
$\eta_{i;2r-1}\mapsto e_{i,i}t^{2r-1}\in\mathfrak{c}_{n}(\sigma)^{\tau}$, and
sending
$\theta_{i;r}\mapsto\theta_{i,i+1}t^{r}\in\mathfrak{c}_{n}(\sigma)^{\tau}$.
One can check that this determines a surjective Lie algebra homomorphism
$\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}\twoheadrightarrow\mathfrak{c}_{n}(\sigma)^{\tau}$,
indeed, checking that relations (3.27)–(3.35) hold amongst the corresponding
elements of $\mathfrak{c}_{n}(\sigma)^{\tau}$ is a routine calculation using
(3.8).

In order to complete the proof of (2) it is sufficient to show that this map
is an isomorphism. For $1\leq i<j\leq n$ and $r\in\mathbb{Z}_{\geq s_{i,j}}$
we inductively define elements

(3.36)
$\displaystyle\theta_{i,j;r}:=\\{\theta_{i,i+1;s_{i,i+1}},\theta_{i+1,j;r-s_{i,i+1}}\\}\in\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}.$

where $\theta_{i,i+1;r}:=\theta_{i;r}$. It remains to check that
$\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$ is spanned by the elements

(3.38) $\displaystyle\begin{array}[]{l}\\{\eta_{i;2r-1}\mid 1\leq i\leq n,\
r\in\mathbb{Z}_{>0}\\}\cup\\{\theta_{i,j;r}\mid\ 1\leq i<j\leq n,\
r-s_{i,j}\in\mathbb{Z}_{\geq 0}\\}\end{array}$

We define a filtration
$\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}=\bigcup_{i>0}\operatorname{F}_{i}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$
by placing the generators (3.26) in degree 1 and satisfying
$\operatorname{F}_{d}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}=\sum_{d_{1}+d_{2}=d}\\{\operatorname{F}_{d_{1}}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau},\operatorname{F}_{d_{2}}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}\\}$.
By convention
$\operatorname{F}_{0}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}=0$. The
associated graded Lie algebra
$\operatorname{gr}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}=\bigoplus_{i>0}\operatorname{F}_{i}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}/\operatorname{F}_{i-1}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$
is generated by elements

(3.39)
$\displaystyle\overline{\eta}_{i;2r-1}:=\eta_{i}t^{2r-1}+\operatorname{F}_{0}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}\text{
for }1\leq i\leq n,\ r\in\mathbb{Z}_{\geq 0}$ (3.40)
$\displaystyle\overline{\theta}_{i;r}:=\theta_{i}t^{r}+\operatorname{F}_{0}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}\text{
for }1\leq i<n,\ r\in\mathbb{Z}_{\geq s_{i,i+1}}.$

These generators of
$\operatorname{gr}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$ satisfy the top
graded components of the relations (3.27)–(3.35).

Let $\mathfrak{a}$ be an abelian Lie algebra with basis $\\{d_{i}t^{r}\mid
1\leq i\leq n,\ r\in\mathbb{Z}_{>0}\\}$. Let $\mathfrak{u}_{n}(\sigma)$ be the
Lie subalgebra of $\mathfrak{c}_{n}(\sigma)$ described in Lemma 3.1. Then
$\mathfrak{a}\oplus\mathfrak{u}_{n}(\sigma)$ is a Lie algebra with
$\mathfrak{a}$ an abelian ideal. Comparing the top components (3.27)–(3.35) to
(3.5)–(3.7) we see that there is surjective Lie algebra homomorphism
$\mathfrak{a}\oplus\mathfrak{u}_{n}(\sigma)\twoheadrightarrow\operatorname{gr}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$
defined by $d_{i}t^{r}\mapsto\overline{\eta}_{i;2r-1}$ and
$e_{i,i+1}t^{r}\mapsto\overline{\theta}_{i,r}$. The algebra
$\mathfrak{a}\oplus\mathfrak{u}_{n}(\sigma)$ has basis consisting of element
$d_{i}t^{r},e_{j,k}t^{s}$ where $i=1,...,n,\ r\in\mathbb{Z}_{>0},\ 1\leq
j<k\leq n,\ s\in\mathbb{Z}_{\geq s_{j,k}}$. It follows that
$\operatorname{gr}\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$ is spanned by
elements $\overline{\eta}_{i;2r-1},\overline{\theta}_{j,k;s}$ where the
indexes vary in the ranges specified in (3.38), and
$\overline{\theta}_{j,k;s}$ defined inductively from (3.40), analogously to
(3.36). We deduce that $\widehat{\mathfrak{c}}_{n}(\sigma)^{\tau}$ is spanned
by the required elements, which completes the proof. ∎

####  3.3. The semiclassical shifted Yangian

In this Section we fix $n>0$, an even shift matrix $\sigma$ of size $n$ and
$\varepsilon\in\\{\pm 1\\}$. The (semiclassical) shifted Yangian
$y_{n}(\sigma)$ is the Poisson algebra generated by the set

(3.41) $\begin{array}[]{c}\\{d_{i}^{(r)}\mid 1\leq i\leq
n,r>0\\}\cup\\{e_{i}^{(r)}\mid 1\leq i<n,r>s_{i,i+1}\\}\\\
\cup\,\\{f_{i}^{(r)}\mid 1\leq i<n,r>s_{i+1,i}\\}\end{array}$

subject to the following relations

(3.42) $\displaystyle\big{\\{}d_{i}^{(r)},d_{j}^{(s)}\big{\\}}=0,$ (3.43)
$\displaystyle\big{\\{}e_{i}^{(r)},f_{j}^{(s)}\big{\\}}=-\delta_{i,j}\sum_{t=0}^{r+s-1}d_{i+1}^{(r+s-1-t)}\widetilde{d}_{i}^{(t)},$
(3.44) $\displaystyle\big{\\{}d_{i}^{(r)},e_{j}^{(s)}\big{\\}}$
$\displaystyle=$
$\displaystyle(\delta_{i,j}-\delta_{i,j+1})\sum_{t=0}^{r-1}d_{i}^{(t)}e_{j}^{(r+s-1-t)},$
(3.45) $\displaystyle\big{\\{}d_{i}^{(r)},f_{j}^{(s)}\big{\\}}$
$\displaystyle=$
$\displaystyle(\delta_{i,j+1}-\delta_{i,j})\sum_{t=0}^{r-1}f_{j}^{(r+s-1-t)}d_{i}^{(t)},$
(3.46) $\displaystyle\big{\\{}e_{i}^{(r)},e_{i}^{(s)}\big{\\}}$
$\displaystyle=$
$\displaystyle\sum_{t=r}^{s-1}e_{i}^{(t)}e_{i}^{(r+s-1-t)}\hskip
31.29802pt\text{if $r<s$},$ (3.47)
$\displaystyle\big{\\{}f_{i}^{(r)},f_{i}^{(s)}\big{\\}}$ $\displaystyle=$
$\displaystyle\sum_{t=s}^{r-1}f_{i}^{(r+s-1-t)}f_{i}^{(t)}\hskip
31.29802pt\text{if $r>s$},$ (3.48)
$\displaystyle\big{\\{}e_{i}^{(r+1)},e_{i+1}^{(s)}\big{\\}}$ $\displaystyle-$
$\displaystyle\big{\\{}e_{i}^{(r)},e_{i+1}^{(s+1)}\big{\\}}=e_{i}^{(r)}e_{i+1}^{(s)},$
(3.49) $\displaystyle\big{\\{}f_{i}^{(r)},f_{i+1}^{(s+1)}\big{\\}}$
$\displaystyle-$
$\displaystyle\big{\\{}f_{i}^{(r+1)},f_{i+1}^{(s)}\big{\\}}=f_{i+1}^{(s)}f_{i}^{(r)},$
(3.50) $\displaystyle\big{\\{}e_{i}^{(r)},e_{j}^{(s)}\big{\\}}$
$\displaystyle=$ $\displaystyle 0\quad\text{ if }|i-j|>1,$ (3.51)
$\displaystyle\big{\\{}f_{i}^{(r)},f_{j}^{(s)}\big{\\}}$ $\displaystyle=$
$\displaystyle 0\quad\text{ if }|i-j|>1,$ (3.52)
$\displaystyle\Big{\\{}e_{i}^{(r)},\big{\\{}e_{i}^{(s)},e_{j}^{(t)}\big{\\}}\Big{\\}}+\Big{\\{}e_{i}^{(s)},\big{\\{}e_{i}^{(r)},e_{j}^{(t)}\big{\\}}\Big{\\}}$
$\displaystyle=$ $\displaystyle 0\quad\text{ if }|i-j|=1,r\neq s,$ (3.53)
$\displaystyle\Big{\\{}f_{i}^{(r)},\big{\\{}f_{i}^{(s)},f_{j}^{(t)}\big{\\}}\Big{\\}}+\Big{\\{}f_{i}^{(s)},\big{\\{}f_{i}^{(r)},f_{j}^{(t)}\big{\\}}\Big{\\}}$
$\displaystyle=$ $\displaystyle 0\quad\text{ if }|i-j|=1,r\neq s.$

for all admissible $i,j,r,s,t$. In these relations, the notation
$d_{i}^{(0)}=\widetilde{d}_{i}^{(0)}:=1$ is used, and the elements
$\widetilde{d}_{i}^{(r)}$ for $r>0$ are defined recursively by

(3.54)
$\displaystyle\widetilde{d}_{i}^{(r)}:=-\sum_{t=1}^{r}d_{i}^{(t)}\widetilde{d}_{i}^{(r-t)}$

In order to describe the structure of $y_{n}(\sigma)$ as a commutative algebra
we make the notation

(3.55) $\displaystyle e_{i,i+1}^{(r)}:=e_{i}^{(r)}\text{ for }1\leq i<n,\
s_{i,i+1}<r;$ (3.56) $\displaystyle f_{i,i+1}^{(r)}:=f_{i}^{(r)}\text{ for
}1\leq i\leq n,\ s_{i+1,i}<r,$

and inductively define

(3.57) $\displaystyle
e_{i,j}^{(r)}:=\\{e_{i,j-1}^{(r-s_{j-1,j})},e_{j-1}^{(s_{j-1,j}+1)}\\}\text{
for }1\leq i<j\leq n,\ s_{i,j}<r;$ (3.58) $\displaystyle
f_{i,j}^{(r)}:=\\{f_{j-1}^{(s_{j,j-1}+1)},f_{i,j-1}^{(r-s_{j,j-1})}\\}\text{
for }1\leq i<j\leq n,\ s_{j,i}<r.$

The shifted Yangian admits a Poisson grading $y_{n}(\sigma)=\bigoplus_{r\geq
0}y_{n}(\sigma)_{r}$, which we call the canonical grading. It places
$d_{i}^{(r)},e_{i}^{(r)},f_{i}^{(r)}$ in degree $r$ and the bracket lies in
degree $-1$, meaning $\\{\cdot,\cdot\\}:y_{n}(\sigma)_{r}\times
y_{n}(\sigma)_{s}\to y_{n}(\sigma)_{r+s-1}$. There is also an important
Poisson filtration $y_{n}(\sigma)=\bigcup_{i\geq
0}\operatorname{F}_{i}y_{n}(\sigma)$, called the loop filtration, which places
$d_{i}^{(r)},e_{i}^{(r)},f_{i}^{(r)}$ in degree $r-1$. Again the bracket is in
degree $-1$ so that
$\operatorname{F}_{r}y_{n}(\sigma)\times\operatorname{F}_{s}y_{n}(\sigma)\to\operatorname{F}_{r+s-1}y_{n}(\sigma)$.
The associated graded Poisson algebra is denoted
$\operatorname{gr}y_{n}(\sigma)$.

The following theorem is a semiclassical analogue of [BK06, Theorem 2.1],
which we ultimately deduce from the noncommutative setting.

######  Theorem 3.4.

There is a Poisson isomorphism
$S(\mathfrak{c}_{n}(\sigma))\overset{\sim}{\longrightarrow}\operatorname{gr}y_{n}(\sigma)$
defined by

(3.62) $\displaystyle\begin{array}[]{rcl}e_{i;r-1}\longmapsto
e_{i}^{(r)}+\operatorname{F}_{r-2}y_{n}(\sigma)&\text{ for }&1\leq i<n,\
r\in\mathbb{Z}_{>s_{i,i+1}};\\\ f_{i;r-1}\longmapsto
f_{i}^{(r)}+\operatorname{F}_{r-2}y_{n}(\sigma)&\text{ for }&1\leq i<n,\
r\in\mathbb{Z}_{>s_{i+1,i}};\\\ d_{i;r-1}\longmapsto
d_{i}^{(r)}+\operatorname{F}_{r-2}y_{n}(\sigma)&\text{ for }&1\leq i\leq n,\
r\in\mathbb{Z}_{>0}.\end{array}$

As a consequence $y_{n}(\sigma)$ is isomorphic to the polynomial algebra on
infinitely many variables

(3.63) $\begin{array}[]{c}\\{d_{i}^{(r)}\mid 1\leq i\leq
n,r>0\\}\cup\\{e_{i,j}^{(r)}\mid 1\leq i<j\leq n,r>s_{i,j}\\}\\\
\cup\,\\{f_{i,j}^{(r)}\mid 1\leq i<j\leq n,r>s_{j,i}\\}.\end{array}$

###### Proof.

Comparing the top graded components of the relations (3.42)–(3.53) with
respect to the loop filtration, with relations
(LABEL:e:shiftedcurrentfreerels1)–(3.20), it is straightforward to see that
(3.62) gives a surjective Poisson homomorphism. To prove the theorem we
demonstrate that the ordered monomials in the elements (3.63) are linearly
independent.

Consider the set

$X:=\\{E_{i}^{(r)},F_{i}^{(s)},D_{j}^{(t)}\mid 1\leq i<n,\ 1\leq j\leq n,\
s_{i,i+1}<r,\ s_{i+1,i}<s,\ 0<t\\}.$

In [BK06, §2] the shifted Yangian $Y_{n}(\sigma)$ is defined as a quotient of
the free algebra $\mathbb{C}\langle X\rangle$ by the ideal generated by the
relations [BK06, (2.4)–(2.15)]. Let $L:=L_{X}$ be the free Lie algebra on $X$
and define a grading $L=\bigoplus_{i\geq 0}L_{i}$ by placing
$E_{i}^{(r)},F_{i}^{(r)},D_{i}^{(r)}$ in degree $r-1$. Then we place a
filtration on the enveloping algebra $U(L)$ so that $L_{i}$ lies in degree
$i+1$. By the PBW theorem for $U(L_{X})$ we see that
$\operatorname{gr}U(L)\cong S(L)$.

The universal property of $U(L)$ ensures that there is a surjective algebra
homomorphism $U(L)\twoheadrightarrow\mathbb{C}\langle X\rangle$ and by [Ser06,
I, Ch. IV, Theorem 4.2] this is an isomorphism. Identifying these algebras,
the filtration on $U(L)$ descends to $Y_{n}(\sigma)=\bigcup_{i\geq
0}\operatorname{F}^{\prime}_{i}Y_{n}(\sigma)$, and this resulting filtration
is commonly referred to as the canonical filtration [BK06, §5]. The associated
graded algebra $\operatorname{gr}Y_{n}(\sigma)$ is equipped with a Poisson
structure in the usual manner. Comparing relations (3.42)–(3.53) with the top
graded components of relations [BK06, (2.4)–(2.15)] we see that the Poisson
surjection $S(L)\twoheadrightarrow\operatorname{gr}Y_{n}(\sigma)$ factors
through $S(L)\to y_{n}(\sigma)$. As a result there is a surjective Poisson
homomorphism
$\pi:y_{n}(\sigma)\twoheadrightarrow\operatorname{gr}Y_{n}(\sigma)$ given by
$e_{i}^{(r)}\mapsto
E_{i}^{(r)}+\operatorname{F}^{\prime}_{r-1}Y_{n}(\sigma),f_{i}^{(r)}\mapsto
F_{i}^{(r)}+\operatorname{F}^{\prime}_{r-1}Y_{n}(\sigma),d_{i}^{(r)}\mapsto
D_{i}^{(r)}+\operatorname{F}^{\prime}_{r-1}Y_{n}(\sigma)$. Following [BK06,
(2.18), (2.19)] we introduce elements $E_{i,j}^{(r)},F_{i,j}^{(r)}$ of
$Y_{n}(\sigma)$ lying in filtered degree $r$. By the definition of the
filtration and the elements (3.57), (3.58) we have

$\displaystyle\begin{array}[]{rcl}\pi(e_{i,j}^{(r)})&=&E_{i,j}^{(r)}+\operatorname{F}^{\prime}_{r-1}Y_{n}(\sigma);\\\
\pi(f_{i,j}^{(r)})&=&F_{i,j}^{(r)}+\operatorname{F}^{\prime}_{r-1}Y_{n}(\sigma).\\\
\end{array}$

By [BK06, Theorem 2.3(iv)] the ordered monomials in
$E_{i,j}^{(r)},F_{i,j}^{(r)},D_{i}^{(r)}$ are linearly independent in
$Y_{n}(\sigma)$ and so we deduce that the images of these monomials in
$\operatorname{gr}Y_{n}(\sigma)$ are linearly independent. This completes the
proof. ∎

We record two formulas for future use, which are semiclassical analogues of
[BT18, (4.32)]

######  Lemma 3.5.

The following hold for $i=1,...,n$, $j=1,...,n-1$, $r>0$ and $s>s_{j,j+1}$:

(3.65)
$\displaystyle\\{\widetilde{d}_{i}^{(r)},e_{j}^{(s)}\\}=(\delta_{i,j+1}-\delta_{i,j})\sum_{t=0}^{r-1}\widetilde{d}_{i}^{(t)}e_{j}^{(r+s-1-t)},$
(3.66)
$\displaystyle\\{\widetilde{d}_{i}^{(r)},f_{j}^{(s)}\big{\\}}=(\delta_{i,j}-\delta_{i,j+1})\sum_{t=0}^{r-1}f_{j}^{(r+s-1-t)}\widetilde{d}_{i}^{(t)}.$

###### Proof.

We only sketch (3.65), as the proof of (3.66) is almost identical. The
argument is by induction based on (3.54). Note that
$\widetilde{d}_{i}^{(1)}=-d_{i}^{(1)}$ and so (3.65) is equivalent to (3.44)
in this case. We have
$\\{\widetilde{d}_{i}^{(r)},e_{j}^{(s)}\\}=\\{\sum_{t=0}^{r-1}d_{i}^{(r-t)}\widetilde{d}_{i}^{(t)},e_{j}^{(s)}\\}$
and relation (3.44) together with the inductive hypothesis imply that the
coefficient of $e_{j}^{(s+t)}$ is
$(\delta_{i,j+1}-\delta_{i,j})(d_{i}^{(r-1-t)}+\sum_{m=t+1}^{r-1}(\widetilde{d}_{i}^{(r-m)}d_{i}^{(m-t-1)}-d_{i}^{(r-m)}\widetilde{d}_{i}^{(m-t-1)}))$.
Using $d_{i}^{(0)}=\widetilde{d}_{i}^{(0)}=1$ we see that all quadratic terms
cancel, whilst the linear term equates to
$(\delta_{i,j+1}-\delta_{i,j})\widetilde{d}_{i}^{(r-1-t)}$, which concludes
the induction. ∎

####  3.4. The Dirac reduction of the shifted Yangian

In this Section we suppose that $\sigma$ is symmetric and even. Examining the
relations (3.42)–(3.53) we see that there is unique Poisson automorphism
$\tau$ of $y_{n}(\sigma)$ determined by

(3.70)
$\displaystyle\begin{array}[]{l}\tau(d_{i}^{(r)}):=(-1)^{r}d_{i}^{(r)};\\\\[3.0pt]
\tau(e_{i}^{(r)}):=(-1)^{r+s_{i,i+1}}f_{i}^{(r)};\\\\[3.0pt]
\tau(f_{i}^{(r)}):=(-1)^{r+s_{i+1,i}}e_{i}^{(r)}.\end{array}$

Examining the relations (3.42)–(3.53) we see that $\tau$ extends to an
involutive Poisson automorphism of $y_{n}(\sigma)$. Our present goal is to
give a complete description of the Dirac reduction $R(y_{n}(\sigma),\tau)$.

For $i=1,...,n$ and $r\in\mathbb{Z}_{>s_{i,i+1}}$ we write

(3.73)
$\displaystyle\begin{array}[]{rcl}\widehat{e}_{i}^{(r)}&:=&e_{i}^{(r)}+(-1)^{r+s_{i,i+1}}f_{i}^{(r)}\in
y_{n}(\sigma);\vspace{4pt}\\\
\widecheck{e}_{i}^{(r)}&:=&e_{i}^{(r)}-(-1)^{r+s_{i,i+1}}f_{i}^{(r)}\in
y_{n}(\sigma),\end{array}$

so that $\widehat{e}_{i}^{(r)}$ and $d_{j}^{(2s)}$ are $\tau$-invariants,
whilst $\widecheck{e}_{i}^{(r)}$ and $d_{j}^{(2s-1)}$ each span a non-trivial
representation of the cyclic group of order 2 generated by $\tau$. We can
recover $e_{i}^{(r)}$ and $f_{i}^{(r)}$ via

(3.76)
$\displaystyle\begin{array}[]{l}e_{i}^{(r)}=\frac{1}{2}(\widehat{e}_{i}^{(r)}+\widecheck{e}_{i}^{(r)});\\\
f_{i}^{(r)}=\frac{1}{2}(-1)^{r+s_{i,i+1}}(\widehat{e}_{i}^{(r)}-\widecheck{e}_{i}^{(r)}).\end{array}$

Now let $\widecheck{y}_{n}(\sigma)$ be the ideal of $y_{n}(\sigma)$ generated
by

$\\{\widecheck{e}_{i}^{(r)},d_{j}^{(2s-1)}\mid i=1,...,n-1,\ j=1,...,n,\
r\in\mathbb{Z}_{>s_{i,i+1}},\ s\in\mathbb{Z}_{>0}\\}.$

Also write $\widecheck{y}_{n}(\sigma)^{\tau}:=\widecheck{y}_{n}(\sigma)\cap
y_{n}(\sigma)^{\tau}$. By Lemma 2.2 we see that
$R(y_{n}(\sigma),\tau)=y_{n}(\sigma)^{\tau}/\widecheck{y}_{n}(\sigma)^{\tau}$
is Poisson generated by elements

(3.79)
$\displaystyle\begin{array}[]{rcl}\theta_{i}^{(r)}&:=&\widehat{e}_{i}^{(r)}+\widecheck{y}_{n}(\sigma)^{\tau}\text{
for }i=1,...,n-1,\ r>s_{i,i+1},\vspace{4pt}\\\
\eta_{j}^{(2r)}&:=&d_{j}^{(2s)}+\widecheck{y}_{n}(\sigma)^{\tau}\text{ for
}i=1,...,n,\ r>0,\end{array}$

with Poisson brackets induced by the bracket on $y_{n}(\sigma)$. Furthermore
$R(y_{n}(\sigma),\tau)$ is generated as a commutative algebra by elements

(3.80) $\displaystyle\\{\eta_{i}^{(2r)}\mid 1\leq i\leq n,\
r\in\mathbb{Z}_{>0}\\}\cup\\{\theta_{i,j}^{(r)}\mid 1\leq i<j\leq
n,r-s_{i,j}\in\mathbb{Z}_{>0}\\}$

where the
$\theta_{i,j}^{(r)}:=e_{i,j}^{(r)}+(-1)^{r+s_{i,j}}f_{i,j}^{(r)}+\widecheck{y}_{n}(\sigma)^{\tau}\in
R(y_{n}(\sigma),\tau)$. Using an inductive argument and (3.43) we see the
elements $\theta_{i,j}^{(r)}$ can also be defined via the following recursion

(3.83)
$\displaystyle\begin{array}[]{rcl}\theta_{i,i+1}^{(r)}:=\theta_{i}^{(r)}&\text{
for }&1\leq i<n,r-s_{i,j}\in\mathbb{Z}_{>0}\vspace{4pt}\\\
\theta_{i,j}^{(r)}:=\\{\theta_{i,j-1}^{(r-s_{j-1,j})},\theta_{j-1}^{(s_{j-1,j}+1)}\\}&\text{
for }&1\leq i<j\leq n,\ r-s_{i,j}\in\mathbb{Z}_{>0}.\end{array}$

The next lemma can be deduced from (3.44), (3.45), (3.65), (3.66).

######  Lemma 3.6.

The following equalities hold in $R(y_{n}(\sigma),\tau)$:

(3.84)
$\displaystyle\\{d_{i}^{(2r-1)},\widecheck{e}_{j}^{(s)}\\}+\widecheck{y}_{n}(\sigma)^{\tau}=(\delta_{i,j}-\delta_{i,j+1})\sum_{t=0}^{r-1}\eta_{i}^{(2t)}\theta_{j}^{(2(r-t-1)+s)}$
(3.85)
$\displaystyle\\{\widetilde{d}_{i}^{(2r-1)},\widecheck{e}_{j}^{(s)}\\}+\widecheck{y}_{n}(\sigma)^{\tau}=(\delta_{i,j+1}-\delta_{i,j})\sum_{t=0}^{r-1}\widetilde{\eta}_{i}^{(2t)}\theta_{j}^{(2(r-t-1)+s)}$

Thanks to Theorem 3.4 that $R(y_{n}(\sigma),\tau)$ comes equipped with the
canonical grading $R(y_{n}(\sigma),\tau)=\bigoplus_{i\geq
0}R(y_{n}(\sigma),\tau)_{i}$ which places $\theta_{i}^{(r)},\eta_{i}^{(r)}$ in
degree $r$ and the Poisson bracket in degree $-1$. Another crucial feature is
the loop filtration $R(y_{n}(\sigma),\tau)=\bigcup_{i\geq
0}\operatorname{F}_{i}R(y_{n}(\sigma),\tau)$ which places
$\theta_{i}^{(r)},\eta_{i}^{(r)}$ in degree $r-1$ and the bracket in degree
$-1$. Both of these structures are naturally inherited from $y_{n}(\sigma)$.

The following is our main structural result on the Dirac reduction of the
shifted Yangian.

######  Theorem 3.7.

The Dirac reduction $R(y_{n}(\sigma),\tau)$ is Poisson generated by elements

(3.86) $\displaystyle\\{\eta_{i}^{(2r)}\mid 1\leq i\leq n,\
r\in\mathbb{Z}_{>0}\\}\cup\\{\theta_{i}^{(r)}\mid 1\leq i<n,\
r-s_{i,i+1}\in\mathbb{Z}_{>0}\\}$

together with the following relations

(3.87) $\displaystyle\\{\eta_{i}^{(2r)},\eta_{j}^{(2s)}\\}=0$ (3.88)
$\displaystyle\big{\\{}\eta_{i}^{(2r)},\theta_{j}^{(s)}\big{\\}}=(\delta_{i,j}-\delta_{i,j+1})\sum_{t=0}^{r-1}\eta_{i}^{(2t)}\theta_{j}^{(2r+s-1-2t)}$
(3.89) $\displaystyle\ \ \ \
\big{\\{}\theta_{i}^{(r)},\theta_{i}^{(s)}\big{\\}}=0$ $\displaystyle\text{
for }r+s-1\text{ odd}$ (3.90) $\displaystyle\ \ \ \
\big{\\{}\theta_{i}^{(r)},\theta_{i}^{(s)}\big{\\}}=\sum_{t=r}^{s-1}\frac{1}{2}\theta_{i}^{(t)}\theta_{i}^{(2m-t)}+(-1)^{s+s_{i,i+1}}\sum_{t=0}^{m}2\eta_{i+1}^{(2m-2t)}\widetilde{\eta}_{i}^{(2t)}$
$\displaystyle\text{ for }r<s,r+s-1=2m$ (3.91)
$\displaystyle\big{\\{}\theta_{i}^{(r+1)},\theta_{i+1}^{(s)}\big{\\}}-\big{\\{}\theta_{i}^{(r)},\theta_{i+1}^{(s+1)}\big{\\}}=\frac{1}{2}\theta_{i}^{(r)}\theta_{i+1}^{(s)}$
(3.92) $\displaystyle\big{\\{}\theta_{i}^{(r)},\theta_{j}^{(s)}\big{\\}}=0$
$\displaystyle\text{ for }|i-j|>1$ (3.93)
$\displaystyle\Big{\\{}\theta_{i}^{(r)},\big{\\{}\theta_{i}^{(s)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}+\Big{\\{}\theta_{i}^{(s)},\big{\\{}\theta_{i}^{(r)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}=0$
$\displaystyle\text{ for }|i-j|=1\text{ and }r+s\text{ odd}$ (3.94)
$\displaystyle\Big{\\{}\theta_{i}^{(r)},\big{\\{}\theta_{i}^{(s)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}+\Big{\\{}\theta_{i}^{(s)},\big{\\{}\theta_{i}^{(r)},\theta_{j}^{(t)}\big{\\}}\Big{\\}}=$
$\displaystyle\ \ \ \ \ \ \ \ \ \
2(-1)^{s+s_{i,i+1}-1}\delta_{i,j+1}\sum_{m_{1}=0}^{m-1}\sum_{m_{2}=0}^{m_{1}}\eta_{i+1}^{(2(m-m_{1}-1))}\widetilde{\eta}_{i}^{(2m_{2})}\theta_{j}^{(2(m_{1}-m_{2})+t)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+2(-1)^{s+s_{i,i+1}-1}\delta_{i+1,j}\sum_{m_{1}=0}^{m-1}\sum_{m_{2}=0}^{m-m_{1}-1}\widetilde{\eta}_{i}^{(2m_{1})}\eta_{i+1}^{(2m_{2})}\theta_{j}^{(2(m-m_{1}-m_{2}-1)+t)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{ for }|i-j|=1\text{ and }r+s=2m\text{ even}$

where we adopt the convention $\eta_{i}^{(0)}=\widetilde{\eta}_{i}^{(0)}=1$
and the elements $\\{\widetilde{\eta}_{i}^{(r)}\mid r\in 2\mathbb{Z}_{\geq
0}\\}$ are defined via the recursion

(3.95)
$\displaystyle\widetilde{\eta}_{i}^{(2r)}:=-\sum_{t=1}^{r}\eta_{i}^{(2t)}\widetilde{\eta}_{i}^{(2r-2t)}.$

###### Proof.

First we show that the elements
$\widetilde{\eta}_{i}^{(2r)}:=\widetilde{d}_{i}^{(2r)}+\widecheck{y}_{n}(\sigma)^{\tau}\in
R(y_{n}(\sigma),\tau)$ satisfy the recursion (3.95). By Theorem 3.4 the
subalgebra of $y_{n}(\sigma)$ generated by $\\{d_{i}^{(r)}\mid i=1,...,n,\
r>0\\}$ is a graded polynomial ring with $d_{i}^{(r)}$ in degree $r$. By
induction $\widetilde{d}_{i}^{(r)}$ lies in degree $r$. This forces
$\widetilde{d}_{i}^{(2r-1)}\in\widecheck{y}_{n}(\sigma)$ for all $r>0$ and so
(3.54) equals (3.95) modulo $\widecheck{y}_{n}(\sigma)^{\tau}$, confirming the
claim.

We now deduce relations (3.87)–(3.94) from (3.42)–(3.53). First of all (3.87)
follows immediately from (3.42). By (3.44) and (3.45) we have

$\displaystyle\\{d_{i}^{(2r)},\widehat{e}_{j}^{(s)}\\}$ $\displaystyle=$
$\displaystyle(\delta_{i,j}-\delta_{i,j+1})\sum_{t=0}^{2r-1}d_{i}^{(t)}\Big{(}e_{j}^{(2r+s-1-t)}-(-1)^{s+s_{j,j+1}}f_{j}^{(2r+s-1-t)}\Big{)}$
$\displaystyle=$
$\displaystyle(\delta_{i,j}-\delta_{i,j+1})\Big{(}\sum_{t=0}^{r-1}d_{i}^{(2t)}\widehat{e}_{j}^{(2r+s-1-2t)}+\sum_{t=0}^{r-1}d_{i}^{(2t+1)}\widecheck{e}_{j}^{(2r+s-2t-2)}\Big{)}.$

Projecting to
$R(y_{n}(\sigma),\tau)=y_{n}(\sigma)^{\tau}/\widecheck{y}_{n}(\sigma)^{\tau}$
we see that the elements (3.79) satisfy (3.88).

Note that (3.43) implies that

(3.96)
$\displaystyle\\{e_{i}^{(r)},f_{i}^{(s)}\\}=\\{e_{i}^{(s)},f_{i}^{(r)}\\}\text{
for }r,s>s_{i,i+1}.$

Together with (3.46) and (3.47) we deduce for all $r<s$

$\displaystyle\\{\widehat{e}_{i}^{(r)},\widehat{e}_{i}^{(s)}\\}$
$\displaystyle=$
$\displaystyle\\{e_{i}^{(r)},e_{i}^{(s)}\\}+(-1)^{r+s}\\{f_{i}^{(r)},f_{i}^{(s)}\\}+(-1)^{s+s_{i,i+1}}\\{e_{i}^{(r)},f_{i}^{(s)}\\}-(-1)^{r+s_{i,i+1}}\\{e_{i}^{(s)},f_{i}^{(r)}\\}$
$\displaystyle=$
$\displaystyle\sum_{t=r}^{s-1}e_{i}^{(t)}e_{i}^{(r+s-1-t)}-\sum_{t=r}^{s-1}f_{i}^{(r+s-1-t)}f_{i}^{(t)}-((-1)^{s+s_{i,i+1}}-(-1)^{r+s_{i,i+1}})\\{e_{i}^{(r)},f_{i}^{(s)}\\}$

Substituting in (3.76) we see that

$e_{i}^{(t)}e_{i}^{(r+s-1-t)}-f_{i}^{(r+s-1-t)}f_{i}^{(t)}=\frac{1}{4}(1-(-1)^{r+s-1})\widehat{e}_{i}^{(t)}\widehat{e}_{i}^{(r+s-1-t)}+\widecheck{y}_{n}(\sigma)^{\tau}.$

Combining with (3.43) we have now deduced (3.89) and (3.90).

Using (3.2), (3.43), (3.48), (3.49) we calculate

$\displaystyle\\{\widehat{e}_{i}^{(r+1)},\widehat{e}_{i+1}^{(s)}\\}-\\{\widehat{e}_{i}^{(r)},\widehat{e}_{i+1}^{(s+1)}\\}$
$\displaystyle=$
$\displaystyle\\{e_{i}^{(r+1)},e_{i+1}^{(s)}\\}-\\{e_{i}^{(r)},e_{i+1}^{(s+1)}\\}$
$\displaystyle+(-1)^{r+s+1+s_{i,i+2}}(\\{f_{i}^{(r+1)},f_{i+1}^{(s)}\\}-\\{f_{i}^{(r)},f_{i+1}^{(s+1)}\\})$
$\displaystyle=$ $\displaystyle
e_{i}^{(r)}e_{i+1}^{(s)}+(-1)^{r+s+s_{i,i+2}}f_{i+1}^{(s)}f_{i}^{(r)}$

Substituting in (3.76) we see that this is equal to the right hand side of
(3.91) modulo $\widecheck{y}_{n}(\sigma)^{\tau}$.

Finally take $|i-j|=1$. Expanding in terms of generators (3.41) and applying
relations (3.43), (3.52), (3.53) together with the Jacobi identity and (3.96)
we obtain

$\displaystyle\\{\widehat{e}_{i}^{(r)},\\{\widehat{e}_{i}^{(s)},\widehat{e}_{j}^{(t)}\\}\\}+\\{\widehat{e}_{i}^{(s)},\\{\widehat{e}_{i}^{(r)},\widehat{e}_{j}^{(t)}\\}\\}$
$\displaystyle\ \ \ \
=(-1)^{s+t+s_{i,i+1}+s_{j,j+1}}\\{e_{i}^{(r)},\\{f_{i}^{(s)},f_{j}^{(t)}\\}\\}+(-1)^{r+t+s_{i,i+1}+s_{j,j+1}}\\{e_{i}^{(s)},\\{f_{i}^{(r)},f_{j}^{(t)}\\}\\}$
$\displaystyle\ \ \ \ \ \ \ \ \
+(-1)^{r+s_{i,i+1}}\\{f_{i}^{(r)},\\{e_{i}^{(s)},e_{j}^{(t)}\\}\\}+(-1)^{s+s_{i,i+1}}\\{f_{i}^{(s)},\\{e_{i}^{(r)},e_{j}^{(t)}\\}\\}$
$\displaystyle\ \ \ \
=(-1)^{s+t+s_{i,i+1}+s_{j,j+1}}(1+(-1)^{r+s})\\{\\{e_{i}^{(r)},f_{i}^{(s)}\\},f_{j}^{(t)}\\}-(-1)^{r+s_{i,i+1}}(1+(-1)^{r+s})\\{\\{e_{i}^{(r)},f_{i}^{(s)}\\},e_{j}^{(t)}\\}$

If $r+s$ is odd this vanishes, which proves (3.93). Now assume that $r+s=2m$
is even. Using (3.43) and Jacobi the last line of the previous equation
reduces to

$\displaystyle
2(-1)^{s+s_{i,i+1}-1}\big{\\{}\\{e_{i}^{(r)},f_{i}^{(r)}\\},\widecheck{e}_{j}^{(t)}\big{\\}}$
$\displaystyle
2(-1)^{s+s_{i,i+1}-1}\big{\\{}\sum_{m_{1}=0}^{2m-1}d_{i+1}^{(2m-1-m_{1})}\widetilde{d}_{i}^{(m_{1})},\widecheck{e}_{j}^{(t)}\big{\\}}$
$\displaystyle\ \ \ \
=2(-1)^{s+s_{i,i+1}-1}\sum_{m_{1}=0}^{2m-1}\big{(}d_{i+1}^{(2m-1-m_{1})}\\{\widetilde{d}_{i}^{(m_{1})},\widecheck{e}_{j}^{(t)}\\}+\\{d_{i+1}^{(2m-1-m_{1})},\widecheck{e}_{j}^{(t)}\\}\widetilde{d}_{i}^{(m_{1})}\big{)}.$

Using the fact that $d_{i}^{(2r-1)}\in\widecheck{y}_{n}(\sigma)^{\tau}$ for
all $i,r$ we simplify this expression modulo
$\widecheck{y}_{n}(\sigma)^{\tau}$ to get

$\displaystyle
2(-1)^{s+s_{i,i+1}-1}\left(\sum_{m_{1}=0}^{m-1}d_{i+1}^{(2m-2m_{1}-2)}\\{\widetilde{d}_{i}^{(2m_{1}+1)},\widecheck{e}_{j}^{(t)}\\}+\sum_{m_{1}=0}^{m-1}\\{d_{i+1}^{(2(m-m_{1})-1)},\widecheck{e}_{j}^{(t)}\\}\widetilde{d}_{i}^{(2m_{1})}\right)+\widecheck{y}_{n}(\sigma)^{\tau}$

Finally using (3.84) and (3.85) this expression coincides with the right hand
side of (3.94).

We have shown that the generators (3.86) satisfy (3.87)–(3.94), and it remains
to show that these are a complete set of relations.

Let $\widehat{R}(y_{n}(\sigma),\tau)$ denote the Poisson algebra with
generators (3.86) and relations (3.87)–(3.94). We have shown that there is a
Poisson homomorphism $\widehat{R}(y_{n}(\sigma),\tau)\twoheadrightarrow
R(y_{n}(\sigma),\tau)$ sending the elements (3.86) to the elements (3.79) with
the same names. To complete the proof we show that this map sends a spanning
set to a basis.

We define a loop filtration on $\widehat{R}(y_{n}(\sigma),\tau)=\bigcup_{i\geq
0}\operatorname{F}_{i}\widehat{R}(y_{n}(\sigma),\tau)$ by placing
$\theta_{i}^{(r)},\eta_{i}^{(r)}$ in degree $r-1$ and the bracket in degree
$-1$. Examining the top filtered degree pieces of the relations (3.87)–(3.94)
with respect to the loop filtration, we see from Theorem 3.3 that there is a
surjective Poisson homomorphism
$S(\mathfrak{c}_{n}(\sigma)^{\tau})\twoheadrightarrow\operatorname{gr}\widehat{R}(y_{n}(\sigma),\tau)$.
We deduce that $\operatorname{gr}\widehat{R}(y_{n}(\sigma),\tau)$ is generated
as a commutative algebra by elements
$\theta_{i,j}^{(r)}+\operatorname{F}_{r-2}\widehat{R}(y_{n}(\sigma),\tau),\eta_{i}^{(2r)}+\operatorname{F}_{r-2}\widehat{R}(y_{n}(\sigma),\tau)$
with indexes varying in the same ranges as (3.80). Again the elements
$\theta_{i,j}^{(r)}\in\widehat{R}(y_{n}(\sigma),\tau)$ are defined via the
recursion (3.83). By a standard filtration argument we deduce that the
elements of $\widehat{R}(y_{n}(\sigma),\tau)$ with the same names as (3.80)
generate $\widehat{R}(y_{n}(\sigma),\tau)$ as a commutative algebra. We have
shown that $\widehat{R}(y_{n}(\sigma),\tau)\twoheadrightarrow
R(y_{n}(\sigma),\tau)$ maps a spanning set to a basis. This completes the
proof. ∎

Let $\operatorname{gr}R(y_{n}(\sigma),\tau)$ denote the graded algebra for the
loop filtration. In the last paragraph of the proof of Theorem 3.7 we obtained
the following important result.

######  Corollary 3.8.

There is a Poisson isomorphism
$S(\mathfrak{c}_{n}(\sigma)^{\tau})\overset{\sim}{\longrightarrow}\operatorname{gr}R(y_{n}(\sigma),\tau)$
given by

(3.99)
$\displaystyle\begin{array}[]{lcl}\theta_{i;r-1}\longmapsto\theta_{i}^{(r)}+\operatorname{F}_{r-2}y_{n}(\sigma)&\text{
for }&1\leq i<n,\ r\in\mathbb{Z}_{>s_{i,i+1}};\\\
\eta_{i;2r-1}\longmapsto\eta_{i}^{(2r)}+\operatorname{F}_{2r-2}y_{n}(\sigma)&\text{
for }&1\leq i\leq n,\ r\in\mathbb{Z}_{>0}.\end{array}$

###  4\. Finite $W$-algebras for classical Lie algebras

In this section $\mathbb{C}$ continues to be an algebraically closed field of
characteristic zero, and we fix the following additional notation:

  * •

$N>0$ is an integer and $\varepsilon\in\\{\pm 1\\}$ such that
$\varepsilon^{N}=1$.

  * •

$\mathfrak{g}=\mathfrak{gl}_{N}(\mathbb{C})$ and
$\mathfrak{g}_{0}\subseteq\mathfrak{g}$ is a classical Lie subalgebra such
that

$\mathfrak{g}_{0}\cong\left\\{\begin{array}[]{rc}\mathfrak{so}_{N}&\text{ if
}\varepsilon=1,\\\ \mathfrak{sp}_{N}&\text{ if
}\varepsilon=-1.\end{array}\right.$

  * •

$G=\operatorname{GL}_{N}(\mathbb{C})$ and $G_{0}\subseteq G$ is the connected
algebraic subgroup satisfying $\mathfrak{g}_{0}=\operatorname{Lie}(G_{0})$.

  * •

$\kappa:\mathfrak{g}_{0}\times\mathfrak{g}_{0}\to\mathbb{C}$ is the trace form
associated to the natural representation $\mathbb{C}^{N}$ of $G$.

####  4.1. Partitions for nilpotent orbits in classical Lie algebras

We refer to the $G_{0}$-orbits in $\mathfrak{g}_{0}$ consisting of nilpotent
elements as nilpotent orbits. Nilpotent $G$-orbits in $\mathfrak{g}$ are
classified by partitions, corresponding to the sizes of Jordan blocks, and for
each nilpotent $G$-orbit $\mathcal{O}_{\lambda}\subseteq\mathfrak{g}$ the
intersection $\mathcal{O}_{\lambda}\cap\mathfrak{g}_{0}$ is a union of either
one or two nilpotent $G_{0}$-orbits. Therefore an approximate classification
of nilpotent $G_{0}$-orbits is achieved by describing the partitions
$\lambda\vdash N$ for which
$\mathcal{O}_{\lambda}\cap\mathfrak{g}_{0}\neq\emptyset$. The set of such
partitions is denoted $\mathcal{P}_{\varepsilon}(N)$, and they are
characterised as follows.

######  Lemma 4.1.

Let $\lambda=(\lambda_{1},...,\lambda_{n})\vdash N$. Then
$\lambda\in\mathcal{P}_{\varepsilon}(N)$ if and only if there exists an
involution $i\mapsto i^{\prime}$ on the set $\\{1,...,n\\}$ such that:

  1. (1)

$\lambda_{i}=\lambda_{i^{\prime}}$ for all $i=1,...,n$;

  2. (2)

$i=i^{\prime}$ if and only if $\epsilon(-1)^{\lambda_{i}}=1$;

  3. (3)

$i^{\prime}\in\\{i-1,i,i+1\\}$. $\hfill\qed$

Whenever we choose $\lambda\in\mathcal{P}_{\varepsilon}(N)$ we will assume
that a choice of involution has been fixed in accordance the Lemma. We also
adopt the convention that our partitions are non-decreasing
$\lambda_{1}\leq\cdots\leq\lambda_{n}$. For completeness we mention that
$\lambda\in\mathcal{P}_{\varepsilon}(N)$ corresponds to one orbit unless
$\varepsilon=1$ and all parts of $\lambda$ are even, which case there are two
$G_{0}$-orbits of nilpotent elements of $\mathfrak{g}_{0}$ with Jordan blocks
given by $\lambda$ and these are permuted by the outer automorphism group of
$\mathfrak{g}_{0}$, see [CM93, Section 5.1] for example.

####  4.2. The Dynkin pyramid and the centraliser

Choose $\lambda=(\lambda_{1},...,\lambda_{n})\in\mathcal{P}(N)$. We recalled
the notion of a shift matrix in Section 3.2, following [BK06]. The symmetric
shift matrix for $\lambda$ is the even, symmetric shift matrix
$\sigma=(s_{i,j})_{1\leq i,j\leq n}$ defined by

(4.1) $\displaystyle s_{i,j}:=\frac{|\lambda_{i}-\lambda_{j}|}{2}$

For $1\leq i\leq n$ and $1\leq j\leq\lambda_{i}$ we let $b_{i,j}$ denote the
$j$th box in the $i$th row and we identify $\mathbb{C}^{N}$ with the vector
space spanned by the symbols $b_{i,j}$. The general linear Lie algebra
$\mathfrak{g}=\mathfrak{gl}_{N}$ has a basis

(4.2) $\\{e_{i,j;k,l}\mid 1\leq i,k\leq n,\ 1\leq j\leq\lambda_{i},\ 1\leq
l\leq\lambda_{k}\\}$

where $e_{i,j;k,l}b_{r,s}=\delta_{k,r}\delta_{l,s}e_{i,j;r,s}$, so that

(4.3)
$\displaystyle[e_{i_{1},j_{1};k_{1},l_{1}},e_{i_{2},j_{2};k_{2},l_{2}}]=\delta_{k_{1},i_{2}}\delta_{l_{1},j_{2}}e_{i_{1},j_{1};k_{2},l_{2}}-\delta_{k_{2},i_{1}}\delta_{l_{2},j_{1}}e_{i_{2},j_{2};k_{1},l_{1}}$

We pick the nilpotent element in $\mathfrak{g}(2)$ by the rule

(4.4) $\displaystyle e:=\sum_{i=1}^{n}\sum_{j=1}^{\lambda_{i}-1}e_{i,j;i,j+1}$

It has Jordan blocks of sizes $\lambda_{1},...,\lambda_{n}$. We also define a
semisimple element $h\in\mathfrak{gl}_{N}$ by

(4.5) $\displaystyle
h:=\sum_{i=1}^{n}\sum_{j=1}^{\lambda_{i}}(\lambda_{i}-1-2j)e_{i,j;i,j}$

It is easy to see that the pair $\\{e,h\\}$ can be completed to an
$\mathfrak{sl}_{2}$ triple $\\{e,h,f\\}$. We refer to the grading
$\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}}\mathfrak{g}(i)$ induced by
$\operatorname{ad}(h)$ as the Dynkin grading for $e$. It is well-known that
this is a good grading in the sense of [BG07]. It satisfies

(4.6) $\displaystyle\deg(e_{i,j;k,l})=2(l-j)+\lambda_{i}-\lambda_{k}.$

For $1\leq i,k\leq n$ and
$r=s_{i,k},s_{i,k}+1,...,s_{i,k}+\min(\lambda_{i},\lambda_{k})-1$, we define
elements

(4.7) $\displaystyle
c_{i,k}^{(r)}=\sum_{2r=2(l-j)+\lambda_{i}-\lambda_{k}}e_{i,j;k,l}$

The following fact is well known. See [BK06, Lemma 7.3] where the notation
$c_{i,j}^{(r)}$ differs from ours by a shift in $r$.

######  Lemma 4.2.

The centraliser $\mathfrak{g}^{e}$ has basis

(4.8) $\displaystyle\\{c_{i,k}^{(r)}\mid 1\leq i,k\leq n,\
r-s_{i,k}=0,1,...,\min(\lambda_{i},\lambda_{k})-1\\}$

with Lie bracket

(4.9)
$[c_{i,j}^{(r)},c_{k,l}^{(s)}]=\delta_{j,k}c_{i,k}^{(r+s)}-\delta_{i,l}c_{k,j}^{(r+s)}.$

Furthermore $\mathfrak{g}^{e}$ is a Dynkin graded Lie subalgebra with
$c_{i,k}^{(r)}$ lying in degree $2r$. $\hfill\qed$

####  4.3. Symplectic and orthogonal subalgebras

In this Section we assume that
$\lambda=(\lambda_{1},...,\lambda_{n})\in\mathcal{P}_{\varepsilon}(N)$ and
that the involution on $\\{1,...,n\\}$ coming from Lemma 4.1 is trivial.

Consider the matrix

(4.10) $\displaystyle
J:=\sum_{i=1}^{n}\sum_{j=1}^{\lambda_{i}}(-1)^{j}e_{i,j;i,\lambda_{i}+1-j}$

This block diagonal matrix can be described as follows. For each index
$i=1,...,n$ there is a block of size $\lambda_{i}$, each of these blocks has
entries $\pm 1$ on the antidiagonal and zeroes elsewhere. Thus when
$\varepsilon=1$ the matrix $J$ is symmetric and when $\varepsilon=-1$ it is
anti-symmetric. It follows that

(4.11) $\displaystyle J^{-1}=\varepsilon J.$

Now define an involution of $\mathfrak{gl}_{N}$ by the rule

(4.12) $\displaystyle\tau:X\mapsto-J^{-1}X^{\top}J$

######  Lemma 4.3.

For all admissible indexes we have:

  1. (i)

$\tau(e_{i,j;k,l})=(-1)^{j-l-1}e_{k,\lambda_{k}+1-l;i,\lambda_{i}+1-j}$;

  2. (ii)

$\tau(e)=e$ and $\tau(h)=h$;

  3. (iii)

$\tau(c_{i,k}^{(r)})=(-1)^{r-\frac{\lambda_{k}-\lambda_{i}}{2}-1}c_{k,i}^{(r)}.$

###### Proof.

Using (4.11) and multiplying matrices we have
$\tau(e_{i,j;k,l})=-\varepsilon(-1)^{\lambda_{k}+j-l}e_{k,\lambda_{k}+1-l;i,\lambda_{i}+1-j}$
and so (i) follows from Lemma 4.1(2). Now (ii) and (iii) follow by applying
(i) to (4.4), (4.5) and (4.7). ∎

We decompose $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ into
eigenspaces for $\tau$ so that $\mathfrak{g}_{0}$ is the space of
$\tau$-invariants. Then we have

(4.15)
$\displaystyle\mathfrak{g}_{0}\cong\left\\{\begin{array}[]{cl}\mathfrak{so}_{N}&\text{
if }\varepsilon=1;\\\ \mathfrak{sp}_{N}&\text{ if
}\varepsilon=-1.\par\end{array}\right.$

By Lemma 4.3(ii) we have $\\{e,h\\}\subseteq\mathfrak{g}_{0}$, and
$\mathfrak{g}_{1}$ is a $\mathfrak{g}_{0}$-module. Write $G_{0}$ for the
connected component of the group of elements $g\in\operatorname{GL}_{N}$ such
that $gJg^{\top}=J$. This is a classical group satisfying
$\operatorname{Lie}(G_{0})=\mathfrak{g}_{0}$.

As $\tau$ preserves $\mathfrak{g}(-2)$ we also have that
$\mathfrak{g}(-2)=\mathfrak{g}_{0}(-2)\oplus\mathfrak{g}_{1}(-2)$. Writing
$f=f_{1}+f_{2}$ with $f_{1}\in\mathfrak{g}_{0}(-2)$ and
$f_{2}\in\mathfrak{g}_{1}(-2)$ and using the fact that $h\in\mathfrak{g}$ we
deduce that $h=[e,f_{1}]$ and $[e,f_{2}]=0$. Since the Dynkin grading is good
for $e$ in $\mathfrak{g}$ this yields $f_{2}=0$. We conclude that the
$\mathfrak{sl}_{2}$-triple $\\{e,h,f\\}$ is contained in $\mathfrak{g}_{0}$.

Our next result explains the relationship between the shifted current algebras
and the centralisers described above.

######  Lemma 4.4.

  1. (1)

There is a surjective Lie algebra homomorphism

$\displaystyle\mathfrak{c}_{n}(\sigma)\relbar\joinrel\twoheadrightarrow\mathfrak{g}^{e};$
$\displaystyle e_{i,j}t^{r}\longmapsto c_{i,j}^{(r)}$

where $c_{i,j}^{(r)}:=0$ for $r\geq s_{i,j}+\lambda_{1}$. The kernel is
Poisson generated by $e_{1,1}t^{r}$ with $r\geq\lambda_{1}$.

  2. (2)

If $\sigma$ is even and symmetric the homomorphism from (1) restricts to
$\mathfrak{c}_{n}(\sigma)^{\tau}\twoheadrightarrow\mathfrak{g}^{e}_{0}$.

     * (i)

For $\varepsilon=1$ the kernel is Poisson generated by $\\{\eta_{1;2r-1}\mid
2r-1\geq\lambda_{1}\\}$.

     * (ii)

For $\varepsilon=-1$ the kernel is Poisson generated by those same elements
along with

$\\{\theta_{i;\lambda_{i}+s_{i,i+1}}\mid i=1,...,n-1\\}.$

###### Proof.

Part (1) is explained in [GT19c, Lemma 2.6]. For the first claim in part (2)
compare (3.24) with Lemma 4.3(iii) to see that the map is $\tau$-equivariant.

We go on to describe the kernel in (2). First of all observe that the elements
listed there are $\tau$-fixed elements of the kernel of
$\mathfrak{c}_{n}(\sigma)\twoheadrightarrow\mathfrak{g}^{e}$; for
$\varepsilon=-1$ we use (3.18), (3.18). Now observe that
$\mathfrak{g}_{0}^{e}$ has a spanning set consisting of elements of the form
$c_{i,j}^{(r)}+\tau(c_{i,j}^{(r)})$. By Lemma 4.2 we can show that the
elements in (2) generate the kernel by checking that the ideal $\mathfrak{i}$
which they generate contains

(4.16) $\displaystyle\\{\eta_{i;r}\mid i=1,...,n,\
r\geq\lambda_{i}\\}\cup\\{\theta_{i;r}\mid i=1,..,n-1,\ r\geq
s_{i,i+1}+\lambda_{i}\\}.$

First take $\varepsilon=1$ so that $\lambda_{1}$ is odd, by Lemma 4.1. Using
relation (3.88) we see that $\mathfrak{i}$ contains
$\\{\eta_{1;\lambda_{1}},\theta_{1;r}\\}=\theta_{1;\lambda_{1}+r}$ for $r\geq
s_{1,2}$, whilst (3.91) shows that
$\eta_{1;\lambda_{2}+r}-\eta_{2;\lambda_{2}+r}\in\mathfrak{i}$ for all $r\geq
0$. In particular $\eta_{2;\lambda_{2}+r}\in\mathfrak{i}$ for $r\geq 0$. By
Theorem 3.3 the subalgebra of $\mathfrak{c}_{n}(\sigma)^{\tau}$ generated by
$\theta_{i,r},\eta_{i;r}$ with $2\leq i$ is isomorphic to a current Lie
algebra of smaller rank so the description of the kernel follows by induction.

For $\varepsilon=-1$, $\lambda_{1}$ is even and the ideal $\mathfrak{i}$
contains
$\\{\eta_{1;\lambda_{1}+1},\theta_{1;s}\\}=\theta_{1;\lambda_{1}+1+s}$ for
$s\geq s_{1,2}$. Together with $\theta_{1;\lambda_{1}+s_{1,2}}$ this gives all
elements of the form $\theta_{1;r}$ listed in (4.16). Now the induction
proceeds in the same was as the case $\varepsilon=1$. ∎

####  4.4. Generators of the $W$-algebra

Here we recall formulas for generators of $S(\mathfrak{gl}_{N},e)$, due to
Brundan and Kleshchev [BK06]. Their notation is slightly different, however it
is a simple exercise to translate between the two settings. In this Section we
continue to assume that the involution on $\\{1,...,n\\}$ coming from Lemma
4.1 is trivial. This implies that $\lambda$ is even.

Now for $1\leq i,j\leq n$, $0\leq x<n$ and $r>0$, we let

(4.17)
$t_{i,k;x}^{(r)}:=\sum_{s=1}^{r}(-1)^{r-s}\sum_{\begin{subarray}{c}(i_{m},j_{m},k_{m},l_{m})\\\
m=1,...,s\end{subarray}}(-1)^{\hash\\{q=1,\dots,s-1\mid k_{q}\leq
x\\}}e_{i_{1},j_{1};k_{1},l_{1}}\cdots e_{i_{s},j_{s};k_{s},l_{s}}\in
S(\mathfrak{g}(\operatorname{\geq\\!0}))$

where the sum is taken over all indexes such that for $m=1,...,s$

(4.18) $\displaystyle 1\leq i_{1},...,i_{s},k_{1},...,k_{s}\leq n,1\leq
j_{m}\leq\lambda_{i_{m}}\text{ and }1\leq l_{m}\leq\lambda_{k_{m}}$

satisfying the following six conditions:

  * (a)

$\sum_{m=1}^{s}(2l_{m}-2j_{m}+\lambda_{i_{m}}-\lambda_{k_{m}})=2(r-s)$;

  * (b)

$2l_{m}-2j_{m}+\lambda_{i_{m}}-\lambda_{k_{m}}\geq 0$ for each $m=1,\dots,s$;

  * (c)

if $k_{m}>x$, then $l_{m}<j_{m+1}$ for each $m=1,\dots,s-1$;

  * (d)

if $k_{m}\leq x$ then $l_{m}\geq j_{m+1}$ for each $m=1,\dots,s-1$;

  * (e)

$i_{1}=i$, $k_{s}=k$;

  * (f)

$k_{m}=i_{m+1}$ for each $m=1,\dots,s-1$.

Keeping $r>0$ fixed and choosing $s\in\\{1,...,r\\}$ we make the notation
$\mathcal{X}_{i,k}^{(r,s)}$ for the set of ordered sets
$(i_{m},j_{m},k_{m},l_{m})_{m=1}^{s}$ in the range (4.18) satisfying the
conditions (a)–(f) and consider the map

(4.19)
$\displaystyle\upsilon(i_{m},j_{m},k_{m},l_{m}):=(k_{s+1-m},\lambda_{k_{s+1-m}}+1-l_{s+1-m},i_{s+1-m},\lambda_{i_{s+1-m}}+1-j_{s+1-m})$

defined on ordered sets in the range (4.18).

######  Lemma 4.5.

The map $\upsilon$ defines a bijection
$\mathcal{X}_{i,k}^{(r,s)}\overset{1\text{-}1}{\longrightarrow}\mathcal{X}_{k,i}^{(r,s)}.$

###### Proof.

The index bounds (4.18) ensure that $\mathcal{X}_{i,k}^{(r,s)}$ is finite.
Since $\upsilon^{2}$ is the identity map it will suffice to show that
$\upsilon$ is well defined
$\mathcal{X}_{i,k}^{(r,s)}\to\mathcal{X}_{k,i}^{(r,s)}$. We fix
$(i_{m},j_{m},k_{m},l_{m})_{m=1}^{s}\in\mathcal{X}_{i,k}^{(r,s)}$ and check
that conditions (a)–(f) hold for $\upsilon(i_{m},j_{m},k_{m},l_{m})$. For
example,

$\displaystyle\sum_{m=1}^{s}\big{(}2(\lambda_{i_{s+1-m}}+1-j_{s+1-m})-2(\lambda_{l_{s+1-m}}+1-k_{s+1-m})+(\lambda_{k_{s+1-m}}-\lambda_{i_{s+1-m}})\big{)}\
\ \ \ \ \ \ \ $ $\displaystyle\ \ \ \ \ \ \ \ \ \
=\sum_{m=1}^{s}(2l_{m}-2j_{m}+\lambda_{i_{m}}-\lambda_{k_{m}})=2(r-s)$

This shows that (a) for $(i_{m},j_{m},k_{m},l_{m})_{m=1}^{s}$ implies (a) for
$\upsilon(i_{m},j_{m},k_{m},l_{m})_{m=1}^{s}$. Conditions (b)–(f) can be
checked similarly. ∎

Now we define a map $\theta:\mathfrak{g}^{e}\to
S(\mathfrak{g}(\operatorname{\geq\\!0}))$ by

(4.23) $\displaystyle\begin{array}[]{rcl}c_{i,i}^{(r)}\longmapsto
t_{i,i;i-1}^{(r-1)}&&\text{ for }i=1,...,n\text{ and }0\leq r<\lambda_{i};\\\
c_{i,k}^{(r)}\longmapsto t_{i,k;i}^{(r-1)}&&\text{ for }1\leq i<k\leq n\text{
and }r-s_{i,k}=0,...,\min(\lambda_{i},\lambda_{k})-1;\\\
c_{k,i}^{(r)}\longmapsto t_{k,i;i}^{(r-1)}&&\text{ for }1\leq i<k\leq n\text{
and }r-s_{j,k}=0,...,\min(\lambda_{i},\lambda_{k})-1.\end{array}$

Comparing (4.6) with Lemma 4.2 and Lemma 4.3 we see that $\tau$ induces
involutions on $\mathfrak{g}^{e}$ and
$S(\mathfrak{g}(\operatorname{\geq\\!0}))$, and these will all be denoted
$\tau$. The hardest part of the following result is the assertion that the
image of the map $\theta$ described in (4.23) lies in $S(\mathfrak{g},e)$,
which follows from a result of Brundan and Kleshchev. We add to this the
observation that $\theta$ is $\tau$-equivariant.

######  Proposition 4.6.

$\theta:\mathfrak{g}^{e}\to S(\mathfrak{g},e)$ is $\tau$-equivariant. In
particular we have

(4.24)
$\displaystyle\tau(t_{i,k;m}^{(r-1)})=(-1)^{r-\frac{\lambda_{k}-\lambda_{i}}{2}-1}t_{k,i;m}^{(r-1)}$
$\displaystyle\text{ for }1\leq i\leq k\leq n\text{ and all }m,r.$

###### Proof.

The assertion that the image of $\theta$ lies in $S(\mathfrak{g},e)$ is an
immediate consequence of [BK06, Corollary 9.4], upon taking the top graded
term with respect to the Kazhdan filtration of the finite $W$-algebra (see
also [GT19b, §4] for a short summary). Thanks to Lemma 4.3(iii) the
$\tau$-equivariance will follow from (4.24). Fix indexes $1\leq i\leq k\leq
n,\ 0\leq x<n$ and $0<r$. Choose $s\in\\{1,...,r\\}$. Let
$(i_{m},j_{m},k_{m},l_{m})_{m=1}^{s}\in\mathcal{X}_{i,i}^{(r,s)}$. Thanks to
Lemma 4.3(i) we have

(4.25) $\displaystyle\tau(\prod_{m=1}^{s}e_{i_{m},j_{m};k_{m},l_{m}})$
$\displaystyle=$
$\displaystyle(-1)^{\sum_{m=1}^{s}(j_{m}-l_{m}+1)}\prod_{t=1}^{s}e_{\upsilon(i_{m},j_{m},k_{m},s_{m})}$

Conditions (a), (e) (f) imply that
$\sum_{m}(j_{m}-l_{m}+1)=\frac{\lambda_{i_{1}}-\lambda_{k_{s}}}{2}-r.$ Hence
the sign on the right hand side of (4.25) is equal to
$(-1)^{r-\frac{\lambda_{k}-\lambda_{i}}{2}}$. Furthermore condition (f)
ensures that $\hash\\{m=1,...,s-1\mid k_{m}\leq x\\}=\hash\\{m=2,...,s\mid
i_{m}\leq x\\}.$ Combining these observations together with Lemma 4.5 we have

$\displaystyle\tau(t_{i,k;x}^{(r)})$ $\displaystyle=$
$\displaystyle\sum_{s=1}^{r}(-1)^{r-s}\sum_{(i_{m},j_{m},k_{m},l_{m})_{m}\in\mathcal{X}_{i,k}^{(r,s)}}(-1)^{\hash\\{m=1,...,s-1\mid
k_{m}\leq x\\}}\tau(\prod_{m}e_{i_{m},j_{m},k_{m},l_{m}})$ $\displaystyle=$
$\displaystyle(-1)^{r-\frac{\lambda_{k}-\lambda_{i}}{2}}\sum_{s=1}^{r}(-1)^{r-s}\sum_{(i_{m},j_{m},k_{m},l_{m})_{m}\in\mathcal{X}_{k,i}^{(r,s)}}(-1)^{\hash\\{m=1,...,s-1\mid
k_{m}\leq x\\}}\prod_{m}e_{i_{m},j_{m},k_{m},l_{m}}$ $\displaystyle=$
$\displaystyle(-1)^{r-\frac{\lambda_{k}-\lambda_{i}}{2}}t_{k,i;i-1}^{(r)}$

This completes the proof. ∎

######  Remark 4.7.

Comparing the linear terms appearing in (4.17) with the basis (4.7) for
$\mathfrak{g}^{e}$ one can check that $\theta$ is a Poincaré–Birkhoff–Witt map
satisfying the properties of Theorem 2.4.

####  4.5. The semiclassical Brundan–Kleshchev homomorphism

Recall that in (3.70) we defined a Poisson involution of $y_{n}(\sigma)$ which
is denoted $\tau$.

######  Proposition 4.8.

There is a unique $\tau$-equivariant surjective Poisson homomorphism

(4.26) $\displaystyle\varphi:y_{n}(\sigma)\relbar\joinrel\twoheadrightarrow
S(\mathfrak{g},e)$

determined by

(4.30) $\displaystyle\begin{array}[]{rcl}d_{i}^{(r)}\longmapsto
t_{i,i;i-1}^{(r)}&\text{ for }&1\leq i\leq n,\ r\in\mathbb{Z}_{>0};\\\\[3.0pt]
e_{i}^{(r)}\longmapsto t_{i,i+1;i}^{(r)}&\text{ for }&1\leq i<j\leq n,\
r-s_{i,j}\in\mathbb{Z}_{>0};\\\\[3.0pt] f_{i}^{(r)}\longmapsto
t_{i+1,i;i}^{(r)}&\text{ for }&1\leq i<j\leq n,\
r-s_{j,i}\in\mathbb{Z}_{>0}.\end{array}$

The kernel is the Poisson ideal generated by $\\{d_{1}^{(r)}\mid
r>\lambda_{1}\\}$. The canonical grading on $y_{n}(\sigma)$ in Theorem 3.4
corresponds to the Kazhdan grading on $S(\mathfrak{g},e)$.

###### Proof.

The fact that (4.30) defines a graded Poisson homomorphism follows from [BK06,
Theorem 10.1] upon taking the associated graded map with respect to the
canonical and Kazhdan filtrations. The $\tau$-equivariance can be checked by
comparing Proposition 4.6 with formulas (3.70).

Another consequence of [BK06, Theorem 10.2] is that the elements
$\\{d_{1}^{(r)}\mid r>\lambda_{1}\\}$ lie in the kernel of the homomorphism.
Write $I$ for the Poisson ideal generated by these elements. Combining Theorem
2.4 and Lemma 4.2 we can show that $I=\operatorname{Ker}\varphi$ by
demonstrating that the quotient $y_{n}(\sigma)/I$ is generated as a
commutative algebra by

(4.33) $\displaystyle\begin{array}[]{rcl}&\\{e_{i,j}^{(r)}\mid 1\leq i<j\leq
n,\ s_{i,j}<r\leq s_{i,j}+\lambda_{\min(i,j)}\\}\cup\\{d_{i}^{(r)}\mid 1\leq
i\leq n,\ 0<r\leq\lambda_{i}\\}&\\\ &\cup\\{f_{i,j}^{(r)}\mid 1\leq i<j\leq
n,\ s_{j,i}<r\leq s_{j,i}+\lambda_{\min(i,j)}\\}.&\end{array}$

By Theorem 3.4 we can identify
$S(\mathfrak{c}_{n}(\sigma))=\operatorname{gr}y_{n}(\sigma)$ as Poisson
algebras, with respect to the loop filtration. The associated graded ideal
$\operatorname{gr}I$ contains elements $\\{d_{i;r}\mid r\geq\lambda_{1}\\}$,
and so applying Lemma 4.4(1) we see that $\operatorname{gr}y_{n}(\sigma)/I$ is
generated by by the top graded components of the elements (4.33). Applying a
standard argument on filtered algebras we deduce that $y_{n}(\sigma)/I$ is
generated by the requisite elements. ∎

Recall that the Dirac reduction functor is right exact. Applying Theorem 2.6
we obtain an approximate description of the $W$-algebra for
$\mathfrak{g}_{0}$.

######  Theorem 4.9.

There is a surjective Poisson homomorphism
$\varphi:R(y_{n}(\sigma),\tau)\twoheadrightarrow S(\mathfrak{g}_{0},e)$.

  * •

If $\varepsilon=1$ the kernel of $\varphi$ is Poisson generated by
$\\{\eta_{1}^{(2r)}\mid 2r>\lambda_{1}\\}$.

  * •

If $\varepsilon=-1$ the kernel of $\varphi$ is Poisson generated by

(4.34) $\displaystyle\Big{\\{}\eta_{i}^{(2r)}\mid
2r>\lambda_{1}\Big{\\}}\cup\Big{\\{}\sum_{t=0}^{\lambda_{i}/2}\eta_{i}^{(2t)}\theta_{i}^{(\lambda_{i}-2t+s_{1,2}+1)}\mid
i=1,...,n-1\Big{\\}}.$

###### Proof.

The elements $d_{1}^{(2r)}$ with $2r>\lambda_{1}$ lie in the kernel of
$y_{n}(\sigma)\to S(\mathfrak{g},e)$ and project to $\eta_{1}^{(2r)}$ under
$y_{n}(\sigma)^{\tau}\twoheadrightarrow R(y_{n}(\sigma),\tau)$. Similarly for
$\varepsilon=-1$ we can use (3.84) to see that the elements
$\\{d_{i}^{(\lambda_{i}+1)},\widecheck{e}_{i}^{(s)}\\}+y_{n}(\sigma)^{\tau}=\sum_{t=0}^{\lambda_{i}/2}\eta_{i}^{(2t)}\theta_{i}^{(\lambda_{i}-2t+s_{1,2}+1)}$
lie in the kernel.

To complete the proof we use a loop filtration argument identical to the proof
of Proposition 4.8 to show that the ideal generated by the stated elements is
large enough. For this one can apply Corollary 3.8 and Lemma 4.4(2), and we
leave the details to the reader ∎

Thanks to Lemma 4.2 the Lie algebra $\mathfrak{g}_{0}$ is spanned by elements
$c_{i,j}^{(r)}+\tau(c_{i,j}^{(k)})$. Using (4.17) along with Theorem 2.4 and
Theorem 4.9 we see that $S(\mathfrak{g},e)$ is generated (as a commutative
algebra) by the images of elements

(4.35) $\displaystyle\\{\eta_{i}^{(2r)}\mid 1\leq i\leq n,1\leq
2r\leq\lambda_{i}\\}\cup\\{\theta_{i,j}^{(r)}\mid 1\leq i<j\leq
n,s_{i,j}<r\leq s_{i,j}+\lambda_{i}\\}$

By slight abuse of notation we denote the images by the same names.

######  Corollary 4.10.

If $\mathfrak{m}_{0}$ is the unique graded maximal ideal of
$S(\mathfrak{g}_{0},e)$ then we have a Poisson isomorphism
$S(\mathfrak{g}_{0}^{e})\overset{\sim}{\longrightarrow}\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g}_{0},e)$
to the $\mathfrak{m}_{0}$-adic graded algebra defined by

(4.36) $\displaystyle
c_{i,i+1}^{(r)}+\tau(c_{i,i+1}^{(r)})\longmapsto\theta_{i}^{(r+1)}+\mathfrak{m}_{0}^{2}$
for $\displaystyle 1\leq i<n,\ s_{i,j}\leq r<s_{i,j}+\lambda_{i};$ (4.37)
$\displaystyle
c_{i,i}^{(r)}+\tau(c_{i,i}^{(r)})\longmapsto\eta_{i}^{(r+1)}+\mathfrak{m}_{0}^{2}$
for $\displaystyle 1\leq i\leq n,\ 0\leq r<\lambda_{i}.$

###### Proof.

The graded algebra $\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g}_{0},e)$
is Poisson, and it is straightforward to check, using Theorem 3.7, that (4.36)
gives a Lie algebra homomorphism
$\mathfrak{g}_{0}^{e}\twoheadrightarrow\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g}_{0},e)$.
By the universal property of the Lie–Poisson algebra we get a Poisson
homomorphism
$S(\mathfrak{g}_{0}^{e})\twoheadrightarrow\operatorname{gr}_{\mathfrak{m}_{0}}S(\mathfrak{g}_{0},e)$.
By Theorem 2.4 we see that this map is an isomorphism. ∎

##  Part II One dimensional representations of $W$-algebras

###  5\. Generalities on Poisson schemes

####  5.1. Conic degenerations of affine schemes

We begin the second Part of the paper by recording two basic results, allowing
us to compare the number of irreducible components of certain complex schemes
of finite type. The first records an obvious bound on the dimensions of
components of subschemes, whilst the second allows us to bound the number of
components of a deformation of a reduced conic affine scheme.

We define $\mathcal{P}$ be the set of all finite sequences of all non-negative
integers of arbitrary length. We define a dominance (partial) order on
$\mathcal{P}$ as follows. If $d=(d_{1},...,d_{n})$ and
$d^{\prime}=(d_{1}^{\prime},...,d_{m}^{\prime})$ then we say that
$d>d^{\prime}$ if there is an index $j=1,...,\min(n,m)$ such that
$d_{i}=d^{\prime}_{i}$ for $i=1,...,j$ and $d_{j+1}>d^{\prime}_{j+1}$. Here we
adopt the convention $d_{m+1}^{\prime}=d_{n+1}=0$

If $X$ is a complex scheme of finite type then we write
$X=\bigcup_{i=1}^{l}X_{i}$ for the decomposition into irreducible components,
ordered so that $\dim X_{i}\geq\dim X_{j}$ for $i<j$. We define the dimension
vector of $X$ to be the sequence $d(X):=(\dim X_{1},\dim X_{2},...,\dim
X_{l})\in\mathcal{P}$.

The following useful fact can be checked by considering the dimension vector
of all proper closed subschemes of a scheme $X$.

######  Lemma 5.1.

Let $X,Y$ be complex schemes of finite type with a closed embedding $Y\to X$.
Then $d(X)\geq d(Y)$ with equality if and only if the underlying reduced
schemes are isomorphic.

Let $A$ be a finitely generated $\mathbb{C}$-algebra. We say that
$A=\bigcup_{i\geq 0}\mathcal{F}_{i}A$ is a standard filtration of $A$ if there
is a surjection $\mathbb{C}[x_{1},...,x_{n}]\twoheadrightarrow A$ for some
$n$, along with integers $m_{1},...,m_{n}\geq 0$ such that $\mathcal{F}_{m}A$
is spanned by the image of $\\{x_{1}^{k_{1}}\cdots+x_{n}^{k_{n}}\mid
k_{1}m_{1}+\cdots+k_{n}m_{n}\leq m\\}$. If $X=\operatorname{Spec}(A)$ is the
affine scheme the we refer to
$\mathbb{C}X=\operatorname{Spec}(\operatorname{gr}A)$ as the asymptotic cone
of $X$.

######  Lemma 5.2.

If $\mathbb{C}X$ is reduced then
$\hash\operatorname{Comp}(X)\leq\hash\operatorname{Comp}(\mathbb{C}X)$.

###### Proof.

If $\operatorname{gr}A$ is reduced then so is $A$. We begin by choosing a
presentation for the standard filtration. Let
$R=\mathbb{C}[x_{1},...,x_{n}]=\bigoplus_{i\geq 0}R_{i}$ be a graded
polynomial ring with $x_{i}$ in degree $m_{i}$. Write
$\operatorname{F}_{i}R=\bigoplus_{j=0}^{i}R_{j}$. We let $\phi:R\to A$ be the
homomorphism inducing the standard filtration, set
$I:=\operatorname{Ker}(\phi)$ and denote the minimal prime ideals over $I$ by
$\mathfrak{p}_{1},...,\mathfrak{p}_{m}\subseteq A$, so that
$I=\bigcap_{i}\mathfrak{p}_{i}$. Using [MR01, Proposition 7.6.13], for
example, we see that there is a natural isomorphism $\operatorname{gr}A\cong
R/\operatorname{gr}I$ and we view $X$ and $\mathbb{C}X$ as subschemes of
$\operatorname{Spec}R=\mathbb{A}^{n}$.

For $g\in\operatorname{F}_{i}R$ we write
$\bar{g}=g+\operatorname{F}_{i-1}R\in\operatorname{gr}R$. For any ideal $J$ of
$R$ we write $V(J)\subseteq\mathbb{A}^{n}$ for the corresponding variety of
closed points. We have inclusions
$\operatorname{gr}(J)\operatorname{gr}(K)\subseteq\operatorname{gr}(JK)\subseteq\operatorname{gr}(J)\cap\operatorname{gr}(K)$
for any ideals $J,K\subseteq R$ (see [Pr02, §5.2]), and so

(5.1) $\displaystyle
V(\operatorname{gr}I)=\bigcup_{i=1}^{m}V(\operatorname{gr}\mathfrak{p}_{i}).$

We prove the contrapositive of the lemma, so assume that
$m=\hash\operatorname{Comp}(X)>\hash\operatorname{Comp}(\mathbb{C}X)$. Then, a
fortiori, we must have
$V(\operatorname{gr}\mathfrak{p}_{j})\subseteq\bigcup_{i\neq
j}V(\operatorname{gr}\mathfrak{p}_{i})$ for some $j$, say $j=1$. Equivalently
$\bigcap_{i\neq
1}\sqrt{\operatorname{gr}\mathfrak{p}_{i}}\subseteq\sqrt{\operatorname{gr}\mathfrak{p}_{1}}$,
which implies $\bigcap_{i\neq
1}{\operatorname{gr}\mathfrak{p}_{i}}\subseteq\sqrt{\operatorname{gr}\mathfrak{p}_{1}}$.
Since $I\subsetneq\mathfrak{p}_{2}\mathfrak{p}_{3}\cdots\mathfrak{p}_{m}$ it
follows that
$\operatorname{gr}I\subsetneq\operatorname{gr}(\mathfrak{p}_{2}\cdots\mathfrak{p}_{m})$
and so we may choose $g_{2},...,g_{m}$ with $g_{i}\in\mathfrak{p}_{i}$ such
that $g:=g_{2}\cdots g_{m}$ satisfies $\bar{g}\notin\operatorname{gr}I$. On
the other hand, we have
$\bar{g}\in\operatorname{gr}(\mathfrak{p}_{2}\cdots\mathfrak{p}_{m})\subseteq\bigcap_{i=2}^{m}\operatorname{gr}(\mathfrak{p}_{i})\subseteq\bigcap_{i=1}^{m}\sqrt{\operatorname{gr}(\mathfrak{p}_{i})}=\sqrt{\operatorname{gr}(I)}$,
where the last equality follows from (5.1). Thus we conclude that
$\bar{g}\in\sqrt{\operatorname{gr}(I)}\setminus\operatorname{gr}I$, so that
$\mathbb{C}X$ is not reduced. This completes the proof. ∎

####  5.2. Completions and nilpotent elements of graded algebras

If $\mathfrak{p}\in\operatorname{Spec}(A)$ then, as usual, $A_{\mathfrak{p}}$
denotes the localisation and $A_{\mathfrak{p}}^{\\!\wedge}$ the completion at
the maximal ideal of $A_{\mathfrak{p}}$. The kernel of $A\to A_{\mathfrak{p}}$
is the set of elements annihilating some element of $A\setminus\mathfrak{p}$,
whilst the kernel of $A\to A_{\mathfrak{p}}^{\\!\wedge}$ is
$\bigcap_{k>0}\mathfrak{p}^{k}$. When $\mathfrak{p}$ is a maximal ideal it
follows from Krull’s intersection theorem that these kernels are equal (see
the remark following [AM69, Theorem 10.17]).

Now let $A=\bigoplus_{i\geq 0}A_{i}$ be a finitely generated, connected graded
algebra with unique maximal graded ideal $\mathfrak{m}_{0}$. The connected
grading induces a one parameter family of automorphisms of $A$, and a
$\mathbb{C}^{\times}$-action on $\operatorname{m-Spec}(A)$ contracting to the
unique fixed point $\mathfrak{m}_{0}$. For $a\in A$ make the notation

$\Xi_{a}:=\Big{\\{}\mathfrak{m}\in\operatorname{m-Spec}(A)\mid
a\notin\operatorname{Ker}(A\to A_{\mathfrak{m}}^{\\!\wedge})\Big{\\}}.$

######  Lemma 5.3.

The following hold:

  1. (1)

$\bigcap_{\mathfrak{m}}\bigcap_{i>0}\mathfrak{m}^{i}=0$ where the intersection
is taken over all maximal ideals of $A$.

  2. (2)

If $a\in A_{i}$ for some $i$ then $\Xi_{a}$ is closed and conic for the
contracting $\mathbb{C}^{\times}$-action.

###### Proof.

Since $A$ is finitely generated over $\mathbb{C}$ the intersection
$\bigcap_{\mathfrak{m}\in\operatorname{m-Spec}(A)}\mathfrak{m}=\operatorname{Rad}(A)$
is a graded ideal, however for every $i>0$ we have
$A_{i}\cap\mathfrak{m}_{0}^{i+1}=0$ and so (1) follows.

If $\mathfrak{m}\in\operatorname{m-Spec}(A)$, we remarked above that the
kernel of $A\to A_{\mathfrak{m}}^{\\!\wedge}$ is equal to the kernel of $A\to
A_{\mathfrak{m}}$. Writing $\operatorname{Ann}(a)=\\{b\in A\mid ab=0\\}$, it
follows that $\operatorname{Ker}(A\to A_{\mathfrak{m}}^{\\!\wedge})=\\{a\in
A\mid ab=0\text{ for some }b\notin\mathfrak{m}\\}=\\{a\in
A\mid\operatorname{Ann}(a)\nsubseteq\mathfrak{m}\\}$. We have shown that
$\Xi_{a}=\\{\mathfrak{m}\in\operatorname{m-Spec}(A)\mid\operatorname{Ann}(a)\subseteq\mathfrak{m}\\}$,
which is conic and closed since $\operatorname{Ann}(a)$ is a graded ideal (see
[Pr02, 5.3] for more detail). ∎

######  Lemma 5.4.

Suppose that $A=\bigoplus_{i>0}A_{i}$ is a finitely generated, connected
graded algebra with unique maximal graded ideal $\mathfrak{m}_{0}$. The
following are equivalent:

  1. (i)

$A$ is reduced.

  2. (ii)

$A_{\mathfrak{m}_{0}}^{\\!\wedge}$ is reduced.

  3. (iii)

$A_{\mathfrak{m}}^{\\!\wedge}$ is reduced for every maximal ideal
$\mathfrak{m}\in\operatorname{Spec}(A)$.

Furthermore if $\operatorname{gr}_{\mathfrak{m}_{0}}A$ is reduced then these
equivalent conditions hold.

###### Proof.

We prove (ii) $\Rightarrow$ (i) by contraposition. So suppose that $0\neq
f\in\operatorname{Rad}(A)$ is a nonzero element. Without loss of generality we
may assume $f$ is homogeneous. By Lemma 5.3(1) we have
$\operatorname{Ker}(A\to\prod_{\mathfrak{m}\in\operatorname{m-Spec}(A)}A_{\mathfrak{m}}^{\\!\wedge})=0$,
which means that $f$ maps to a nonzero element of
$A_{\mathfrak{m}}^{\\!\wedge}$ for some $\mathfrak{m}$, implying
$\mathfrak{m}\in\Xi_{f}$. Using Lemma 5.3(2) we see $\Xi_{f}$ is conic and
closed, so it must contain $\mathfrak{m}_{0}$, which implies that
$A_{\mathfrak{m}_{0}}^{\\!\wedge}$ has nilpotent elements. This proves (ii)
$\Rightarrow$ (i).

To see (i) $\Rightarrow$ (iii) we first of all observe that the property of
being reduced is preserved by localisation for any commutative ring, and then
apply [Ma86, §32, Remark 1]. Clearly (iii) $\Rightarrow$ (ii).

We prove that that if $A$ is non-reduced then
$\operatorname{gr}_{\mathfrak{m}_{0}}A$ is non-reduced. Let $0\neq
f\in\operatorname{Rad}(A)$ and suppose (without loss of generality) that $f$
is homogeneous. There are two cases: either
$f\in\mathfrak{m}_{0}^{i}\setminus\mathfrak{m}_{0}^{i+1}$ for some $i>0$, or
$f\in\bigcap_{i>0}\mathfrak{m}_{0}^{i}$. In the first case
$f+\mathfrak{m}_{0}^{i+1}$ is a nonzero nilpotent element of
$\operatorname{gr}_{\mathfrak{m}_{0}}A$, which proves that
$\operatorname{gr}_{\mathfrak{m}_{0}}A$ is non-reduced.

We now show that the second case cannot happen. Suppose
$f\in\bigcap_{i>0}\mathfrak{m}_{0}^{i}=\operatorname{Ker}(A\to
A_{\mathfrak{m}_{0}}^{\\!\wedge})$, or equivalently,
$\mathfrak{m}_{0}\notin\Xi_{f}$. By Lemma 5.3(2) we must have
$\Xi_{f}=\emptyset$ which implies that $f\in\operatorname{Ker}(A\to
A_{\mathfrak{m}}^{\\!\wedge})$ for all maximal ideals $\mathfrak{m}$. By Lemma
5.3(1) we see $f\in\bigcap_{\mathfrak{m}}\bigcap_{i>0}\mathfrak{m}^{i}=0$.
This contradiction completes the proof. ∎

####  5.3. Poisson schemes

If $A$ is a Poisson algebra then the abelianisation $A^{\operatorname{ab}}$ is
the largest Poisson commutative quotient of $A$. It is constructed by
factoring by the ideal generated by $\\{A,A\\}$.

When $X$ is a complex manifold it can be stratified into immersed submanifolds
known as symplectic leaves (see [We83] for an introduction). Therefore the
closed points of a regular complex Poisson scheme can be decomposed into a
disjoint union of symplectic leaves. More generally the closed points of a
singular affine Poisson scheme can be decomposed into symplectic leaves by an
iterative procedure [BG03, 3.5].

The stratification by leaves can be coarsened to a stratification by rank,
which is especially transparent in the case of affine schemes: if $A$ is
finitely generated by elements $x_{1},...,x_{n}$ and carries a Poisson
structure, then we can form the matrix $\pi=(\\{x_{i},x_{j}\\})_{1\leq i,j\leq
n}$ and consider the subschemes $X_{0}\subseteq X_{1}\subseteq\cdots X$ where
$X_{k}$ is defined by the ideal of $A$ generated by all $(k+1)\times(k+1)$
minors of $\pi$.

It follows from [BG03, Proposition 3.6] that the locally closed set
$X_{k}\setminus X_{k-1}$ is the union of all symplectic leaves of dimension
$k$, and it is known as the $k$th rank stratum. In particular $X_{0}$ is the
union of symplectic leaves of dimension zero. Note that
$X_{0}=\operatorname{Spec}(A^{\operatorname{ab}})$ by definition. These
observations prove the following lemma.

######  Lemma 5.5.

The following reduced subschemes of $X$ coincide:

  1. (1)

$\operatorname{Spec}(A^{\operatorname{ab}})_{\operatorname{red}}$;

  2. (2)

The union of all zero dimensional symplectic leaves of
$\operatorname{Spec}(A)$. $\hfill\qed$

If $A$ is a Poisson algebra and $I$ is an ideal then the completion
$A_{I}^{\\!\wedge}$ can be equipped with a Poisson structure in a unique way
such that $A\to A_{I}^{\\!\wedge}$ is Poisson. In particular, for
$a:=(a_{i}+I^{i})_{i>0},b:=(b_{i}+I^{i})_{i>0}\in
A_{I}^{\\!\wedge}\subseteq\prod_{i\in\mathbb{Z}_{>0}}A/I^{i}$ we define
$\\{a,b\\}=(c_{i}+I^{i})_{i>0}$ where $c_{i}=\\{a_{j},b_{k}\\}$ for any choice
of $j,k\geq i+1$. The definition of $A_{I}^{\\!\wedge}$ ensures that this does
not depend on $j,k$ modulo $I^{i}$. When $I$ is Poisson this structure has an
easier description: here every $A/I^{i}$ is a Poisson algebra and the bracket
on $A_{I}^{\\!\wedge}$ coincides with the inverse limit in the category of
Poisson algebras.

The next lemma states that Poisson abelianisation and completion commute.

######  Lemma 5.6.

Let $A$ be finitely generated and Poisson, and pick
$\mathfrak{m}\in\operatorname{m-Spec}(A^{\operatorname{ab}})$. Then there is a
natural isomorphism

$(A_{\mathfrak{m}}^{\wedge})^{\operatorname{ab}}\overset{\sim}{\longrightarrow}(A^{\operatorname{ab}})_{\mathfrak{m}}^{\wedge}.$

###### Proof.

According to Lemma 5.5 the point $\mathfrak{m}$ is a zero dimensional
symplectic leaf, and it follows that $\mathfrak{m}$ is a Poisson ideal. In
particular, $A_{\mathfrak{m}}^{\\!\wedge}$ is a projective limit of Poisson
algebras. We let $B$ denote the ideal of $A$ generated by the Poisson brackets
$\\{A,A\\}$, let $\widehat{B}_{\mathfrak{m}}$ be the ideal of
$A^{\\!\wedge}_{\mathfrak{m}}$ generated by
$\\{A^{\\!\wedge}_{\mathfrak{m}},A^{\\!\wedge}_{\mathfrak{m}}\\}$ and let
$B_{\mathfrak{m}}^{\\!\wedge}$ be the ideal
$A^{\\!\wedge}_{\mathfrak{m}}\otimes_{A}B$, which we identify with an ideal of
$A_{\mathfrak{m}}^{\\!\wedge}$ using [AM69, Proposition 10.13]. Considering
the exact sequence of $A$-modules $B\to A\to A^{\operatorname{ab}}$, we deduce
$(A^{\operatorname{ab}})_{\mathfrak{m}}^{\\!\wedge}\cong
A_{\mathfrak{m}}^{\\!\wedge}/B_{\mathfrak{m}}^{\\!\wedge}$ from [AM69,
Proposition 10.12].

We must show that $B_{\mathfrak{m}}^{\\!\wedge}=\widehat{B}_{\mathfrak{m}}$.
We certainly have an inclusion
$B_{\mathfrak{m}}^{\\!\wedge}\subseteq\widehat{B}_{\mathfrak{m}}$ and so the
claim will follow from the universal property of
$(A_{\mathfrak{m}}^{\\!\wedge})^{\operatorname{ab}}$ if we can show that
$A_{\mathfrak{m}}^{\\!\wedge}/B_{\mathfrak{m}}^{\\!\wedge}$ is an abelian
Poisson algebra. As we noted earlier, the latter is isomorphic to
$(A^{\operatorname{ab}})_{\mathfrak{m}}^{\\!\wedge}$. Since this is a
projective limit of abelian Poisson algebras the claim follows. ∎

###  6\. Sheets and induction

####  6.1. Lusztig–Spaltenstein induction, sheets and slices

Let $G$ be a complex connected reductive group and $P\subseteq G$ a parabolic
subgroup with Levi decomposition $P=LN$, and write
$\mathfrak{l}=\operatorname{Lie}(L)$. If $\mathfrak{v}\subseteq\mathfrak{g}$
is any quasi-affine subvariety then we write
$\mathfrak{v}_{\operatorname{reg}}$ for the set of elements $v\in\mathfrak{v}$
such that $\dim\operatorname{Ad}(G)v$ attains the maximal value.

For any choice of nilpotent orbit $\mathcal{O}_{0}\subseteq\mathfrak{l}$ and
$z\in\mathfrak{z}(\mathfrak{l})$ it is not hard to see that
$\operatorname{Ad}(G)(z+\mathcal{O}_{0}+\mathfrak{n})$ contains a dense
$G$-orbit. This orbit is (Lusztig–Spaltenstein) induced from
$(\mathfrak{l},\mathcal{O}_{0},z)$ and is denoted
$\operatorname{Ind}_{\mathfrak{l}}^{\mathfrak{g}}(z+\mathcal{O}_{0})$.
Remarkably the induced orbit depends only on the $G$-conjugacy class of
$(\mathfrak{l},\mathcal{O}_{0},z)$, not on the choice of $P$ admitting $L$ as
a Levi factor [Lo16, Lemma 4.1]. We call this $G$-orbit an induction datum,
and when $z=0$ we say that the orbit of $(\mathfrak{l},\mathcal{O}_{0})$ is
the induction datum, suppressing $z$.

If an orbit cannot be obtained by induction from a proper Levi subalgebra then
it is called rigid, otherwise it is called induced. Rigid orbits are
necessarily nilpotent. We say that an induction datum
$(\mathfrak{l},\mathcal{O}_{0})$ is rigid if $\mathcal{O}_{0}$ is a rigid
orbit in $\mathfrak{l}$. When $G$ is almost simple of classical type the
conjugacy classes of Levi subalgebras and nilpotent orbits can be described
combinatorially, and induction can be totally understood in terms of
partitions [CM93, §7].

The sheets of a Lie algebra are the irreducible components of the rank
varieties which consist of points whose orbit has a fixed dimension. These
were classified in terms of rigid induction data by Borho [Bo81]. For each
induction datum $(\mathfrak{l},\mathcal{O}_{0})$ define

$\displaystyle\mathcal{D}(\mathfrak{l},\mathcal{O}_{0}):=\operatorname{Ad}(G)(\mathcal{O}_{0}+\mathfrak{z}(\mathfrak{l})_{\operatorname{reg}})$
$\displaystyle\mathcal{S}(\mathfrak{l},\mathcal{O}_{0}):=\overline{\mathcal{D}(\mathfrak{l},\mathcal{O}_{0})}_{\operatorname{reg}}.$

######  Theorem 6.1.

Then the sheets of $\mathfrak{g}$ are the sets