# The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Alexey V. Petukhov and Susan J. Sierra  Sierra: School of Mathematics, The
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom<EMAIL_ADDRESS>Petukhov: Institute for Information Transmission Problems, Bolshoy Karetniy
19-1, Moscow 127994, Russia<EMAIL_ADDRESS>
###### Abstract.

Let $W=\mathbb{C}[t,t^{-1}]\partial_{t}$ be the Witt algebra  of algebraic
vector fields on $\mathbb{C}^{\times}$ and let $V\\!ir$ be the Virasoro
algebra, the unique nontrivial central extension of $W$. In this paper, we
study the Poisson ideal structure of the symmetric algebras of $V\\!ir$ and
$W$, as well as several related Lie algebras. We classify prime Poisson ideals
and Poisson primitive ideals of $\operatorname{S}(V\\!ir)$ and
$\operatorname{S}(W)$. In particular, we show that the only functions in
$W^{*}$ which vanish on a nontrivial Poisson ideal (that is, the only maximal
ideals of $\operatorname{S}(W)$ with a nontrivial Poisson core) are given by
linear combinations of derivatives at a finite set of points; we call such
functions local. Given a local function $\chi\in W^{*}$, we construct the
associated Poisson primitive ideal through computing the algebraic symplectic
leaf of $\chi$, which gives a notion of coadjoint orbit in our setting.

As an application, we prove a structure theorem for subalgebras of $V\\!ir$ of
finite codimension and show in particular that any such subalgebra of $V\\!ir$
contains the central element $z$, substantially generalising a result of
Ondrus and Wiesner on subalgebras of codimension 1. We thus show that
$\operatorname{S}(V\\!ir)/(z-\lambda)$ is Poisson simple if and only if
$\lambda\neq 0$.

###### Key words and phrases:

Virasoro algebra, Witt algebra, pseudo-orbit, symmetric algebra, Poisson
ideal, Poisson primitive ideal

###### 2010 Mathematics Subject Classification:

Primary: 17B68, 17B63; Secondary 17B08, 14L99

###### Contents

  1. 1 Introduction
  2. 2 Poisson ideals and pseudo-orbits
    1. 2.1 Some results on pseudo-orbits
    2. 2.2 Pseudo-orbits versus orbits
  3. 3 Primitive ideals and local functions
    1. 3.1 Local functions on $W_{\geq-1}$ and $W_{\geq 1}$
    2. 3.2 Local functions on $W$ and applications
    3. 3.3 Local functions on $V\\!ir$
  4. 4 Pseudo-orbits and Poisson primitive ideals for the algebras of interest
    1. 4.1 An algebraic group acting on local functions
    2. 4.2 Explicit description of ${\it DLoc}_{x}^{\leq n}$-orbits
    3. 4.3 Implicit description of pseudo-orbits
  5. 5 Prime Poisson ideals
    1. 5.1 Chevalley’s theorem
    2. 5.2 Parameterising Poisson primes
    3. 5.3 The Poisson Dixmier-Moeglin equivalence
    4. 5.4 Radical Poisson ideals contain Poisson primitive ideals
  6. 6 Subalgebras of finite codimension
    1. 6.1 Notation and concepts
    2. 6.2 Subalgebras of codimension 2
    3. 6.3 Subalgebras of codimension 3
    4. 6.4 Some general comments on finite codimension subalgebras of $W$
  7. 7 The orbit method for two-dimensional pseudo-orbits in $W^{*}$
    1. 7.1 Pseudo-orbits of $\chi_{x;\alpha,\gamma}$
    2. 7.2 Constructing primitive ideals through the orbit method
  8. 8 Conjectures for $\operatorname{U}(\mathfrak{g})$

##  1\. Introduction

Let $G$ be a connected algebraic group over $\mathbb{C}$ with Lie algebra
$\mathfrak{g}$, and consider the coadjoint action of $G$ on
$\mathfrak{g}^{*}$. This is a beautiful classical topic, with profound
connections to areas from geometric representation theory to combinatorics to
physics. Algebraic geometry tells us that coadjoint orbits in
$\mathfrak{g}^{*}$ correspond to $G$-invariant radical ideals in the symmetric
algebra $\operatorname{S}(\mathfrak{g})$. These can be alternately defined
using the Kostant-Kirillov Poisson bracket on
$\operatorname{S}(\mathfrak{g})$:

$\\{f,g\\}=\sum_{i,j}\frac{\partial f}{\partial e_{i}}\frac{\partial
f}{\partial e_{j}}[e_{i},e_{j}]$

where $\\{e_{i}\\}$ is a basis of $\mathfrak{g}$. Recall that an ideal $I$ of
$\operatorname{S}(\mathfrak{g})$ is a Poisson ideal if $I$ is also a Lie ideal
for the Poisson bracket. A basic fact is that $I$ is $G$-invariant if and only
if $I$ is Poisson.

Thus to compute the closure of the coadjoint orbit of
$\chi\in\mathfrak{g}^{*}$, let $\mathfrak{m}_{\chi}$ be the kernel of the
evaluation morphism

${\rm ev}_{\chi}:{\rm S}(\mathfrak{g})\to\mathbb{{\Bbbk}},$

and let $\operatorname{P}(\chi)$ be the Poisson core of $\mathfrak{m}_{\chi}$:
the maximal Poisson ideal contained in $\mathfrak{m}_{\chi}$. By definition,
an ideal of the form $\operatorname{P}(\chi)$ is called Poisson primitive; by
a slight abuse of notation, we refer to $\operatorname{P}(\chi)$ as the
Poisson core of $\chi$. Then the closure of the coadjoint orbit of $\chi$ is
defined by $\operatorname{P}(\chi)$:

(1.1)
$\overline{G\cdot\chi}=V(\operatorname{P}(\chi)):=\\{\nu\in\mathfrak{g}^{*}\
|\ \operatorname{ev}_{\nu}(\operatorname{P}(\chi))=0\\},$

and so $\chi,\nu\in\mathfrak{g}^{*}$ are in the same $G$-orbit if and only if
$\operatorname{P}(\chi)=\operatorname{P}(\nu)$. In the case of algebraic Lie
algebras over $\mathbb{C}$ or $\mathbb{R}$, coadjoint orbits are symplectic
leaves for the respective Poisson structure.

In this paper, we investigate how this theory extends to the Witt algebra
$W=\mathbb{C}[t,t^{-1}]\partial_{t}$ of algebraic vector fields on
$\mathbb{C}^{\times}$, and to its central extension the Virasoro algebra
$V\\!ir=\mathbb{C}[t,t^{-1}]\partial_{t}\oplus\mathbb{C}z$, with Lie bracket
given by

$[f\partial_{t},g\partial_{t}]=(fg^{\prime}+f^{\prime}g)\partial_{t}+\operatorname{Res}_{0}(f^{\prime}g^{\prime\prime}-g^{\prime}f^{\prime\prime})z,\quad\mbox{$z$
is central.}$

(We also consider some important Lie subalgebras of $W$.) These infinite-
dimensional Lie algebras, of fundamental importance in representation theory
and in physics, have no adjoint group [L], but one can still study the Poisson
cores of maximal ideals, and more generally the Poisson ideal structure of
$\operatorname{S}(W)$ and $\operatorname{S}(V\\!ir)$. Motivated by (1.1), we
will say that functions $\chi,\nu\in V\\!ir^{*}$ or in $W^{*}$ are in the same
pseudo-orbit if $\operatorname{P}(\chi)=\operatorname{P}(\nu)$. These
(coadjoint) pseudo-orbits can be considered as algebraic symplectic leaves in
$V\\!ir^{*}$ or $W^{*}$.

Our focus is on prime Poisson ideals and Poisson primitive ideals of
$\operatorname{S}(V\\!ir)$ and $\operatorname{S}(W)$. A central goal is to
compute the pseudo-orbits in $V\\!ir^{*}$ and $W^{*}$ and the corresponding
Poisson primitive ideals. A key insight here is provided by a result of Iyudu
and the second author [IS, Theorem 1.3]: if $I$ is a nontrivial Poisson ideal
of $\operatorname{S}(W)$ (respectively, a non-centrally generated Poisson
ideal of $\operatorname{S}(V\\!ir)$), then $\operatorname{S}(W)/I$
(respectively, $\operatorname{S}(V\\!ir)/I$) has polynomial growth. This
further suggests that if $\operatorname{P}(\chi)$ is nontrivial for some
$\chi\in V\\!ir^{*}$, one may be able to investigate the finite-dimensional
subvariety $V(\operatorname{P}(\chi))$ using tools from affine algebraic
geometry. We will see that this is indeed the case.

From the discussion above, it is important to characterise which functions in
$V\\!ir^{*}$ have nontrivial Poisson cores. One striking result, proved in
this paper, is that such $\chi$ must vanish on the central element $z$.
Further, the induced function $\overline{\chi}\in W^{*}$ is given by
evaluating local behaviour on a proper (that is, finite) subscheme of
$\mathbb{C}^{\times}$. We have:

######  Theorem 1.2 (Theorem 3.20).

Let $\chi\in V\\!ir^{*}$. The following are equivalent:

  1. (1)

The Poisson core of $\chi$ is nontrivial: that is,
$\operatorname{P}(\chi)\supsetneqq(z-\chi(z))$.

  2. (2)

$\chi(z)=0$ and the induced function $\overline{\chi}\in W^{*}$ is a linear
combination of functions of the form

$f\partial_{t}\mapsto\alpha_{0}f(x)+\ldots+\alpha_{n}f^{(n)}(x)$

where $x\in\mathbb{C}^{\times}$ and
$\alpha_{0},\dots,\alpha_{n}\in\mathbb{C}$.

  3. (3)

The isotropy subalgebra $V\\!ir^{\chi}$ of $\chi$ has finite codimension in
$V\\!ir$.

We call functions $\chi\in V\\!ir^{*}$ satisfying the equivalent conditions of
Theorem 1.2 local functions as by condition $(2)$ they are defined by local
data.

As an immediate corollary of Theorem 1.2, we show:

######  Corollary 1.3 (Corollary 3.24).

If $0\neq\lambda\in\mathbb{C}$, then $\operatorname{S}(V\\!ir)/(z-\lambda)$ is
Poisson simple: it has no nontrivial Poisson ideals.

We then study the pseudo-orbits of local functions on $V\\!ir$, $W$, and
related Lie algebras; we describe our results for $V\\!ir$ in the
introduction. We show that pseudo-orbits of local functions in $V\\!ir^{*}$
are in fact orbits of a finite-dimensional solvable algebraic (Lie) group
acting on an affine subvariety of $V\\!ir^{*}$, and we describe these orbits
explicitly (Section 4.2). This allows us to completely determine the pseudo-
orbit of an arbitrary local function in $V\\!ir^{*}$ (Theorem 4.21) and thus
also determine the Poisson primitive ideals of $\operatorname{S}(V\\!ir)$
(Remark 4.29).

Through this analysis, we obtain a nice combinatorial description of pseudo-
orbits in $W^{*}$: pseudo-orbits of local functions on $W$, and thus Poisson
primitive ideals of $\operatorname{S}(W)$, correspond to a choice of a
partition $\lambda$ and a point in some finite-dimensionsal affine space
$\mathbb{A}^{k}$, where $k$ can be calculated from $\lambda$. (See Remark
4.23.) In Theorem 5.11 and Remark 5.15, we expand this correspondence to
obtain a parameterisation of all prime Poisson ideals of $\operatorname{S}(W)$
and $\operatorname{S}(V\\!ir)$. We also study the related Lie algebra
$W_{\geq-1}=\mathbb{C}[t]\partial_{t}$, and prove (Corollary 5.16) that
Poisson primitive and prime Poisson ideals of $\operatorname{S}(W_{\geq-1})$
are induced by restriction from $\operatorname{S}(W)$.

Our understanding of prime Poisson ideals allows us to determine exactly which
prime Poisson ideals of $\operatorname{S}(V\\!ir)$ obey the Poisson Dixmier-
Moeglin equivalence, which generalises the characterisation of primitive
ideals in enveloping algebras of finite-dimensional Lie algebras due to
Dixmier and Moeglin. The central question is when a Poisson primitive ideal of
$\operatorname{S}(V\\!ir)$ is Poisson locally closed: that is, locally closed
in the Zariski topology on Poisson primitive ideals. (Note that if
$\dim\mathfrak{g}<\infty$ then a prime Poisson ideal of
$\operatorname{S}(\mathfrak{g})$ is Poisson primitive if and only if it is
Poisson locally closed [LL, Theorem 2].) We show (Theorem 5.17) that $(z)$ is
the only Poisson primitive ideal of $\operatorname{S}(V\\!ir)$ which is not
Poisson locally closed.

Motivated by condition $(3)$ of Theorem 1.2, we further investigate
subalgebras of $V\\!ir$ of finite codimension. We prove:

######  Theorem 1.4 (Proposition 3.22).

Let $\mathfrak{k}\subseteq V\\!ir$ be a subalgebra of finite codimension. Then
there is $f\in\mathbb{C}[t,t^{-1}]\setminus\\{0\\}$ so that
$\mathfrak{k}\supseteq\mathbb{C}z+f\mathbb{C}[t,t^{-1}]\partial_{t}$. In
particular, any subalgebra of $V\\!ir$ of finite codimension contains $z$.

One part of the proof of Theorem 1.4 is to show that, given a subalgebra
$\mathfrak{k}$ of finite codimension in $W$, there are a finite collection of
points $S$ (the “support” of $\mathfrak{k}$) and $n\in\mathbb{Z}_{\geq 0}$
such that all vector fields in $\mathfrak{k}$ vanish at all points of $S$ and
so that $\mathfrak{k}$ contains all vector fields vanishing to order $\geq n$
at every point of $S$. Based on this result, we classify subalgebras of
$V\\!ir$ of codimension $\leq 3$ in Section 6.

Our original motivation for studying Poisson ideals of
$\operatorname{S}(V\\!ir)$ was to study two-sided ideals in the universal
enveloping algebra $\operatorname{U}(V\\!ir)$. In the finite-dimensional
setting, Kirillov’s orbit method gives a correspondence between primitive
ideals of $\operatorname{U}(\mathfrak{g})$ and coadjoint orbits in
$\mathfrak{g}^{*}$. We conjecture that a similar correspondence exists for
$V\\!ir$ and related Lie algebras, and in Section 7 we show that pseudo-orbits
of dimension 2 in $V\\!ir^{*}$ quantise to give a family of primitive ideals,
kernels of well-known maps from $\operatorname{U}(V\\!ir)$ through
$\operatorname{U}(W)$ to the localised Weyl algebra. We end the paper with
some conjectures about (two-sided) ideals of the universal enveloping algebra
$\operatorname{U}(V\\!ir)$ which are motivated by our work on
$\operatorname{S}(V\\!ir)$; these are the subject of further research.

Acknowledgements:  We thank Jason Bell, Ken Brown, Iain Gordon, Ivan Frolov,
Omar León Sánchez, and Toby Stafford for interesting discussion and questions.
The first author was supported by the Russian Foundation for Basic Research,
grant no. 20–01–00091–a. The second author is partially supported by EPSRC
grants EP/M008460/1 and EP/T018844/1. In addition, we are grateful to the
Edinburgh Mathematical Society and the University of Edinburgh School of
Mathematics for supporting a 2019 mini-workshop on enveloping algebras of
infinite-dimensional Lie algebras.

##  2\. Poisson ideals and pseudo-orbits

In this section we recall the general notions of Poisson algebra, Poisson
ideal, and Poisson primitive ideal. We then consider how these concepts behave
for the symmetric algebras of the infinite-dimensional Lie algebras in which
we are interested.

Let ${\Bbbk}$ be an uncountable algebraically closed field of characteristic
0. We define the Lie algebras of interest in this paper. The Witt algebra
$W={\Bbbk}[t,t^{-1}]\partial$ is the Lie algebra of vector fields on
${\Bbbk}^{\times}:={\Bbbk}\setminus\\{0\\}$; here $\partial=\partial_{t}$. It
is graded by setting $\deg t^{n}\partial=n-1$. Then $W_{\geq-1}$ is the
subalgebra ${\Bbbk}[t]\partial$ of $W$, and $W_{\geq 1}$, sometimes called the
positive Witt algebra, is the subalgebra $t^{2}{\Bbbk}[t]\partial$; $W_{\geq
0}$ stands for $t{\Bbbk}[t]\partial$.

The Virasoro algebra  $V\\!ir$ is isomorphic as a vector space to
${\Bbbk}[t,t^{-1}]\partial\oplus{\Bbbk}z$. It is endowed with a Lie algebra
structure by the formula

$[f\partial+c_{1}z,g\partial+c_{2}z]=(fg^{\prime}-f^{\prime}g)\partial+\operatorname{Res}_{0}(f^{\prime}g^{\prime\prime}-f^{\prime\prime}g^{\prime})z.$

(Here $\operatorname{Res}_{0}(f)$ stands for the algebraic residue of $f$ at
0, i.e. the coefficient of $t^{-1}$ in the Laurent expansion of $f$ at 0.) It
is well-known that $V\\!ir$ is the unique nontrivial one-dimensional central
extension of $W$. There is a canonical Lie algebra homomorphism $V\\!ir\to W$
given by factoring out $z$.

Let $V$ be a ${\Bbbk}$-vector space. We will use $\operatorname{S}(V)$ to
denote the symmetric algebra of $V$; that is, polynomial functions on $V^{*}$.
For $\chi\in V^{*}$ we denote by $\operatorname{ev}_{\chi}$ the induced
homomorphism $\operatorname{S}(V)\to{\Bbbk}$ defined by
$\operatorname{ev}_{\chi}(f)=f(\chi)$.

Let $\mathfrak{g}$ be one of the Lie algebras $V\\!ir$, $W$, $W_{\geq-1}$, or
$W_{\geq 1}$. Our assumption on the cardinality of the field ${\Bbbk}$ means
that the following extended Nullstellensatz applies to
$\operatorname{S}(\mathfrak{g})$; see [MR, Corollary 9.1.8, Lemma 9.1.2].

######  Theorem 2.1 (Extended Nullstellensatz).

Let $A$ be a commutative ${\Bbbk}$-algebra such that
$\dim_{\Bbbk}A<|{\Bbbk}|.$ Then:

$\bullet$ $A$ is a Jacobson ring: i.e. every radical ideal is an intersection
of a family of maximal ideals;

$\bullet$ if $\mathfrak{m}\subseteq A$ is a maximal ideal then the canonical
map ${\Bbbk}\to A/\mathfrak{m}$ is an isomorphism;

$\bullet$ if $A=\operatorname{S}(V)$ for a vector space $V$ with $\dim
V<|{\Bbbk}|$, then the maximal ideals of $A$ are all of the form
$\mathfrak{m}_{\chi}=\operatorname{\ker}{\rm ev}_{\chi}$ for some $\chi\in
V^{*}$.

Let $A$ be a commutative algebra over ${\Bbbk}$. (Note that we make no
noetherianity or finiteness assumption on $A$.) Denote by
$\operatorname{MSpec}A$ the set of maximal ideals of $A$, which we consider as
a (potentially infinite-dimensional) variety. We denote by $\mathfrak{m}_{x}$
the maximal ideal corresponding to a point $x\in{\rm MSpec}A$. As usual,
$\operatorname{MSpec}A$ is a topological space under the Zariski topology.
Given an ideal $N$ of $A$, we denote the corresponding closed subset of
$\operatorname{MSpec}A$ by

$V(N):=\\{\mathfrak{m}\in\operatorname{MSpec}A\ |\ N\subseteq\mathfrak{m}\\}.$

Given $X\subseteq\operatorname{MSpec}A$, we denote the corresponding radical
ideal of $A$ by

$I(X):=\bigcap_{x\in X}\mathfrak{m}_{x}.$

As we have enlarged the class of varieties somewhat beyond the usual, we make
the convention that for us an algebraic variety is an integral separated
scheme of finite type over ${\Bbbk}$, as in [H, p. 105]. If $A,B$ are
commutative ${\Bbbk}$-algebras which are domains, an algebraic map or morphism
of varieties $\phi:\operatorname{MSpec}A\to\operatorname{MSpec}B$ has the
usual meaning: a function so that the pullback $\phi^{*}$ defines an algebra
homomorphism $B\to A$.

Let $\\{\cdot,\cdot\\}:A\times A\to A$ be a skew-symmetric ${\Bbbk}$-bilinear
map. We say that $(A,\\{\cdot,\cdot\\})$ is a Poisson algebra if
$\\{\cdot,\cdot\\}$ satisfies the Leibniz rule on each input and the Jacobi
identity. An ideal $I$ of $A$ is Poisson if $\\{I,A\\}\subseteq I$. It is
clear that the sum of all Poisson subideals of any ideal $I$ of $A$ is the
maximal Poisson ideal inside $I$; we denote this ideal by ${\rm Core}(I)$ and
refer to it as the Poisson core of $I$. Note that if $I$ is radical,
respectively prime, then ${\rm Core}(I)$ inherits this property; see [PS,
Lemmata 2.6 and 2.8].

A Poisson ideal $I$ is called Poisson primitive if
$I=\operatorname{P}(\mathfrak{m})$ for a maximal ideal $\mathfrak{m}$ of $A$.
We denote the set of Poisson primitive ideals of $A$ by
$\operatorname{PSpec}_{\rm prim}A$, and the set of prime Poisson ideals by
$\operatorname{PSpec}A$. Both are given the Zariski topology, where the closed
subsets are defined by Poisson ideals of $A$.

Consider a Lie algebra $\mathfrak{g}$ with $\dim\mathfrak{g}<|{\Bbbk}|$. It is
well known that $\operatorname{S}(\mathfrak{g})$ possesses a canonical Poisson
algebra structure, induced by defining $\\{u,v\\}=[u,v]$ for any
$u,v\in\mathfrak{g}$. As $\dim\mathfrak{g}<|{\Bbbk}|$ then by the
Nullstellensatz $\operatorname{MSpec}(\operatorname{S}(\mathfrak{g}))$ can be
canonically identified with $\mathfrak{g}^{*}:$

$\chi\in\mathfrak{g}^{*}\leftrightarrow\mathfrak{m}_{\chi}:=\operatorname{\ker}{\rm
ev}_{\chi}\in\operatorname{MSpec}(\operatorname{S}(\mathfrak{g})).$

Thus any Poisson primitive ideal of $\operatorname{S}(\mathfrak{g})$ is equal
to $\operatorname{Core}(\mathfrak{m}_{\chi})$ for some
$\chi\in\mathfrak{g}^{*}$. Set
$\operatorname{P}(\chi):=\operatorname{Core}(\mathfrak{m}_{\chi})$. For any
Lie algebra $\mathfrak{g}$, the ideal
$\mathfrak{m}_{0}=\operatorname{Core}(\mathfrak{m}_{0})=\operatorname{P}(0)$
(the augmentation ideal) is the Poisson core of $0\in\mathfrak{g}^{*}$ and so
is Poisson primitive.

Even in the absence of an adjoint group to $\mathfrak{g}$, the Poisson
primitives $\operatorname{P}(\chi)$ give analogues of coadjoint orbits.

######  Definition 2.2.

Let $\mathfrak{g}$ be any Lie algebra. The pseudo-orbit of
$\chi\in\mathfrak{g}^{*}$ is

$\mathbb{O}(\chi):=\\{\nu\in\mathfrak{g}^{*}|\operatorname{P}(\nu)=\operatorname{P}(\chi)\\}.$

The dimension of $\mathbb{O}(\chi)$ is defined to be
$\operatorname{GKdim}\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi)$.
(Here if $R$ is a ${\Bbbk}$-algebra then $\operatorname{GKdim}R$ denotes the
Gelfand-Kirillov dimension of $R$; see [KL].)

######  Remark 2.3.

If $\mathfrak{g}={\rm Lie}\ G$ is the Lie algebra of a connected algebraic
group and $\chi\in\mathfrak{g}^{*}$, then $\mathbb{O}(\chi)$ is the coadjoint
orbit of $\chi$. In our setting, $\mathfrak{g}$ is not the Lie algebra of any
algebraic group. However, we will see that we can still define group actions
on pieces of $\mathfrak{g}^{*}$ that allow us to recover pseudo-orbits as
actual orbits.

Part (b) of the next result is an analogue of (1.1) for pseudo-orbits.

######  Lemma 2.4.

Let $\mathfrak{g}$ be a Lie algebra with $\dim\mathfrak{g}<|{\Bbbk}|$.

  * (a)

Any radical Poisson ideal in $\operatorname{S}(\mathfrak{g})$ is equal to an
intersection of Poisson primitive ideals. Explicitly, given a radical Poisson
ideal $I$ we have

$I=\bigcap\\{\operatorname{P}(\chi)\ |\
\chi\in\mathfrak{g}^{*},~{}\operatorname{ev}_{\chi}(I)=0\\}.$

  * (b)

Assume now that $\dim\mathfrak{g}$ is countable, and let
$\chi\in\mathfrak{g}^{*}$ be nonzero. Then

$\operatorname{P}(\chi)=\bigcap_{\nu\in\mathbb{O}(\chi)}\mathfrak{m}_{\nu}.$

###### Proof.

(a) is well-known, but we give a proof for completeness.

As $I$ is radical and Poisson we have

$I\subseteq\bigcap\\{\operatorname{P}(\chi)\ |\
\chi\in\mathfrak{g}^{*},~{}\operatorname{ev}_{\chi}(I)=0\\},$

and this is contained by definition in
$\bigcap_{\chi\in\mathfrak{g}^{*},\operatorname{ev}_{\chi}(I)=0}\mathfrak{m}_{\chi}$.
By the Extended Nullstellensatz, this last is equal to $I$.

For (b), set $R:=\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi)$.
Thanks to the proof of [LS, Theorem 6.3], which uses only that $\mathfrak{g}$
is countably generated (see also [BLLM, Theorem 3.2]), there is a sequence of
non-zero Poisson ideals $L_{1},L_{2},\ldots$ of $R$ such that if some
$\mu\in\mathfrak{g}^{*}$ has that $\operatorname{P}(\mu)$ strictly contains
$\operatorname{P}(\chi)$, then $\operatorname{P}(\mu)/\operatorname{P}(\chi)$
contains $L_{i}$ for at least one $i$.

This is equivalent to the following statement: there is a sequence
$f_{1},f_{2},\ldots\in R\setminus\\{0\\}$ such that if $\operatorname{P}(\mu)$
strictly contains $\operatorname{P}(\chi)$ then $f_{i}(\mu)=0$ for at least
one $i$. Therefore
$R^{\prime}=\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi)[f_{1}^{-1},f_{2}^{-1},\ldots]$
is Poisson simple. On the other hand $R^{\prime}$ is clearly at most
countable-dimensional and hence $(0)\subset R^{\prime}$ is an intersection of
a family of maximal ideals of $R^{\prime}$ by the Nullstellensatz. The
definition of $R^{\prime}$ and $f_{i}^{\prime}$ guarantees that
$\operatorname{P}(\mu)=\operatorname{P}(\chi)$ for each
$\mu\in\mathfrak{g}^{*}$ defining such an ideal. We thus obtain a family of
maximal ideals of $\operatorname{S}(\mathfrak{g})$ contained in the pseudo-
orbit of $\chi$ whose intersection is $\operatorname{P}(\chi)$. This implies
the desired result. ∎

###  2.1. Some results on pseudo-orbits

We now give several results which are well known for finite-dimensional
algebraic Lie algebras, but require proof in our setting. Throughout this
subsection, let $\mathfrak{g}$ be a Lie algebra with
$\dim\mathfrak{g}<|{\Bbbk}|$.

For every $\chi\in\mathfrak{g}^{*}$ define the skew-symmetric bilinear form
$B_{\chi}(x,y):=\chi([x,y])$. The kernel of $B_{\chi}$ is a Lie subalgebra of
$\mathfrak{g}$, which we denote by $\mathfrak{g}^{\chi}$. Observe that
$\mathfrak{g}^{\chi}$ is precisely $\\{v\in\mathfrak{g}|v\cdot\chi=0\\}$; that
is $\mathfrak{g}^{\chi}$ is the isotropy subalgebra of $\chi$ under the
(coadjoint) action of $\mathfrak{g}$ on $\mathfrak{g}^{*}$. As $B_{\chi}$
induces a nondegenerate skew-symmetric form on
$\mathfrak{g}/\mathfrak{g}^{\chi}$, we have
$\operatorname{rk}B_{\chi}=\dim\mathfrak{g}/\mathfrak{g}^{\chi}$.

######  Lemma 2.5.

The dimension of $\mathbb{O}(\chi)$ is at least $\operatorname{rk}B_{\chi}$,
i.e.

${\rm GKdim}({\rm
S}(\mathfrak{g})/\operatorname{P}(\chi))\geq\operatorname{rk}B_{\chi}.$

Before proving Lemma 2.5, we establish some notation, which we will need for
several results. For $u_{1}\ldots,u_{n},$ $v_{1},\ldots,v_{n}\in\mathfrak{g}$
set
$D(u_{1},\ldots,u_{n};v_{1},\ldots,v_{n})\in\operatorname{S}(\mathfrak{g})$ to
be the determinant

(2.6) $\begin{vmatrix}[u_{1},v_{1}]&\cdots&[u_{1},v_{n}]\\\
\cdots&\cdots&\cdots\\\ {[u_{n},v_{1}]}&\cdots&{[u_{n},v_{n}]}\\\
\end{vmatrix}.$

###### Proof of Lemma 2.5.

Pick $r\in\mathbb{Z}_{\geq 0}$ such that $r\leq\operatorname{rk}B_{\chi}$.
Then there is an $r$-dimensional subspace $V$ of $\mathfrak{g}$ so that
$\operatorname{rk}(B_{\chi}|_{V})=r$; that is, if $u_{1},\ldots,u_{r}$ is a
basis of $V$ then
$\operatorname{ev}_{\chi}(D(u_{1},\dots,u_{r},u_{1},\dots,u_{r}))\neq 0$. We
will show that $u_{1},\ldots,u_{r}$ are algebraically independent in ${\rm
S}(\mathfrak{g})/\operatorname{P}(\chi)$; that is, that
$\operatorname{GKdim}(\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi))\geq
r$.

Assume to the contrary that the $u_{i}$ are not algebraically independent
modulo $\operatorname{P}(\chi)$. This means that there is some nonzero
$P\in{\Bbbk}[u_{1},\dots,u_{r}]\cap\operatorname{P}(\chi)$, and we may assume
$P$ is of minimal degree among such elements. As $\operatorname{P}(\chi)$ is
Poisson,

(2.7)
$\\{u_{i},P\\}=\sum_{j}\\{u_{i},u_{j}\\}\partial_{j}P\in\operatorname{P}(\chi),$

where $\partial_{j}P=\frac{\partial P}{\partial u_{j}}$.

Let $Q$ be the field of fractions of
$\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi)$ (recall that
$\operatorname{P}(\chi)$ is prime). We may rewrite (2.7) as the matrix
equation

$\Bigl{(}\\{u_{i},u_{j}\\}\Bigr{)}_{ij}\begin{pmatrix}\partial_{1}P\\\
\vdots\\\ \partial_{r}P\end{pmatrix}=\vec{0}$

over $Q$. By minimality of $\deg P$, the vector
$\begin{pmatrix}\partial_{1}P\\\ \vdots\\\ \partial_{r}P\end{pmatrix}\in
Q^{r}$ is nonzero. Thus $\Bigl{(}\\{u_{i},u_{j}\\}\Bigr{)}_{ij}\in M_{r\times
r}(Q)$ is singular, so
$D(u_{1},\dots,u_{r};u_{1},\dots,u_{r})\in\operatorname{P}(\chi)\subseteq\mathfrak{m}_{\chi}$.
This contradicts the first paragraph of the proof. ∎

It is not necessarily easy to calculate $\operatorname{P}(\chi)$, but we can
sometimes use GK-dimension to show that a Poisson ideal contained in
$\mathfrak{m}_{\chi}$ is in fact equal to $\operatorname{P}(\chi)$. This is
given by the following lemma.

######  Lemma 2.8.

Let $\mathfrak{g}$ be a Lie algebra and $A$ a commutative ${\Bbbk}$-algebra
that is a domain. Let ${\phi}:\operatorname{S}(\mathfrak{g})\to A$ be an
algebra homomorphism so that $\ker{\phi}$ is Poisson. Let
$\chi\in\mathfrak{g}^{*}$ be such that

(2.9)
$\operatorname{\ker}\phi\subseteq\mathfrak{m}_{\chi}\mathrm{~{}and~{}}\operatorname{rk}B_{\chi}\geq\operatorname{GKdim}A.$

Then $\operatorname{\ker}\phi=\operatorname{P}(\chi)$ and is Poisson
primitive; further, $\operatorname{rk}B_{\chi}=\operatorname{GKdim}A$.

###### Proof.

Certainly $\ker{\phi}\subseteq\operatorname{P}(\chi)$. If the containment is
strict, then as $A$ is a domain,

$\operatorname{GKdim}(\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi))<\operatorname{GKdim}A\leq\operatorname{rk}B_{\chi},$

contradicting Lemma 2.5. The final equality is direct from Lemma 2.5. ∎

######  Corollary 2.10.

If both $\chi_{1},\chi_{2}$ satisfy (2.9) for the same ${\phi}$ then
$\operatorname{P}(\chi_{1})=\operatorname{P}(\chi_{2})$. ∎

We will use Lemma 2.8 and Corollary 2.10 to describe primitive ideals in
$\operatorname{S}(\mathfrak{g})$ and the corresponding pseudo-orbits in
$\mathfrak{g}^{*}$ explicitly. In the situation of Lemma 2.8, $A$ may not
necessarily be a Poisson algebra. We abuse notation slightly, however, and say
that if $\ker{\phi}$ is a Poisson ideal, then ${\phi}$ is a Poisson morphism.
Further, given a morphism of varieties

$\psi:\operatorname{MSpec}(A)\to\operatorname{MSpec}(\operatorname{S}(\mathfrak{g}))=\mathfrak{g}^{*},$

we say that $\psi$ is Poisson if $\psi^{*}:\operatorname{S}(\mathfrak{g})\to
A$ is Poisson.

It is important to know under which conditions maps are Poisson and the answer
is given in Proposition 2.11. To explain the setup we need to use some
concepts from (affine) algebraic geometry. As mentioned, we will, somewhat
loosely, refer to infinite-dimensional vector spaces such as
$\mathfrak{g}^{*}$ as varieties, enlarging the class somewhat from standard
usage. For any variety $X$ and any point $x\in X$ let ${\rm
T}_{x}X=(\mathfrak{m}_{x}/\mathfrak{m}_{x}^{2})^{*}$ denote the tangent space
to $X$ at $x$; this definition makes sense for $X=\mathfrak{g}^{*}$ as well.
We will, without further comment, identify ${\rm T}_{x}X$ with elements of
$\mathfrak{m}_{x}^{*}$ which vanish on $\mathfrak{m}_{x}^{2}$. If there is a
map $\psi:X\to Y$ between algebraic varieties $X$ and $Y$ then, for all $x\in
X$, there is an induced map $\psi_{x}:{\rm T}_{x}X\to{\rm T}_{\psi(x)}Y$.

We canonically identify the tangent space to $\mathfrak{g}^{*}$ at
$\chi\in\mathfrak{g}^{*}$ with $\mathfrak{g}^{*}$. For any derivation $D$ of
$\operatorname{S}(\mathfrak{g})$ and any $\chi\in\mathfrak{g}^{*}$ we denote
by $D_{\chi}$ the tangent vector defined by $D$ at $\chi$; that is
$D_{\chi}(f)=\chi(Df)$ for any $f\in\mathfrak{m}_{\chi}$. Now, $\mathfrak{g}$
acts by derivations on $\operatorname{S}(\mathfrak{g})$. For
$u\in\mathfrak{g}$ and $\chi\in\mathfrak{g}^{*}$ let $u_{\chi}$ be the tangent
vector defined by $u$ at $\chi$. If $v\in\mathfrak{g}$ then
$u_{\chi}(v)=\chi([u,v])$, and thus $u_{\chi}\in{\rm
T}_{\chi}(\mathfrak{g}^{*})=\mathfrak{g}^{*}$ is identified with $u\cdot\chi$.
If $\mathfrak{u}\subseteq\mathfrak{g}$ let $\mathfrak{u}_{\chi}=\\{u_{\chi}\
|\ u\in\mathfrak{u}\\}$. Thus
$\mathfrak{g}_{\chi}\cong\mathfrak{g}\cdot\chi\cong\mathfrak{g}/\mathfrak{g}^{\chi}$
and $\operatorname{rk}B_{\chi}=\dim\mathfrak{g}_{\chi}$.

######  Proposition 2.11.

Let $\mathfrak{g}$ be a Lie algebra with $\dim\mathfrak{g}<|{\Bbbk}|$. Let $X$
be an affine variety and let $\phi:X\to\mathfrak{g}^{*}$ be a morphism of
varieties. Then $\phi$ is Poisson if and only if
$\mathfrak{g}_{\phi(x)}\subseteq\phi_{x}({\rm T}_{x}X)$ for all $x\in X$.

Proposition 2.11 is a direct consequence of the following lemma.

######  Lemma 2.12.

Let $\mathfrak{g},X,\phi$ be as above and let $D$ be a derivation of
$\operatorname{S}(\mathfrak{g})$. Then $\ker\phi^{*}$ is $D$-stable if and
only if $D_{\phi(x)}\in\phi_{x}({\rm T}_{x}X)$ for all $x\in X$.

###### Proof.

Suppose that $D_{\phi(x)}\in\phi_{x}({\rm T}_{x}X)$ for all $x\in X$. Let
$f\in\ker\phi^{*}$. We show that $\phi^{*}(Df)=0$, or, equivalently, that
$Df\in\mathfrak{m}_{\phi(x)}$ for all $x\in X$. Fix $x$, and let $\ell\in{\rm
T}_{x}X$ be such that $D_{\phi(x)}=\phi_{x}(\ell)$. Then

$(Df)(\phi(x))=\ell(\phi^{*}f)=\ell(0)=0,$

as needed.

Conversely, let $x\in X$ and let
$\mathfrak{n}_{x}:=\mathfrak{m}_{x}\cap\phi^{*}(\operatorname{S}(\mathfrak{g}))$.
If $\ker\phi^{*}$ is $D$-stable, then $D$ induces a derivation $\overline{D}$
on $\phi^{*}(\operatorname{S}(\mathfrak{g}))$, defined by

$\overline{D}(\phi^{*}f)=\phi^{*}(Df),$

and thus defines an element
$\overline{D}_{x}\in(\mathfrak{n}_{x}/(\mathfrak{n}_{x})^{2})^{*}$. Let
$\ell\in{\rm T}_{x}(X)$ be any extension of $\overline{D}_{x}$ to
$\mathfrak{m}_{x}/\mathfrak{m}_{x}^{2}$. Then for $f\in\mathfrak{m}_{\phi(x)}$
we have

$D_{\phi(x)}(f)=\operatorname{ev}_{\phi(x)}(Df)={\phi}^{*}(Df)(x)=\overline{D}_{x}\phi^{*}f=\ell\phi^{*}f,$

showing that $D_{\phi(x)}\in\phi_{x}({\rm T}_{x}(X))$. ∎

###  2.2. Pseudo-orbits versus orbits

We wish to relate the pseudo-orbits from the previous subsection with orbits
of an algebraic group acting on an appropriate algebraic variety $X$. The next
result gives us a general technique to do this.

######  Proposition 2.13.

Let $\mathfrak{g}$ be a Lie algebra with $\dim\mathfrak{g}<|{\Bbbk}|$. Let $X$
be an irreducible affine algebraic variety acted on by a connected algebraic
group $H$ with Lie algebra $\mathfrak{h}$ and let $U\subseteq X$ be an open
affine subset. Fix a morphism of varieties $\phi:U\to\mathfrak{g}^{*}$. Assume
that for every $x\in U$ we have
$\mathfrak{g}_{\phi(x)}\subseteq\phi_{x}(\mathfrak{h}_{x})$.

  1. (a)

For all $x\in U$ the pseudo-orbit of $\phi(x)$ is contained in
$\overline{\phi(Hx\cap U)}$, where recall that the topology on
$\mathfrak{g}^{*}$ is the Zariski topology.

  2. (b)

For $x\in U$, let $I^{U}(Hx)$ be the defining ideal in ${\Bbbk}[U]$ of
$\overline{Hx}\cap U$. Let $x\in U$ be such that
$\dim\mathfrak{g}_{\phi(x)}=\dim\mathfrak{h}_{x}$. Then
$\operatorname{P}(\phi(x))$ is equal to the kernel of the induced homomorphism

(2.14)
$\operatorname{S}(\mathfrak{g})\to{\Bbbk}[U]/I^{U}(Hx)={\Bbbk}[\overline{Hx}\cap
U],$

and

(2.15)
$\dim\mathbb{O}(\phi(x))=\operatorname{GKdim}\operatorname{S}(\mathfrak{g})/\operatorname{P}(\phi(x))=\dim\mathfrak{g}_{\phi(x)}.$

In particular, if $y\in U$ is such that $Hx=Hy$ and
$\dim\mathfrak{g}_{\phi(y)}=\dim\mathfrak{h}_{y}$ then
$\operatorname{P}(\phi(x))=\operatorname{P}(\phi(y))$.

  3. (c)

Let $x,y\in U$ be such that $\dim\mathfrak{g}_{\phi(x)}=\dim\mathfrak{h}_{x}$
and $\dim\mathfrak{g}_{\phi(y)}=\dim\mathfrak{h}_{y}$. Then
$\operatorname{P}(\phi(x))=\operatorname{P}(\phi(y))$ if and only if there are
open subsets $U_{x}\subseteq Hx\cap U$ and $U_{y}\subseteq Hy\cap U$ such that
$\phi(U_{x})=\phi(U_{y})$.

###### Proof.

(a). The kernel of (2.14) is Poisson by Proposition 2.11 and is contained in
$\mathfrak{m}_{\phi(x)}$ by definition. Thus it is contained in
$\operatorname{P}(\phi(x))$, which is what we need. Note also that this
statement is completely trivial if $\mathfrak{g}$ is the Lie algebra of a
finite-dimensional algebraic group.

(b). Let $K$ be the kernel of (2.14). Then $K$ is Poisson by (a). We have

$\operatorname{GKdim}\operatorname{S}(\mathfrak{g})/K\leq\dim
Hx=\dim\mathfrak{h}_{x}=\operatorname{rk}B_{\phi(x)},$

so $K=\operatorname{P}(\phi(x))$ by Lemma 2.8. Certainly

$\operatorname{GKdim}\operatorname{S}(\mathfrak{g})/\operatorname{P}(\phi(x))\leq\operatorname{GKdim}{\Bbbk}[U]/I^{U}(Hx)=\dim\mathfrak{h}_{x}=\dim\mathfrak{g}_{\phi(x)},$

and the two are equal by Lemma 2.8. The final statement follows from Corollary
2.10.

(c). For any dense subset $U_{x}\subseteq Hx\cap U$, by (b)
$\operatorname{P}(\phi(x))$ is equal to the kernel of the induced map
$\operatorname{S}(\mathfrak{g})\to{\Bbbk}[U_{x}]$. This is determined by
$\phi(U_{x})$, so if $\phi(U_{x})=\phi(U_{y})$ then
$\operatorname{P}(\phi(x))=\operatorname{P}(\phi(y))$.

Suppose now that $\operatorname{P}(\phi(x))=\operatorname{P}(\phi(y))$.
Consider the induced maps $Hx\to\mathfrak{g}^{*},Hy\to\mathfrak{g}^{*}$ and
the respective fibre product $(Hx)\times_{\mathfrak{g}^{*}}(Hy)$. Note that
$(Hx)\times_{\mathfrak{g}^{*}}(Hy)$ is a closed subset of $(Hx)\times(Hy)$ and
hence the ideal defining $(Hx)\times_{\mathfrak{g}^{*}}(Hy)$ is generated by a
finite collection of elements. This implies that there is a finite-dimensional
subspace $V\subset\mathfrak{g}$ such that

$(Hx)\times_{\mathfrak{g}^{*}}(Hy)=(Hx)\times_{V^{*}}(Hy).$

As $\operatorname{P}(\phi(x))=\operatorname{P}(\phi(y))$, we see that the
images of $Hx$ and $Hy$ in $V^{*}$ have the same closure in $V^{*}$; call it
$Z$. The image of $Hx$ in $V^{*}$ may not be open in $Z$, but it is
constructible and thus contains an open subset $U(x)$ of $Z$. Likewise the
image of $Hy$ in $V^{*}$ contains an open subset $U(y)$ of $Z$. We pick
$U_{x}$ to be the preimage of $U(x)\cap U(y)$ in $Hx$ and let $U_{y}$ be the
preimage of $U(x)\cap U(y)$ in $Hy$; then they satisfy the desired property. ∎

##  3\. Primitive ideals and local functions

We now specialise to let $\mathfrak{g}$ be one of $W$, $W_{\geq 1}$,
$W_{\geq-1}$, or $V\\!ir$. The main goal of this section is to determine which
functions $\chi\in\mathfrak{g}^{*}$ have nontrivial Poisson core
$\operatorname{P}(\chi)$. We will see that these $\chi$ are precisely those
$\chi$ which measure the local behavior of $f\partial\in\mathfrak{g}$ at a
finite collection of points; we call these functions local. We provide several
equivalent characterisations of local functions, and apply these to classify
subalgebras of $\mathfrak{g}$ of finite codimension. As a consequence, we show
that $\operatorname{S}(V\\!ir)/(z-\lambda)$ is Poisson simple for $\lambda\neq
0$.

More formally, consider $\mathfrak{g}=W_{\geq-1}$. Let
$x,\alpha_{0},\dots,\alpha_{n}\in{\Bbbk}$ with
$(\alpha_{0},\dots,\alpha_{n})\neq\vec{\hskip 2.0pt0}$. Define a linear
function $\chi_{x;\alpha_{0},\dots,\alpha_{n}}\in W_{\geq-1}^{*}$ by

(3.1)
$\chi_{x;\alpha_{0},\ldots,\alpha_{n}}:f\partial\mapsto\alpha_{0}f(x)+\ldots+\alpha_{n}f^{(n)}(x).$

The same formula defines elements of $W_{\geq 1}^{*}$ and $W^{*}$, although in
the last case we need to require that $x\neq 0$.

We now formally define local functions.

######  Definition 3.2.

  1. (a)

A local function on $W_{\geq-1}$ or $W_{\geq 1}$ is a sum of finitely many
functions of the form (3.1) with (possibly) distinct $x$;

  2. (b)

A local function on $W$ is a sum of finitely many functions of the form (3.1)
with (possibly) distinct $x\neq 0$;

  3. (c)

A local function on $V\\!ir$ is the pullback of a local function on $W$ via
the canonical map $V\\!ir\to W$.

A local function of the form (3.1) is called a one-point local function. Let
$\chi=\chi_{x;\alpha_{0},\ldots,\alpha_{n}}$ be a one-point local function. We
say that $\\{x\\}$ is the support of $\chi$ and that $x$ is the base point of
$\chi$. If $\alpha_{n}\neq 0$ we say that $n$ is the order of $\chi$.

Let $\chi$ be an arbitrary local function. The support of $\chi$ is the union
of the supports of the component one-point local functions. Further, the
orders of the component one-point local functions give rise to a partition
$\lambda(\chi)$. More explicitly, write $\chi=\chi_{1}+\dots+\chi_{r}$, where
the $\chi_{i}$ are one-point local functions based at distinct points. Let
$m_{i}$ be the order of $\chi_{i}$. By reordering the $\chi_{i}$ if necessary,
we may assume that $m_{1}\geq m_{2}\geq\dots\geq m_{r}$. The partition

$\lambda(\chi):=(m_{1},\dots,m_{r})$

is called the order partition of $\chi$. We call $m_{1}$ the order of $\chi$.

It follows from the Chinese remainder theorem that a local function is 0 if
and only if it is 0 pointwise and thus any local function $\chi$ on $W$,
$W_{\geq-1}$, or $V\\!ir$ has a unique presentation as a sum of nonzero one-
point local functions with distinct base points. This also shows that the
partition $\lambda(\chi)$ and the order of $\chi$ are well-defined. For
$W_{\geq 1}$ it is easy to see that $\chi_{0;1}=\chi_{0;0,1}=0$ and the
presentation is unique under the assumption that the coefficients of $f(0)$
and $f^{\prime}(0)$ are 0.

######  Remark 3.3.

Let $\mathfrak{g}$ be $V\\!ir$ or $W$ or $W_{\geq-1}$ or $W_{\geq 1}$. Then
$\mathfrak{g}^{*}$ and the subspace of local functions are both uncountable-
dimensional vector spaces; for local functions observe that any set of one-
point local functions with distinct base points is linearly independent. On
the other hand, clearly “most” elements of $\mathfrak{g}^{*}$ are not local.
In fact, we will see in Remark 5.6 that local functions are parameterised by a
countable union of algebraic varieties.

For a specific example of a non-local function, let
$\alpha_{0},\alpha_{1},\dots\in{\Bbbk}$ be algebraically independent over
$\mathbb{Q}$, and define $\varkappa\in W_{\geq-1}^{*}$ by
$\varkappa(t^{i}\partial)=\alpha_{i}$.

As local functions are defined similarly for $W$, $W_{\geq 1}$, $W_{\geq-1}$,
and $V\\!ir$ we sometimes discuss all of them simultaneously. When we do so,
we will assume without comment whenever we talk about a one-point local
function on $W$ or $V\\!ir$ that $x\neq 0$.

Pick $\chi\in W^{*}$ or $W_{\geq 1}^{*}$ or $W_{\geq-1}^{*}$. In this section
we will show that $\operatorname{P}(\chi)$ is nonzero iff $\chi$ is local and
prove a similar statement for $V\\!ir^{*}$. The starting point here is the
following result, due in its strongest form to Iyudu and the second author.

######  Theorem 3.4.

Let $\mathfrak{g}$ be $W$ or $W_{\geq 1}$ or $W_{\geq-1}$ and let $I$ be a
nonzero Poisson ideal of $\operatorname{S}(\mathfrak{g})$. Then

$\operatorname{GKdim}(\operatorname{S}(\mathfrak{g}))/I<\infty.$

In particular, if $\chi\in\mathfrak{g}^{*}$ is such that
$(0)\neq\operatorname{P}(\chi)$, then
$\operatorname{GKdim}(\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi))<\infty$.

Further, if $\chi\in V\\!ir^{*}$ is such that
$\operatorname{P}(\chi)\neq(z-\chi(z))$, then

$\operatorname{GKdim}(\operatorname{S}(V\\!ir)/\operatorname{P}(\chi))<\infty.$

###### Proof.

See [PS, Theorem 1.4] for $W_{\geq 1}$ and [IS, Theorem 1.3] for all other
$\mathfrak{g}$.∎

Theorem 3.4 has the following extremely useful consequence:

######  Corollary 3.5.

Let $\mathfrak{g}$ be as above and let $\chi\in\mathfrak{g}^{*}$. If
$\mathfrak{g}=W$ or $W_{\geq-1}$ or $W_{\geq 1}$ assume that
$\operatorname{P}(\chi)\neq(0)$, and if $\mathfrak{g}=V\\!ir$ assume that
$\operatorname{P}(\chi)\neq(z-\chi(z))$. Then
$\dim\mathfrak{g}\cdot\chi<\infty$.

###### Proof.

Combine Theorem 3.4 and Lemma 2.5. ∎

We also recall:

######  Proposition 3.6.

[LL, Corollary 5.1] Let $\mathfrak{g}$ be $V\\!ir$ or $W$ or $W_{\geq-1}$ or
$W_{\geq 1}$. Then $\operatorname{S}(\mathfrak{g})$ satisfies the ascending
chain condition on radical Poisson ideals and every Poisson ideal has finitely
many minimal primes, each of which is Poisson.

Although we use similar notation for $W_{\geq-1}$, $W_{\geq 1}$, $W$, and
$V\\!ir$, the details are slightly different, so we analyse local functions in
each of these cases separately.

###  3.1. Local functions on $W_{\geq-1}$ and $W_{\geq 1}$

In this subsection we set $\mathfrak{g}=W_{\geq-1}={\Bbbk}[t]\partial$. It
will be useful to consider Lie subalgebras of $W_{\geq-1}$ of a particular
form. For any $f\in{\Bbbk}[x]\setminus\\{0\\}$ denote by $W_{\geq-1}(f)$ the
space of vector fields of the form

$\\{gf\partial\\}_{g\in{\Bbbk}[x]}.$

In other words, $W_{\geq-1}(f)=fW_{\geq-1}$ under the obvious notation. It is
clear that $W_{\geq-1}(f)$ is a Lie subalgebra of $W_{\geq-1}=W_{\geq-1}(1)$.

We give five equivalent conditions for local functions. Similar conditions
will hold for the other Lie algebras we consider, see Theorems 3.13, 3.20.

######  Theorem 3.7.

Let $f\in{\Bbbk}[t]\setminus\\{0\\}$ and $\chi\in W_{\geq-1}(f)^{*}$. Then the
following conditions are equivalent:

  1. (0)

$\chi$ is the restriction of a local function on $W_{\geq-1}$;

  2. (1)

$\operatorname{P}(\chi)\neq(0)$;

  3. (2)

$\dim W_{\geq-1}(f)/W_{\geq-1}(f)^{\chi}=\dim W_{\geq-1}(f)\cdot\chi<\infty$;

  4. (3)

there exists $h\in{\Bbbk}[t]$ such that $\chi|_{W_{\geq-1}(fh)}=0$;

  5. (4)

$W_{\geq-1}(f)^{\chi}\neq(0)$.

######  Remark 3.8.

Consider a local function $\chi\in W_{\geq-1}(f)^{*}$ and fix a basis
$\\{ft^{i}\partial\ |\ i\geq-1\\}$ of $W_{\geq-1}(f)$. Then $\chi$ can be
identified with a sequence

(3.9)
$\chi_{0}=\chi(f\partial),\chi_{1}=\chi(ft\partial),\chi_{2}=\chi(ft^{2}\partial),\chi_{3}=\chi(ft^{3}\partial),\cdots\in{\Bbbk}.$

Condition $(3)$ above can be restated as follows:

$a_{n}\chi_{m+n}+...+a_{0}\chi_{m}=0$

for all $m\geq 0$, where $h=a_{n}t^{n}+\cdots+a_{0}$. Therefore, local
functions on $W_{\geq-1}(f)$ can be identified with sequences (3.9) obeying a
linear recurrence relation. This shows, in particular, that the function
$\varkappa$ defined in Remark 3.3 is not local.

Part of the proof of Theorem 3.7 is a general technique that can allow us to
show that Poisson cores of elements of $\mathfrak{g}^{*}$ are nontrivial for
any Lie algebra $\mathfrak{g}$.

######  Lemma 3.10.

Let $\mathfrak{g}$ be an arbitrary Lie algebra and let $n\in\mathbb{Z}_{\geq
0}$. There is a Poisson ideal $I(n)$ with the property that

(3.11) $I(n)\subseteq\mathfrak{m}_{\chi}\iff\dim\mathfrak{g}\cdot\chi\leq n$

for any $\chi\in\mathfrak{g}^{*}$.

###### Proof.

Recall the determinant $D(u_{1},\ldots,u_{n};v_{1},\ldots,v_{n})$ from (2.6).
If $\dim\mathfrak{g}\cdot\chi\leq~{}n,$ then (2.6) evaluated at $\chi$ is
degenerate for all $u_{1},\ldots,u_{n+1},v_{1},\ldots,v_{n+1}\in\mathfrak{g}$,
i.e.

${\rm ev}_{\chi}(D(u_{1},u_{2},\ldots,u_{n+1};v_{1},v_{2},\ldots,v_{n+1}))=0.$

Conversely, if

${\rm ev}_{\chi}(D(u_{1},u_{2},\ldots,u_{n+1};v_{1},v_{2},\ldots,v_{n+1}))=0$

for all $u_{1},\ldots,u_{n+1},v_{1},\ldots,v_{n+1}\in\mathfrak{g}$, then
$\dim\mathfrak{g}\cdot\chi\leq n$.

Let $I(n)$ be the ideal generated by the
$D(u_{1},u_{2},\ldots,u_{n+1};v_{1},v_{2},\ldots,v_{n+1})$ for all possible
tuples

$u_{1},\ldots,u_{n+1},v_{1},\ldots,v_{n+1}\in\mathfrak{g}.$

By the previous paragraph,

$I(n)\subseteq\mathfrak{m}_{\chi}\iff\dim\mathfrak{g}\cdot\chi\leq~{}n.$

Let $w\in\mathfrak{g}$. It is easy to check that

$\\{D(u_{1},u_{2},\ldots;v_{1},v_{2},\ldots),w\\}=\\\
D([u_{1},w],u_{2},u_{3},\ldots;v_{1},v_{2},\ldots)+D(u_{1},[u_{2},w],u_{3},\ldots;v_{1},v_{2},\ldots)+\ldots\\\
+D(u_{1},u_{2},\ldots;[v_{1},w],v_{2},v_{3},\ldots)+D(u_{1},u_{2},\ldots;v_{1},[v_{2},w],v_{3},\ldots)+\ldots.$

It follows that $I(n)$ is Poisson. ∎

###### Proof of Theorem 3.7.

That $(0)$ $\iff$ $(3)$ is a straightforward application of the Chinese
remainder theorem. It is clear that $(2)$ $\Rightarrow$ $(4)$; Corollary 3.5
gives that $(1)$ $\Rightarrow$ $(2)$. We will show that $(4)$ $\Rightarrow$
$(3)$ $\Rightarrow$ $(2)$ $\Rightarrow$ $(1)$. This will complete the proof.

We first show that $(4)$ implies $(3)$. Let $hf\partial\in
W_{\geq-1}(f)^{\chi}\setminus\\{0\\}$ with $h\in{\Bbbk}[t]$. Then

(3.12) $0=\chi([hf\partial,hfr\partial])=\chi(h^{2}f^{2}r^{\prime}\partial)$

for all $r\in{\Bbbk}[t]$. This is equivalent to
$\chi|_{W_{\geq-1}(h^{2}f^{2})}=0$ as needed.

Next, we will show that $(3)$ implies $(2)$. Let $h$ satisfy condition $(3)$.
As

$[W_{\geq-1}(f^{2}h^{2}),W_{\geq-1}]\subseteq W_{\geq-1}(fh),$

we have $W_{\geq-1}(h^{2}f^{2})\subseteq W_{\geq-1}(f)^{\chi}$ and thus $\dim
W(f)/W(f)^{\chi}<\infty$ as needed.

Finally we show that $(2)$ implies $(1)$. Suppose that $\chi\in W_{\geq-1}(f)$
satisfies condition $(2)$. Let $n=\dim W_{\geq-1}(f)/W_{\geq-1}(f)^{\chi}=\dim
W_{\geq-1}(f)\cdot\chi$ and let $I(n)$ be the ideal defined in Lemma 3.10. By
that lemma, $I(n)$ is Poisson and $I(n)\subseteq\mathfrak{m}_{\chi}$, so
$I(n)\subseteq\operatorname{P}(\chi)$.

Therefore if $I(n)\neq 0$ then $\operatorname{P}(\chi)\neq 0$. To show that
$I(n)\neq 0$ it suffices to find $\omega\in W_{\geq-1}(f)^{*}$ with $\dim
W_{\geq-1}(f)\cdot\omega>n$. We claim that, in fact, for the majority of
elements $\omega\in W_{\geq-1}^{*}$ we have

$\dim W_{\geq-1}(f)\cdot\omega=\infty.$

Indeed, as $(2)$ $\Rightarrow$ $(3)$, if $\dim
W_{\geq-1}(f)\cdot\omega<\infty$ then $\omega$ can be represented by a
linearly recurrent sequence by Remark 3.8, and the majority of sequences are
not linearly recurrent (e.g., $1,\frac{1}{2},\frac{1}{3},\ldots.$). ∎

Note that $W_{\geq-1}(t^{2})=W_{\geq 1}$ and that $W_{\geq-1}(t)$ is equal to
the non-negative Witt algebra $W_{\geq 0}$. Therefore Theorem 3.7 gives a
complete characterization of local functions on $W_{\geq 1}$ and $W_{\geq 0}$.

###  3.2. Local functions on $W$ and applications

In this subsection we set $\mathfrak{g}={\Bbbk}[t,t^{-1}]\partial$ and define
$W(f)$ similarly to $W_{\geq 1}(f)$. A partial analogue of Theorem 3.7 holds
for $W$.

######  Theorem 3.13.

For any $f\in{\Bbbk}[t]\setminus\\{0\\}$ and $\chi\in W(f)^{*}$ the following
conditions are equivalent:

  1. (0)

$\chi$ is the restriction of a local function on $W$;

  2. (1)

$\operatorname{P}(\chi)\neq(0)$;

  3. (2)

$\dim W(f)/W(f)^{\chi}<\infty$;

  4. (3)

there exists $h\in{\Bbbk}[x]$ such that $\chi|_{W(fh)}=0$.

######  Remark 3.14.

The reason that Theorem 3.13 differs slightly from Theorem 3.7 is that the
function $\operatorname{Res}_{0}(\cdot)\in W^{*}$ satisfies condition (4) of
Theorem 3.7 but does not satisfy the other conditions (0), (1), (2), (3).

Before proving Theorem 3.13 we give two lemmata on functions defined by
residues. Denote by ${\Bbbk}((t))$ the field of formal Laurent power series in
$t$. Fix $f\in{\Bbbk}((t))$ and consider the map

(3.15)
$(a,b)\mapsto(a,b)_{f}:=\operatorname{Res}_{0}(f(ab^{\prime}-a^{\prime}b))$

which defines a skew-symmetric bilinear form on ${\Bbbk}((t))$.

######  Lemma 3.16.

The kernel of $(a,b)_{f}$ is one-dimensional if $f$ is a perfect square in
${\Bbbk}((t))$ and is trivial otherwise. In the first case the kernel is
generated by $\frac{1}{\sqrt{f}}$.

###### Proof.

Let $a$ be in the kernel of $(\cdot,\cdot)_{f}$. Then

(3.17)
$\operatorname{Res}_{0}(fa^{2}r^{\prime})=\operatorname{Res}_{0}(f(a(ar)^{\prime}-a^{\prime}(ar)))=(a,ar)_{f}=0$

for all $r\in{\Bbbk}((t))$. This implies that $fa^{2}$ is constant. Thus $f$
is a perfect square in ${\Bbbk}((t))$ and $a$ is proportional to
$\frac{1}{\sqrt{f}}$. Equation (3.17) also gives that if $f$ is a perfect
square, then $\frac{1}{\sqrt{f}}$ belongs to the kernel of
$(\cdot,\cdot)_{f}$. ∎

The second lemma proves part of (a more general version of) Remark 3.14.

######  Lemma 3.18.

Let $g\in{\Bbbk}[t,t^{-1}]\setminus\\{0\\}$ and define $\omega\in
W(g^{2})^{*}$ by

$\omega=\operatorname{Res}_{0}(\frac{\cdot}{g^{2}}).$

Then $\dim W(g^{2})\cdot\omega=\infty$.

###### Proof.

For $a,b\in{\Bbbk}[t,t^{-1}]$ we have

$B_{\omega}(g^{2}a\partial,g^{2}b\partial)=\operatorname{Res}_{0}(g^{2}(ab^{\prime}-a^{\prime}b)).$

Suppose that $a\not\in{\Bbbk}\cdot\frac{1}{g}$. Then by Lemma 3.16 there is a
formal Laurent series $\hat{b}$ so that

$(a,b)_{g^{2}}=\operatorname{Res}_{0}(g^{2}(a\hat{b}^{\prime}-a^{\prime}\hat{b}))\neq
0.$

However, the computation of
$\operatorname{Res}_{0}(g^{2}(a\hat{b}^{\prime}-a^{\prime}\hat{b}))$ needs
only finitely many terms in the Laurent expansion of $\hat{b}$ and so we may
replace $\hat{b}$ by $b\in{\Bbbk}[t,t^{-1}]$ so that
$\operatorname{Res}_{0}(g^{2}(ab^{\prime}-a^{\prime}b))\neq 0$. Thus
$ag^{2}\not\in\ker B_{\omega}=W(g^{2})^{\omega}$. This means that
$W(g^{2})^{\omega}\subseteq{\Bbbk}\cdot g\partial$ and $\dim
W(g^{2})\cdot\omega=\infty$. ∎

###### Proof of Theorem 3.13.

The proofs of $(0)$ $\iff$ $(3)$, $(1)$ $\Rightarrow$ $(2)$ and $(3)$
$\Rightarrow$ $(2)$ $\Rightarrow$ $(1)$ are very similar to the corresponding
steps of the proof of Theorem 3.7. The only part which is significantly
different is (2)$\Rightarrow(3)$.

Pick $\chi$ satisfying $(2)$ and $h\in W(f)^{\chi}\setminus\\{0\\}$. Then
(3.12) holds for all $r\in{\Bbbk}[t,t^{-1}]$. Unfortunately, this is not
enough to show that $\chi$ vanishes on $W(h^{2}f^{2})$, as the linear map
$r\to r^{\prime}$ is not surjective on ${\Bbbk}[t,t^{-1}]$.

Consider $\chi|_{W(f^{2}h^{2})}$: we have

$\chi(f^{2}h^{2}p\partial)=\chi(\frac{f^{2}h^{2}}{t}\partial)\operatorname{Res}_{0}(p)$

for all $p\in{\Bbbk}[t]$. Suppose that $\chi(\frac{f^{2}h^{2}}{t}\partial)\neq
0$. Then $\dim W(f^{2}h^{2})\cdot\chi|_{W(f^{2}h^{2})}=\infty$ by Lemma 3.18.
This implies that $\dim W(h^{2}f^{2})\cdot\chi=\infty$, as $\dim
W(f)/W(f^{2}h^{2})<\infty$. This contradicts our assumption that $\chi$
satisfies $(2)$ and so $\chi(\frac{f^{2}h^{2}}{t}\partial)=0$, i.e.
$\chi|_{W(f^{2}h^{2})}=0$. ∎

To end the subsection, we apply Theorem 3.13 to obtain a structure result on
finite codimension subalgebras of $W$. For a polynomial
$f\in{\Bbbk}[t,t^{-1}]$ set

${\rm rad}(f):=\prod\\{(t-x)|x\in{\Bbbk}^{\times},f(x)=0\\}.$

######  Proposition 3.19.

Let $\mathfrak{k}$ be a subalgebra of $W$ of finite codimension. Then

  1. (a)

there exists $f\in{\Bbbk}[t,t^{-1}]$ so that $W({\rm
rad}(f))\supseteq\mathfrak{k}\supseteq W(f)$;

  2. (b)

we can choose $f$ satisfying $\mathrm{(a)}$ so that $f\in{\Bbbk}[t]$, $f$ is
monic and $f(0)\neq 0$;

  3. (c)

if we assume that $f$ is of minimal degree then such a choice of $f$ is
unique.

###### Proof.

The inclusion $\mathfrak{k}\subseteq W$ induces the dual map
$W^{*}\to\mathfrak{k}^{*}$. We identify $(W/\mathfrak{k})^{*}$ with the kernel
of this map; that is, with elements of $W^{*}$ which vanish on $\mathfrak{k}$.
Then $\mathfrak{k}$ is the set of common zeroes of
$(W/\mathfrak{k})^{*}\subseteq W^{*}$. Let $\chi_{1},\dots,\chi_{s}$ be a
basis of $(W/\mathfrak{k})^{*}$. Fix $i$; by definition we have
$B_{\chi_{i}}(\mathfrak{k},\mathfrak{k})=0$, so $\mathfrak{k}$ is an isotropic
subspace of $W$ with respect to $B_{\chi_{i}}$. Hence the rank of
$B_{\chi_{i}}$ is at most $2\dim(W/\mathfrak{k})$ and thus is finite. By
Theorem 3.13, therefore, $\chi_{i}$ is local.

Theorem 3.13 implies that for all $i$ there is
$h_{i}\in{\Bbbk}[t,t^{-1}]\setminus\\{0\\}$ with $\chi_{i}(W(h_{i}))=0$.
Therefore $W(h_{1}\cdots h_{s})$ is annihilated by all $\chi_{i}$ and
therefore $W(h_{1}\cdots h_{s})\subseteq\mathfrak{k}$ as desired.

Let $f\in{\Bbbk}[t,t^{-1}]\setminus\\{0\\}$ with $\mathfrak{k}\supseteq W(f)$,
we may assume without loss of generality that $f\in{\Bbbk}[t]$, $f$ is monic
and $f(0)\neq 0$ as $W(f)$ corresponds to an ideal of ${\Bbbk}[t,t^{-1}]$.
Suppose in addition that $f$ has minimal degree among all such polynomials
with $\mathfrak{k}\supseteq W(f)$. Thus if $\mathfrak{k}\supseteq W(h)$, then
by the Euclidean algorithm $f|h$. This justifies uniqueness of $f$.

Write $f=\prod_{i=1}^{n}(t-x_{i})^{a_{i}}$ with the $x_{i}\neq 0$ distinct,
$a_{i}>0$; set $h:={\rm rad}(f)=(t-x_{1})\cdots(t-x_{n}).$ It is clear that
$h\mid f\mid h^{\max(a_{i})}$. Let $k\partial\in\mathfrak{k}$. We wish to show
that $h|k$. Indeed, for all $r\in{\Bbbk}[t,t^{-1}]$ the element
$[k\partial,fr\partial]=(k(fr^{\prime}+f^{\prime}r)-frk^{\prime})\partial$ is
in $\mathfrak{k}$. As $\mathfrak{k}\supseteq W(f)$, thus
$kf^{\prime}r\in\mathfrak{k}$ for all $r\in{\Bbbk}[t,t^{-1}]$. Thus
$\mathfrak{k}\supseteq W(kf^{\prime})$ and so $f|f^{\prime}k$. This forces $k$
to vanish at all roots of $f$, which is equivalent to $h|k$. ∎

###  3.3. Local functions on $V\\!ir$

In this subsection we set $\mathfrak{g}=V\\!ir$.

The natural map $V\\!ir\to W$ extends to the morphism
$\operatorname{S}(V\\!ir)\to\operatorname{S}(W)$; the kernel is the Poisson
ideal $(z)$ of $\operatorname{S}(V\\!ir)$. The main goal of the subsection is
to prove the following analogue of Theorem 3.13 for $V\\!ir$.

######  Theorem 3.20.

For $\chi\in V\\!ir^{*}$ we have

$\operatorname{P}(\chi)\neq(z-\chi(z))\quad\iff\quad\chi\mbox{ is local
}\quad\iff\quad\dim V\\!ir/V\\!ir^{\chi}<\infty.$

In particular, if $\operatorname{P}(\chi)\neq(z-\chi(z))$ then $\chi(z)=0$.

######  Remark 3.21.

It follows from Theorems 3.13 and 3.20 that $\chi\in V\\!ir^{*}$ is local if
and only if there is some $f\in{\Bbbk}[t,t^{-1}]$ such that $\chi$ vanishes on
$W(f)+(z)\subseteq V\\!ir$.

Before proving Theorem 3.20, we consider arbitrary subalgebras of $V\\!ir$ of
finite codimension, and show they are strongly constrained.

######  Proposition 3.22.

If $\mathfrak{k}$ is a subalgebra of $V\\!ir$ of finite codimension, then
$z\in[\mathfrak{k},\mathfrak{k}]$. Thus $\dim V\\!ir/\mathfrak{k}<\infty$ if
and only if $\mathfrak{k}$ contains some $W(f)+(z)$, where
$f\in{\Bbbk}[t,t^{-1}]\setminus\\{0\\}$.

This result generalises [OW, Proposition 2.3], which considered subalgebras of
$V\\!ir$ of codimension 1. We also note that [OW] refers to subalgebras of
$V\\!ir$ of the form $W(f)+(z)$ as polynomial subalgebras.

###### Proof of Proposition 3.22.

Let $\bar{\mathfrak{k}}$ be the image of $\mathfrak{k}$ in $W$. By Proposition
3.19 there is some $f\in{\Bbbk}[t,t^{-1}]\setminus\\{0\\}$ so that
$\bar{\mathfrak{k}}\supseteq W(f)$. Thus for all $p\in\mathbb{Z}$, there is
$\lambda_{p}\in{\Bbbk}$ so that the element

$v_{p}:=ft^{p}\partial+\lambda_{p}z$

is in $\mathfrak{k}$. Thus $[\mathfrak{k},\mathfrak{k}]$ contains the elements

(3.23)
$\frac{1}{q-p}[v_{p},v_{q}]=f^{2}t^{p+q-1}\partial+\operatorname{Res}_{0}\Bigl{(}t^{p+q-3}(2t^{2}(f^{\prime})^{2}+ff^{\prime}t(p+q-1)-t^{2}ff^{\prime\prime}+f^{2}pq)\Bigr{)}z.$

Fix $d=p+q$ and consider $p=d-q$ as a function of $q$. The only part of (3.23)
that varies with $q$ is $\operatorname{Res}_{0}(t^{d-3}f^{2}q(d-q))$.

If $\operatorname{Res}_{0}(t^{d-3}f^{2})$ is not zero then

$\frac{1}{d-2q_{1}}[v_{d-q_{1}},v_{q_{1}}]-\frac{1}{d-2q_{2}}[v_{d-q_{2}},v_{q_{2}}]=(q_{1}-q_{2})(d-q_{1}-q_{2})\operatorname{Res}_{0}(t^{d-3}f^{2})z$

is a nonzero scalar multiple of $z$ for almost all $q_{1},q_{2}$. If
$z\not\in[\mathfrak{k},\mathfrak{k}]$, we therefore have
$\operatorname{Res}_{0}(t^{d}f^{2})=0$ for all $d\in\mathbb{Z}$. This implies
that $f^{2}=0$, contradicting our assumption on $f$.

The final sentence is an immediate consequence of Proposition 3.19. ∎

###### Proof of Theorem 3.20.

Let $\chi\in V\\!ir^{*}$. If $\chi$ is local then by definition $\chi$
descends to a local function $\overline{\chi}$ on $V\\!ir/(z)\cong W$. By
Theorem 3.13, $\operatorname{P}(\chi)\supsetneqq(z)$ and $\dim
V\\!ir/V\\!ir^{\chi}=\dim W/W^{\overline{\chi}}<\infty$.

If $\dim V\\!ir/V\\!ir^{\chi}<\infty$ then by Lemma 3.22,
$z\in[(V\\!ir)^{\chi},(V\\!ir)^{\chi}]$ and so $\chi(z)=0$, as $\chi$
vanishes, by definition, on $[(V\\!ir)^{\chi},(V\\!ir)^{\chi}]$. We may thus
factor out $z$ and apply Theorem 3.13 again to conclude that $\chi$ is local.

Finally, suppose that $\operatorname{P}(\chi)\neq(z-\chi(z))$. Then by
Corollary 3.5 $\dim V\\!ir\cdot\chi=\dim V\\!ir/V\\!ir^{\chi}<\infty$. ∎

As an immediate corollary of Theorem 3.20, we obtain a powerful result on
Poisson ideals of $\operatorname{S}(V\\!ir)$.

######  Corollary 3.24.

If $\lambda\neq 0$ then $\operatorname{S}(V\\!ir)/(z-\lambda)$ is Poisson
simple, i.e. contains no nontrivial Poisson ideals.

###### Proof.

If $\operatorname{S}(V\\!ir)/(z-\lambda)$ is not Poisson simple then
$(z-\lambda)$ is strictly contained in some proper Poisson ideal $J$ of
$\operatorname{S}(V\\!ir)$. By the Nullstellensatz there is some $\chi\in
V\\!ir^{*}$ with $J\subseteq\mathfrak{m}_{\chi}$; thus $\chi(z)=\lambda$ as
$z-\lambda\in\mathfrak{m}_{\chi}$. Further $\operatorname{P}(\chi)\supseteq
J\supsetneqq(z-\lambda)$ and so by Theorem 3.20 we have $\lambda=0$. ∎

We thus show that almost all prime Poisson ideals of
$\operatorname{S}(V\\!ir)$ contain $z$.

######  Corollary 3.25.

Let $Q$ be a prime Poisson ideal of $\operatorname{S}(V\\!ir)$. Then either:

  * •

$Q=(0)$;

  * •

$Q=(z-\lambda)$ for some $\lambda\in{\Bbbk}^{\times}$;

  * •

$Q\supseteq(z)$.

###### Proof.

By Corollary 3.24 and primeness of $Q$, it suffices to prove that if
$Q\neq(0)$ then $Q$ contains a nonzero element of ${\Bbbk}[z]$. Let $h\in
Q\setminus\\{0\\}$; using primeness of $Q$ we may assume that $h$ is not a
multiple of any element of ${\Bbbk}[z]\setminus{\Bbbk}$. Let $\chi\in
V\\!ir^{*}$ so that $Q\subseteq\mathfrak{m}_{\chi}$. As
$h\in\operatorname{P}(\chi)$, we see that
$\operatorname{P}(\chi)\neq(z-\chi(z))$. By Theorem 3.20
$\operatorname{P}(\chi)\supseteq(z)$. Thus, applying Lemma 2.4(a),

$Q=\bigcap\\{\operatorname{P}(\chi)\ |\
\operatorname{ev}_{\chi}(Q)=0\\}\supseteq(z).$

∎

Given Corollary 3.24 it is natural to conjecture:

######  Conjecture 3.26.

If $\lambda\neq 0$ then $\operatorname{U}(V\\!ir)/(z-\lambda)$ is simple.

However, we as yet have no proof of Conjecture 3.26. Note that the obvious
strategy of proof by taking the associated graded of an ideal
$J\supsetneqq(z-\lambda)$ does not work; for in this case
$\operatorname{gr}J\ni z$ and Corollary 3.24 is not directly relevant.

##  4\. Pseudo-orbits and Poisson primitive ideals for the algebras of
interest

Let $\mathfrak{g}$ be one of $V\\!ir,W,W_{\geq-1}$, or $W_{\geq 1}$. In this
section we will describe the pseudo-orbits for $\mathfrak{g}$, using the
results on local functions from the previous section and the general strategy
of Proposition 2.13, and derive some consequences for the Poisson primitive
spectrum of $\operatorname{S}(\mathfrak{g})$.

We begin by describing the pseudo-orbits of non-local functions, where the
results of Section 3 quickly give the answer.

######  Proposition 4.1.

For $\mathfrak{g}=W,W_{\geq-1}$, or $W_{\geq 1}$, the non-local functions in
$\mathfrak{g}^{*}$ form a single pseudo-orbit. If $\mathfrak{g}=V\\!ir$, then
for any $\lambda\in{\Bbbk}$ the non-local functions $\chi$ with
$\chi(z)=\lambda$ form a pseudo-orbit.

###### Proof.

This is immediate from Theorems 3.7 ($\mathfrak{g}=W_{\geq-1}$ or $W_{\geq
1}$), 3.13 ($\mathfrak{g}=W$), or 3.20 ($\mathfrak{g}=V\\!ir$). For by those
results, if $\mathfrak{g}=W,W_{\geq-1},$ or $W_{\geq 1}$ then
$\chi\in\mathfrak{g}^{*}$ is not local if and only if
$\operatorname{P}(\chi)=0$; and if $\mathfrak{g}=V\\!ir$, then $\chi$ is not
local if and only if $\operatorname{P}(\chi)=(z-\chi(z))$. ∎

We may thus restrict to considering pseudo-orbits of local functions. By
Proposition 4.1 if $\chi\in\mathfrak{g}^{*}$ is local and
$\omega\in\mathbb{O}(\chi)$ then $\omega$ is also local. Since by definition
local functions on $V\\!ir$ vanish on $z$, the pseudo-orbits for $W$ directly
determine those for $V\\!ir$.

Thus for the rest of the section we let $\mathfrak{g}$ be $W,W_{\geq-1}$, or
$W_{\geq 1}$. In Subsection 4.1 we will introduce a finite dimensional action
which determines the pseudo-orbits of one-point local functions on
$\mathfrak{g}$, in Subsection 4.2 we describe the orbits of this action
explicitly, and in Subsection 4.3 we use this action to describe pseudo-orbits
of arbitrary local functions.

###  4.1. An algebraic group acting on local functions

Set $\mathfrak{g}$ to be $W$, $W_{\geq-1}$, or $W_{\geq 1}$. We fix notation
for the subsection.

######  Definition 4.2.

For $x\in{\Bbbk}$, $n\in\mathbb{Z}_{\geq 0}$, let ${\it Loc}_{x}^{\leq n}$
denote the subspace of $\mathfrak{g}^{*}$ consisting of one-point local
functions based at $x$ and of order $\leq n$. Let ${\it
Loc}_{x}:=\bigcup_{n\geq 0}{\it Loc}_{x}^{\leq n}.$ Define ${\it Loc}^{\leq
n}_{\mathfrak{g}}=\bigcup_{x\in{\Bbbk}^{\times}}{\it Loc}_{x}^{\leq n}$ if
$\mathfrak{g}=W$, and ${\it Loc}^{\leq
n}_{\mathfrak{g}}=\bigcup_{x\in{\Bbbk}}{\it Loc}_{x}^{\leq n}$ if
$\mathfrak{g}=W_{\geq-1}$ or $W_{\geq 1}$. Let

$\widetilde{{\it Loc}}_{\mathfrak{g}}^{\leq
n}=\begin{cases}{\Bbbk}^{\times}\times{\Bbbk}^{n+1}&\mathfrak{g}=W\\\
{\Bbbk}\times{\Bbbk}^{n+1}&\mathfrak{g}=W_{\geq-1}\\\
{\Bbbk}\times{\Bbbk}^{n+1}&\mathfrak{g}=W_{\geq 1}.\end{cases}$

For all $\mathfrak{g}$ there is a canonical map $\pi_{\mathfrak{g}}^{\leq
n}:\widetilde{{\it Loc}}_{\mathfrak{g}}^{\leq n}\to{\it
Loc}_{\mathfrak{g}}^{\leq n}$ (this is in fact birational, cf. Corollary 5.2).
If $\mathfrak{g}=W,W_{\geq-1}$ then $\pi_{\mathfrak{g}}^{\leq n}$ is an
isomorphism away from $\chi=0$. If $\mathfrak{g}=W_{\geq 1}$ it is an
isomorphism away from $\chi=0$ and $x=0$, due to the fact that
$\chi_{0;\beta_{0},\beta_{1}}=0$ for all $\beta_{0},\beta_{1}\in{\Bbbk}$ in
this case. Formally, we let $\phi_{\mathfrak{g}}^{\leq n}:\widetilde{{\it
Loc}}_{\mathfrak{g}}^{\leq n}\to\mathfrak{g}^{*}$ be the composition of
$\pi_{\mathfrak{g}}^{\leq n}$ with the inclusion ${\it
Loc}_{\mathfrak{g}}^{\leq n}\subseteq\mathfrak{g}^{*}$. There is an induced
pullback morphism $(\phi^{\leq
n}_{\mathfrak{g}})^{*}:\operatorname{S}(\mathfrak{g})\to{\Bbbk}[\widetilde{{\it
Loc}}_{\mathfrak{g}}^{\leq n}]$.

If $\chi$ is a local function on $\mathfrak{g}$ and $v\in\mathfrak{g}$, then
$v\cdot\chi$ is local, and in fact the coadjoint action of $\mathfrak{g}$
preserves ${\it Loc}_{x}$ for all $x$. We now study the action of
$\mathfrak{g}$ on ${\it Loc}_{x}$, and relate it to a finite-dimensional
action using Proposition 2.13.

We begin by defining a group action. Fix $x\in{\Bbbk}$ (if $\mathfrak{g}=W$ we
assume that $x\neq 0$). Let $\tilde{t}:=t-x$. Clearly, (3.1) makes sense for
every formal power series $f\partial\in{\Bbbk}[[\tilde{t}]]\partial$, and so
${\it Loc}_{x}^{\leq n}$ also gives elements of
$({\Bbbk}[[\tilde{t}]]\partial)^{*}$. For every $s\in{\Bbbk}[[\tilde{t}]]$
with $s(x)=x$ we introduce a local change of coordinates endomorphism
$\operatorname{End}_{t\to s}(\cdot)$ of ${\Bbbk}[[\tilde{t}]]$ through the
formula $t\to s$. Note that $\operatorname{End}_{t\to s}$ is invertible if
$s^{\prime}(x)\neq 0$. Let ${\it DLoc}_{x}$, the group of formal local
diffeomorphisms at $x$, denote the group of all $\operatorname{End}_{t\to s}$
with $s(x)=x$ and $s^{\prime}(x)\neq 0$. The group ${\it DLoc}_{x}$ has a
subgroup of transformations of the form $t\to\lambda t+(1-\lambda)x$, for
$\lambda\in{\Bbbk}^{\times}$; we also write this transformation as
$\tilde{t}\to\lambda\tilde{t}$ and let ${\it Dil}_{x}$ denote the group of
such transformations, which we term dilations at $x$.

Pick $s\in{\Bbbk}[[\tilde{t}]]$ with $s(x)=x$ and $s^{\prime}(x)\neq 0$. We
extend $\operatorname{End}_{t\to s}$ to an automorphism of
${\Bbbk}[[\tilde{t}]]\partial$ via the formulae

(4.3) $t\mapsto s,\hskip
10.0pt\partial\mapsto\frac{1}{s^{\prime}}\partial~{}(=\partial_{s}).$

This gives an action of ${\it DLoc}_{x}$ on ${\Bbbk}[[\tilde{t}]]\partial$ and
on $({\Bbbk}[[\tilde{t}]]\partial)^{*}$. We may consider ${\it Loc}_{x}^{\leq
n}$ as a subset of $({\Bbbk}[[\tilde{t}]]\partial)^{*}$, and this subset is
preserved by the ${\it DLoc}_{x}$ action.

Denote by ${\it DLoc}_{x}^{\leq n}$ the image of ${\it DLoc}_{x}$ in the group
$\operatorname{Aut}({\it Loc}_{x}^{\leq n})$ of linear automorphisms of ${\it
Loc}_{x}^{\leq n}$. Although ${\it DLoc}_{x}$ is infinite-dimensional, its
image in $\operatorname{Aut}({\it Loc}_{x}^{\leq n})$ is a finite-dimensional
solvable algebraic group. Let us consider the action of the corresponding
(finite-dimensional) Lie algebra, which we denote by
$\operatorname{lie}_{x}^{\leq n}$.

######  Lemma 4.4.

Identify $\operatorname{lie}_{x}^{\leq n}$ with the tangent space to ${\it
DLoc}_{x}^{\leq n}$ at the identity. Let $s\in{\Bbbk}[[t]]$ with $s(x)=0$ and
denote by $\xi_{s}$ the tangent direction at the identity defined by the line
$h\to\operatorname{End}_{t\to t+hs}(\cdot)$ for $h\in{\Bbbk}$. The action of
${\it DLoc}_{x}^{\leq n}$ derives to an action of $\xi_{s}$ on ${\it
Loc}_{x}^{\leq n}$. Then

  1. (a)

$\operatorname{lie}_{x}^{\leq n}$ consists of vectors of the form $\xi_{s}$
with $s(x)=0$;

  2. (b)

$\xi_{s}\cdot\chi=s\partial\cdot\chi$ for all $\chi\in{\it Loc}_{x}$.

###### Proof.

Part (a) is clear from the definition of ${\it DLoc}_{x}^{\leq n}$. For (b),
as the action of $\xi_{s}$ on ${\it Loc}_{x}$ is induced by the action (4.3)
on ${\Bbbk}[[\tilde{t}]]\partial$, it is enough to check that $\xi_{s}\cdot
f\partial=[s\partial,f\partial]$ for all $f\partial\in\mathfrak{g}$. But

$\displaystyle\xi_{s}\cdot f\partial$ $\displaystyle=(\xi_{s}\cdot
f)\partial+f(\xi_{s}\cdot\partial)$
$\displaystyle=(\partial_{h}f(t+hs))|_{h=0}\partial+f(\partial_{h}(\operatorname{End}_{t\to
t+hs}\partial))|_{h=0}$
$\displaystyle=(f^{\prime}(t+hs)s)|_{h=0}\partial+f(\partial_{h}(\frac{1}{1+hs^{\prime}}\partial))|_{h=0}$
$\displaystyle=f^{\prime}s\partial-fs^{\prime}\partial=[s\partial,f\partial],$

as needed. ∎

We now let $\mathfrak{g}=W_{\geq-1}$. For all $x,n$, the coadjoint action of
$W_{\geq-1}(t-x)$ preserves ${\it Loc}_{x}^{\leq n}$, and Lemma 4.4 shows that

(4.5) $W_{\geq-1}(t-x)\cdot\chi=\operatorname{lie}_{x}^{\leq n}\cdot\chi$

for $\chi\in{\it Loc}_{x}^{\leq n}$. For $z\in{\Bbbk}$ define

${\it
Shift}_{z}(\chi_{x;\alpha_{0},\alpha_{1},\ldots}):=\chi_{x+z;\alpha_{0},\alpha_{1},\ldots}.$

The set $\\{{\it Shift}_{z}|z\in{\Bbbk}\\}$ forms a one-dimensional algebraic
group, which we denote by ${\it Shifts}$; clearly ${\it
Shifts}\cong{\Bbbk}^{+}$. Deriving the action of ${\it Shifts}$ on ${\it
Loc}_{W_{\geq-1}}$ and on ${\it Loc}_{W_{\geq-1}}^{\leq n}$, we have

(4.6) $\partial_{h}{\it
Shift}_{h}(\chi_{x;\alpha_{0},\cdots,\alpha_{n}}(f\partial))|_{h=0}\\\
=\partial_{h}(\alpha_{0}f(x+h)+\alpha_{1}f^{\prime}(x+h)+\cdots+\alpha_{n}f^{(n)}(x+h))|_{h=0}\\\
=\alpha_{0}f^{\prime}(x)+\cdots+\alpha_{n}f^{(n+1)}(x)=(\partial\cdot\chi_{x;\alpha_{0},\cdots,\alpha_{n}})(f\partial)$

and thus we may without ambiguity identify the Lie algebra of ${\it Shifts}$
with ${\Bbbk}\cdot\partial$.

For all $x,z\in{\Bbbk}$, ${\it Shift}_{z}$ gives a continuous homomorphism
${\Bbbk}[[t-x]]\to{\Bbbk}[[t-x-z]]$ and hence induces an isomorphism

${\it Shift}_{z}:{\it DLoc}^{\leq n}_{x}\to{\it DLoc}^{\leq n}_{x+z}.$

In particular, we can take $x=1$, and then for $\mathfrak{g}=W_{\geq-1}$ and
for all $n$ the action map ${\it Shifts}\times{\it Loc}^{\leq n}_{1}\to{\it
Loc}_{W_{\geq-1}}^{\leq n}$ is clearly bijective. This allows us to introduce
the action of $\widehat{{\it DLoc}}^{\leq n}:={\it Shifts}\times{\it
DLoc}_{1}^{\leq n}$ and $\widehat{{\it DLoc}}:={\it Shifts}\times{\it
DLoc}_{1}$ on

${\it Loc}_{W_{\geq-1}}^{\leq n}\cong{\Bbbk}\times{\it Loc}_{1}^{\leq n}$

componentwise. Note that $\widehat{{\it DLoc}}$ and $\widehat{{\it
DLoc}}^{\leq n}$ also act on $\widetilde{{\it Loc}}_{W_{\geq-1}}^{\leq n}$ in
such a way that $\pi_{W_{\geq-1}}^{\leq n}$ is equivariant.

Let $\widehat{\operatorname{lie}}^{\leq n}$ denote the Lie algebra of
$\widehat{{\it DLoc}}^{\leq n}$. From Lemma 4.4 we have:

######  Lemma 4.7.

Let $\mathfrak{g}=W,W_{\geq-1}$ or $W_{\geq 1}$ and fix $x$, $n$, and
$\chi\in{\it Loc}_{x}^{\leq n}$ (recall that $x\neq 0$ if $\mathfrak{g}=W$).

  1. (a)

If $\mathfrak{g}=W_{\geq-1}$ then

$\mathfrak{g}\cdot\chi=\operatorname{lie}_{x}^{\leq
n}\cdot\chi+{\Bbbk}\partial_{x}\cdot\chi.$

  2. (b)

If $\mathfrak{g}=W_{\geq-1}$ then $\mathfrak{g}\cdot\chi=\widehat{\rm
lie}^{\leq n}\cdot\chi$ for all $\chi\in{\it Loc}_{\mathfrak{g}}^{\leq n}$.

  3. (c)

If $x\neq 0$ then we can identify ${\it Loc}_{x}^{\leq n}$ for $W,W_{\geq-1},$
and $W_{\geq 1}$. If $\mathfrak{g}=W$ or $\mathfrak{g}=W_{\geq 1}$, under this
identification we have $\mathfrak{g}\cdot\chi=W_{\geq-1}\cdot\chi$.

  4. (d)

If $\mathfrak{g}=W_{\geq 1}$ and $x=0$ then
$\mathfrak{g}\cdot\chi=\operatorname{lie}_{x}^{\leq n}\cdot\chi$.

###### Proof.

Parts (a), (b), and (d) are straightforward from the above discussion. For
part (c) note that the image of $f\partial$ in ${\it Loc}_{x}^{\leq n}$
depends only on the Taylor series expansion of $f$ around $x$ up to degree
$n+1$, which is not affected by the behaviour of $f$ at $0$. ∎

It is useful to have a more detailed description of ${\it DLoc}_{x}$ and ${\it
DLoc}_{x}^{\leq n}$. The main point here is that ${\it DLoc}_{x}^{\leq n}$ is
a solvable algebraic group and thus it has a filtration by normal subgroups
with one-dimensional quotients.

Pick $k\geq 2$. Let $x\in{\Bbbk}$ (or $x\in{\Bbbk}^{\times}$ if
$\mathfrak{g}=W$), and recall that $\tilde{t}$ denotes $t-x$. Elements of the
form

$\operatorname{End}_{\tilde{t}\to\tilde{t}+{\tilde{t}}^{k}h},\hskip
10.0pth\in{\Bbbk}[[\tilde{t}]],$

constitute a subgroup of ${\it DLoc}_{x}$ and we denote this subgroup by ${\it
DLoc}_{x}^{k+}$. We use the notation ${\it DLoc}_{x}^{\leq n,k+}$ for the
image of ${\it DLoc}_{x}^{k+}$ in ${\it DLoc}_{x}^{\leq n}$. The following
lemma is straightforward.

######  Lemma 4.8.

Let $x\in{\Bbbk}$, and let $\mathfrak{g}=W,W_{\geq 1},W_{\geq-1}$ with $x\neq
0$ for $\mathfrak{g}=W$.

  1. (a)

${\it DLoc}_{x}^{2+,\leq n}$ is the unipotent radical of ${\it DLoc}_{x}^{\leq
n}$ and ${\it Dil}_{x}$ is a maximal reductive subgroup of ${\it
DLoc}_{x}^{\leq n}$. In particular, the natural map

${\it Dil}_{x}\to{\it DLoc}_{x}^{\leq n}/{\it DLoc}_{x}^{2+,\leq n}$

is an isomorphism.

  2. (b)

${\it DLoc}_{x}^{k+,\leq n}/{\it DLoc}_{x}^{k+1+,\leq n}\cong{\Bbbk}$ or
$\\{0\\}$ if $2\leq k\leq n$. Moreover, the natural map

${\Bbbk}\to{\it DLoc}_{x}^{k+,\leq n}/{\it DLoc}_{x}^{k+1+,\leq n}$

induced by sending
$h\mapsto\operatorname{End}_{\tilde{t}\to\tilde{t}+h\tilde{t}^{n}}(\cdot)$ is
surjective.

  3. (c)

${\it DLoc}_{x}^{k+,\leq n}$ is trivial for $k\geq n$. ∎

###  4.2. Explicit description of ${\it DLoc}_{x}^{\leq n}$-orbits

We now compute the ${\it DLoc}_{x}$-orbits of one-point local functions; in
the next subsection we will see that this allows us to compute the pseudo-
orbit of an arbitrary local function. If $\mathfrak{g}=W_{\geq-1}$ then the
action of ${\it DLoc}_{x}$ is clearly homogeneous in $x$ for all
$x\in{\Bbbk}$, and similarly for $W$ for $x\neq 0$. If $x=0$ and
$\mathfrak{g}=W_{\geq 1}$ the story is a bit more delicate.

Let $\mathfrak{g}$ be $W_{\geq-1}$ or $W$ or $W_{\geq 1}$, and let
$x\in{\Bbbk}$. If $\mathfrak{g}=W$ or $W_{\geq-1}$ we additionally assume that
$x\neq 0$. For $i\in\mathbb{Z}_{\geq 0}$ we set $e_{i}(x)=(t-x)^{i+1}\partial$
and define $e_{i}(x)^{*}$ by the formula

$f\partial\to\frac{f^{i+1}(x)}{(i+1)!},$

so that $e_{i}(x)^{*}(e_{j}(x))=\delta_{ij}$. (We view the $e_{i}(x)^{*}$ as
elements either of $\mathfrak{g}^{*}$ or $({\Bbbk}[[t-x]]\partial)^{*}$,
depending on context.) The main goal of this subsection is to prove the
following theorem, and to consider its consequences.

######  Theorem 4.9.

Assume $\mathfrak{g}=W_{\geq-1}$ or $W$ or $W_{\geq 1}$, and let
$x\in{\Bbbk}$. If $\mathfrak{g}=W$ or $W_{\geq-1}$ we additionally assume that
$x\neq 0$. Fix $n,\beta_{0},\ldots,\beta_{n}$ with $\beta_{n}\neq 0$ and set
$\chi=\chi_{x;\beta_{0},...,\beta_{n}}$. Let $e_{i}:=e_{i}(x)$ and
$e_{i}^{*}:=e_{i}(x)^{*}$ for all $i\in\mathbb{Z}_{\geq 0}$.

  * (a)

If $n$ is even then ${\it DLoc}_{x}^{\leq n}\chi={\it DLoc}_{x}^{\leq
n}e_{n-1}^{*}$, and $\dim{\it DLoc}_{x}^{\leq n}\chi=n+1$.

  * (b)

If $n$ is odd and $n>1$ then there is $\beta$ such that ${\it DLoc}_{x}^{\leq
n}\chi={\it DLoc}_{x}^{\leq n}(e_{n-1}^{*}+\beta e_{k}^{*})$ where
$k=\frac{n-1}{2}$. We have $\dim{\it DLoc}_{x}^{\leq n}\chi=n$.

  * (b′)

If $n=1$ then ${\it DLoc}_{0}^{\leq n}\chi={\it DLoc}_{x}^{\leq
n}(\beta_{1}e_{0}^{*})$, and $\dim{\it DLoc}_{x}^{\leq n}\chi=1$.

  * (c)

Pick $\beta_{1},\beta_{2}\in{\Bbbk}$ and $k\in\mathbb{Z}_{\geq 1}$. Then ${\it
DLoc}_{x}^{\leq n}(e_{2k}^{*}+\beta_{1}e_{k}^{*})={\it DLoc}_{x}^{\leq
n}(e_{2k}^{*}+\beta_{2}e_{k}^{*})$ if and only if $\beta_{1}=\pm\beta_{2}$.

  * (c′)

Pick $\beta_{1},\beta_{2}\in{\Bbbk}$. Then ${\it DLoc}_{x}^{\leq
1}(\beta_{1}e_{0}^{*})={\it DLoc}_{x}^{\leq 1}(\beta_{2}e_{0}^{*})$ if and
only if $\beta_{1}=\beta_{2}$.

###### Proof.

The proof is the same for any base point; to reduce notation we give the proof
for $\mathfrak{g}=W_{\geq-1}$ and $x=0$.

First we compute the dimensions of the corresponding orbits. Note that

$\dim{\it DLoc}_{0}^{\leq n}\chi=\dim\operatorname{lie}_{0}^{\leq
n}\cdot\chi=\dim W_{\geq 0}\cdot\chi,$

see (4.5). Moreover, (4.6) implies that $W_{\geq-1}\cdot\chi\not\subseteq{\it
Loc}_{0}^{\leq n}$. Thus $\dim W_{\geq-1}\cdot\chi=\dim W_{\geq
0}\cdot\chi+1$. Further $\dim W_{\geq-1}\cdot\chi=\operatorname{rk}B_{\chi}$.
The rank of this form can be evaluated explicitly. Indeed,

$\chi_{0;\beta_{0},\ldots,\beta_{n}}([e_{i},e_{j}])=\begin{cases}0&{\rm~{}if~{}}i+j>n-1\\\
(j-i)(i+j+1)!\beta_{i+j+1}&{\rm~{}if~{}}i+j\leq n-1\end{cases}.$

From this formula it is clear that $B_{\chi}(e_{i},e_{j})=0$ if $i\geq n+1$ or
$j\geq n+1$. Thus the rank of this form can be evaluated on the first
$n+2\times n+2$ entries corresponding to $-1\leq i,j\leq n$. This block is
skew-upper-triangular. If $n$ is even then all values on the skew-diagonal
line are nonzero (because $\beta_{n}\neq 0$); hence the rank is $n+2$ in this
case. If $n$ is odd then this skew-diagonal line contains zero only in the
position corresponding to $i=j=\frac{n-1}{2}$; hence the rank is $n+1$ in this
case. This provides the desired dimensions.

The idea of the rest of the proof is to use the subnormal series in Lemma 4.8
to reduce the number of coefficients which we have to consider step by step.

Let $i\in\mathbb{Z}_{\geq 0}$. For $j\geq 1$ we have

$\operatorname{End}_{t\to t+\alpha t^{j+1}}(e_{i})=\frac{(t+\alpha
t^{j+1})^{i+1}}{1+\alpha(j+1)t^{j}}\partial\\\
=(t^{i+1}+\alpha(i-j)t^{i+j+1}+\mbox{higher})\partial=e_{i}+\alpha(i-j)e_{i+j}+\mbox{higher}.$

Thus

(4.10) $\operatorname{End}_{t\to t+\alpha
t^{j+1}}(e_{i}^{*})=e_{i}^{*}+\alpha(i-2j)e_{i-j}^{*}+\mbox{a linear
combination of $e_{<i-j}^{*}$}.$

Likewise

(4.11) $\operatorname{End}_{t\to\lambda t}(e_{i}^{*})=\lambda^{i}e_{i}^{*}.$

We apply these transformations to

(4.12)
$\chi_{0;\beta_{0},\dots,\beta_{n}}=\beta_{0}e_{-1}^{*}+\dots+\beta_{n}(n!)e_{n-1}^{*},$

noting that the coefficient of $e_{n-1}^{*}$ in (4.12) is nonzero.

By applying $\operatorname{End}_{t\to t+\alpha_{1}t^{2}}$ with appropriate
$\alpha_{1}$ we may cancel the coefficient of $e_{n-2}^{*}$, using (4.10).
This does not affect the coefficient of $e_{n-1}^{*}$. Then by applying the
appropriate $\operatorname{End}_{t\to t+\alpha_{2}t^{3}}$ we can cancel the
coefficient of $e_{n-3}^{*}$ without changing the coefficient of $e_{n-2}^{*}$
or $e_{n-1}^{*}$. Repeating, we may cancel the coefficients of all
$e_{n-1-k}^{*}$ with $1\leq k\leq n$, unless $k=\frac{n-1}{2}$.

Thus in case (a) or (b′) we obtain that $\omega=\beta_{n}e_{n-1}^{*}\in{\it
DLoc}_{0}^{\leq n}(\chi)$. In case (b) we obtain some

$\omega=\alpha e_{\frac{n-1}{2}}^{*}+\beta_{n}e_{n-1}^{*}\in{\it
DLoc}_{0}^{\leq n}(\chi).$

Now applying (4.11) to $\omega$ we may rescale the coefficient of
$e_{n-1}^{*}$ by any $\lambda^{n-1}$, so if $n\neq 1$ we may set the
coefficient of $e_{n-1}^{*}$ to be $1$. This proves rest of (a), (b), (b′).

We now prove (c′). It follows from (4.10) that applying any nontrivial element
of the unipotent radical of ${\it DLoc}_{0}^{\leq 1}$ to $e_{0}^{*}$ will give
a nonzero $e_{-1}^{*}$ term. Thus we must simply consider acting by ${\it
Dil}_{0}$, and rescaling as in (4.11) does not affect the coefficient of
$e_{0}^{*}$.

Finally, we prove (c). Formula (4.11) implies that if $\beta_{1}=\pm\beta_{2}$
then

${\it DLoc}_{0}^{\leq n}(e_{2k}^{*}+\beta_{1}e_{k}^{*})={\it DLoc}_{0}^{\leq
n}(e_{2k}^{*}+\beta_{2}e_{k}^{*}).$

Pick $\beta_{1},\beta_{2}\in{\Bbbk}$ and set
$\chi_{1}=e_{2k}^{*}+\beta_{1}e_{k}^{*},\chi_{2}=e_{2k}^{*}+\beta_{2}e_{k}^{*}$.
Assume that ${\it DLoc}_{0}^{\leq n}(\chi_{1})={\it DLoc}_{0}^{\leq
n}(\chi_{2})$, i.e. there exists $s\in{\Bbbk}[[t]]$ with
$s(0)=0,s^{\prime}(0)\neq 0$ and $\operatorname{End}_{t\to
s}(\chi_{1})=\chi_{2}$.

If

(4.13) $s=s^{\prime}(0)t$

then the statement of (c) is straightforward. Assume to the contrary that
$s\neq s^{\prime}(0)t$. Then

$s(t)\neq s^{\prime}(0)t\mod t^{d}$

for some $d$, and we choose $d$ to be minimal.

If $d\geq 2k+2$ then linear operators $\operatorname{End}_{t\to s}$ and
$\operatorname{End}_{t\to s^{\prime}(0)t}$ coincide after the restriction to
${\it Loc}_{0}^{\leq n}$, i.e. we can replace $s$ by $s^{\prime}(0)t$, which
is case (4.13).

If $d\leq 2k+1$ then

$s=s^{\prime}(0)t+\frac{s^{(d)}(0)}{d!}t^{n}\mod t^{d+1}$

with $s^{(d)}(0)\neq 0$; set
$\gamma=s^{\prime}(0),\tau=\frac{s^{(d)}(0)}{d!}$. Further (4.10), (4.11)
imply

$\operatorname{End}_{t\to
s}(\chi_{1})=\gamma^{-2k}e_{2k}+\gamma^{i-2j-1}\tau(2k-2d)e_{2k-d}\mod{\it
Loc}_{0}^{\leq 2k-d}.$

This cannot give $\chi_{2}$ unless $d=k$.

So suppose $d=k$, and note that

$\operatorname{End}_{t\to t+\alpha
t^{k+1}+(\alpha^{2}(k+1)-\alpha\frac{\beta}{2})t^{2k+1}}(e_{2k}^{*}+\beta
e_{k}^{*})=(e_{2k}^{*}+\beta e_{k}^{*})$

for every $\alpha\in{\Bbbk}$. Thus we can replace $s$ by

(4.14) $s+\alpha s^{k+1}+(\alpha^{2}(k+1)-\alpha\frac{\beta}{2})s^{2k+1}$

for an arbitrary $\alpha$. Pick
$\alpha=-\frac{s^{(d+1)}(0)}{(d+1)!(s^{\prime}(0))^{d+1}}$. Then (4.14) is
equal to $s^{\prime}(0)t\mod t^{d+1}$ and we have reduced this case to the
previous one. ∎

A modification of Theorem 4.9 holds for $\mathfrak{g}=W_{\geq 1}$ and $x=0$
with a very similar proof, which we leave to the reader.

######  Theorem 4.15.

Assume $\mathfrak{g}=W_{\geq 1}$. Fix $n\geq 2$ and
$\beta_{2},\ldots,\beta_{n}$ with $\beta_{n}\neq 0$ and set
$\chi=\chi_{0;0,0,\beta_{2},...,\beta_{n}}$.

  * (a)

If $n$ is even then ${\it DLoc}_{0}^{\leq n}\chi={\it DLoc}_{0}^{\leq
n}e_{n-1}^{*}$. We have $\dim{\it DLoc}_{0}^{\leq n}\chi=n-1.$

  * (b)

If $n$ is odd and $n>1$ then there is $\beta$ such that ${\it DLoc}_{0}^{\leq
n}\chi={\it DLoc}_{0}^{\leq n}(e_{n-1}^{*}+\beta e_{k}^{*})$ where
$k=\frac{n-1}{2}$. We have $\dim{\it DLoc}_{0}^{\leq n}\chi=n-2$.

  * (c)

Pick $\beta_{1},\beta_{2}\in{\Bbbk}$ and $k\in\mathbb{Z}_{\geq 1}$. Then ${\it
DLoc}_{0}^{\leq n}(e_{2k}^{*}+\beta_{1}e_{k}^{*})={\it DLoc}_{0}^{\leq
n}(e_{2k}^{*}+\beta_{2}e_{k}^{*})$ if and only if $\beta_{1}=\pm\beta_{2}$.

∎

We next describe the “Bruhat order” on ${\it DLoc}_{x}^{\leq n}$ orbits (that
is, the inclusions of orbit closures): it turns out that it can be computed
just from the dimension of the orbit.

######  Corollary 4.16.

Let $\mathfrak{g},x$ be as in the statement of Theorem 4.9. Let
$\chi^{-},\chi^{+}$ be local functions on $\mathfrak{g}$ based at $x$ and of
order $\leq n$. Then the following conditions are equivalent:

  1. (1)

$\dim{\it DLoc}_{x}^{\leq n}\cdot\chi^{-}<\dim{\it DLoc}_{x}^{\leq
n}\cdot\chi^{+}$;

  2. (2)

${\it DLoc}_{x}^{\leq n}\cdot\chi^{-}$ is contained in the closure of ${\it
DLoc}_{x}^{\leq n}\cdot\chi^{+}$ and ${\it DLoc}_{x}^{\leq
n}\cdot\chi^{-}\neq{\it DLoc}_{x}^{\leq n}\cdot\chi^{+}$;

  3. (3)

$\operatorname{P}(\chi^{+})\subsetneqq\operatorname{P}(\chi^{-})$.

###### Proof.

We again give the proof for $\mathfrak{g}=W_{\geq-1}$ and $x=0$. We will show
that $(3)\Rightarrow(1)\Leftrightarrow(2)\Rightarrow(3)$.

It is clear that $(2)$ implies $(1)$. Now we will show that $(1)$ implies
$(2)$. Pick $n$ and the related presentation
$\chi_{0;\beta_{0}^{+},\beta_{1}^{+},\ldots,\beta_{n}^{+}}$ for $\chi^{+}$ as
in Theorem 4.9. If $n$ is even then the closure of ${\it DLoc}_{0}^{\leq
n}\cdot\chi^{+}$ equals ${\it Loc}_{0}^{\leq n}$. One the other hand all
orbits of strictly smaller dimension belong to ${\it Loc}_{0}^{\leq n}$, so
this completes the proof in this case.

If $n=2k+1$ is odd then the closure of ${\it DLoc}_{0}^{\leq n}\cdot\chi^{+}$
is an irreducible subvariety of ${\it Loc}_{0}^{\leq n}$ of codimension 1,
i.e. a hypersurface defined by some function $F_{+}$. We claim that this
function is semiinvariant with respect to ${\it DLoc}_{0}^{\leq n}$.

To prove that $F_{+}$ is semiinvariant, we describe it in more detail. Let
$R={\Bbbk}[\beta_{0},\beta_{1},\ldots,\beta_{n-1},\beta_{n}^{\pm 1}]$, where
$\beta_{0},\beta_{1},\ldots,\beta_{n}$ are free variables. Denote by
$(\cdot)_{{\Bbbk}\to R}$ the base change from ${\Bbbk}$ to $R$. The arguments
of Theorem 4.9(b) imply that there exists a group element $g\in({\it
DLoc}_{0}^{\leq n})_{{\Bbbk}\to R}$ and $h\in R$ such that

$g\cdot\chi_{\mbox{$0$};\underbrace{0,\ldots,0}_{k\text{~{}times}},\mbox{$h$},\underbrace{0,\ldots,0}_{k\text{~{}times}},\mbox{$\beta_{n}$}}=\chi_{0;\beta_{0},\beta_{1},\ldots,\beta_{n}}.$

Recall that $k=\frac{n-1}{2}$. By replacing each $\beta_{i}$ by
$\frac{e_{i-1}}{i!}$ (cf. (4.12)) we may identify $R$ with ${\Bbbk}[{\it
Loc}_{0}^{\leq n}][e_{2k}^{-1}]$. (Here we regard the $e_{i}$ as functions on
${\it Loc}_{0}^{\leq n}$ in the obvious way.) Then $h=f/e_{2k}^{\ell}$ for
some $f\in{\Bbbk}[{\it Loc}_{0}^{\leq n}]$ and $\ell\in\mathbb{Z}_{\geq 0}$.

Let $\chi,\chi^{\prime}\in{\it Loc}_{0}^{\leq n}$ with
$e_{2k}(\chi),e_{2k}(\chi^{\prime})\neq 0$. As in the proof of Theorem 4.9(c),
${\it DLoc}_{0}^{\leq n}\cdot\chi={\it DLoc}_{0}^{\leq n}\cdot\chi^{\prime}$
if and only if

$e_{2k}(\chi)^{-1}\frac{f^{2}(\chi)}{e_{2k}(\chi)^{2\ell}}=e_{2k}(\chi^{\prime})^{-1}\frac{f^{2}(\chi^{\prime})}{e_{2k}^{*}(\chi^{\prime})^{2\ell}}.$

The rational function

(4.17) $F:=f^{2}/e_{2k}^{2\ell+1}$

is thus ${\it DLoc}_{0}^{\leq n}$-invariant and separates orbits. Therefore,
${\it DLoc}_{0}^{\leq n}\cdot\chi^{+}$ is the hypersurface in ${\it
Loc}_{0}^{\leq n}\setminus{\it Loc}_{0}^{\leq n-1}$ defined by

$F_{+}:=e_{2k}(\chi^{+})^{2\ell+1}f^{2}-f^{2}(\chi^{+})e_{2k}^{2\ell+1}.$

Note that $F_{+}$ is semiinvariant, as claimed.

All orbits of dimension $<n$ belong to ${\it Loc}_{0}^{\leq n-2}$. We will
check that the closure of ${\it DLoc}_{0}^{\leq n}\cdot\chi^{+}$ contains
${\it Loc}_{0}^{\leq n-2}$. The orbit closure $\overline{{\it DLoc}_{0}^{\leq
n}\chi^{+}}$ is defined in ${\it Loc}_{0}^{\leq n}$ by $F_{+}$, so we must
show that $F_{+}|_{{\it Loc}_{0}^{\leq n-2}}=0.$ The restriction of $F_{+}$ to
${\it Loc}^{\leq n-1}_{0}$ is also a semiinvariant function; in particular,
this restriction is invariant with respect to ${\it DLoc}_{0}^{1+,\leq n}$,
see Lemma 4.8. The arguments of Theorem 4.9 imply that all $n$-dimensional
${\it DLoc}_{0}^{1+,\leq n}$-invariant hypersurfaces of ${\it Loc}_{0}^{\leq
n-1}$ are defined by the equality $e_{n-1}=\beta_{n-1}$ for some
$\beta_{n-1}\in{\Bbbk}$. This implies that the restriction of $F_{+}$ is a
polynomial in $e_{n-1}$. The fact that $F_{+}$ is ${\it DLoc}_{0}^{\leq n}$
semiinvariant implies that $F_{+}|_{{\it Loc}^{\leq
n-1}_{0}}=c(e_{n-1}^{*})^{m}$ for some $m\geq 0$ and $c\in{\Bbbk}$. This
implies the desired condition that $F_{+}|_{{\it Loc}^{\leq n-2}_{0}}=0$.

We now show that $(2)\Rightarrow(3)$. This is an application of Proposition
2.13. Let $X=\widetilde{{\it Loc}}^{\leq n}_{\mathfrak{g}}$ (recall that
$\mathfrak{g}=W_{\geq-1}$) and let $H=\widehat{{\it DLoc}}^{\leq n}$. Let
$\mathfrak{h}=\widehat{\operatorname{lie}}^{\leq n}=\operatorname{Lie}(H)$.
Let $\phi=\phi^{\leq n}_{\mathfrak{g}}:X\to\mathfrak{g}^{*}$. Choose
$x^{+}\in\phi^{-1}(\chi^{+})$ and $x^{-}\in\phi^{-1}(\chi^{-})$; note here
that $\phi$ is bijective onto ${\it Loc}^{\leq
n}_{\mathfrak{g}}\setminus\\{0\\}$ so $x^{\pm}$ may be uniquely determined. By
Lemma 4.7, $\dim\mathfrak{h}_{x^{\pm}}=\dim\mathfrak{g}_{\chi^{\pm}}$. Thus by
Proposition 2.13, $\operatorname{P}(\chi^{+})$ is the kernel of the map
$\operatorname{S}(\mathfrak{g})\to{\Bbbk}[\overline{H\cdot x^{+}}]$ induced
from $\phi^{*}:\operatorname{S}(\mathfrak{g})\to{\Bbbk}[X]$ and
$\operatorname{P}(\chi^{-})$ is the kernel of
$\operatorname{S}(\mathfrak{g})\to{\Bbbk}[\overline{H\cdot x^{-}}]$. However,
by assumption $H\cdot x^{-}\subseteq\overline{H\cdot x^{+}}\subseteq X$,
giving a commutative diagram

$\textstyle{\operatorname{S}(\mathfrak{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1)}$$\scriptstyle{(2)}$$\textstyle{{\Bbbk}[\overline{H\cdot
x^{+}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Bbbk}[\overline{H\cdot
x^{-}}].}$

Thus $\operatorname{P}(\chi^{-})=\ker\ (2)\supseteq\ker\
(1)=\operatorname{P}(\chi^{+})$. The two are clearly distinct.

Finally, if $(3)$ holds then
$\dim\mathbb{O}(\chi^{-})<\dim\mathbb{O}(\chi^{+})$. By the discussion above
and (2.15),

$\dim\mathbb{O}(\chi^{\pm})=\dim\mathfrak{g}_{\chi^{\pm}}=\dim{\it DLoc}^{\leq
n}_{x}\cdot\chi^{\pm}+1.$

Thus (1) holds. ∎

######  Remark 4.18.

If $\chi\in W^{*}$ is a one-point local function, the beginning of the proof
of Theorem 4.9 provides an explicit description of $W^{\chi}$. Indeed, if
$\chi=\chi_{x;\beta_{0},\beta_{1},\ldots,\beta_{n}}$ with $\beta_{n}\neq 0$
then $W^{\chi}=W((t-x)^{n+1})$ if $n$ is even, and $W^{\chi}$ is the sum of
$W((t-x)^{n+1})$ with a one-dimensional subspace if $n$ is odd.

For future reference, we record that Corollary 4.16 also gives information
about $\widehat{{\it DLoc}}^{\leq n}$ orbits. Let

${\it Loc}_{W_{\geq-1}}^{n}:={\it Loc}_{W_{\geq-1}}^{\leq n}\setminus{\it
Loc}_{W_{\geq-1}}^{\leq n-1}$

and similarly define ${\it Loc}_{W}^{n}$. We identify ${\Bbbk}[{\it
Loc}_{W_{\geq-1}}^{n}]$ with
${\Bbbk}[x,\beta_{0},\dots,\beta_{n},\beta_{n}^{-1}]$. Note that each ${\it
Loc}_{W_{\geq-1}}^{n}$ is $\widehat{{\it DLoc}}^{\leq n}$-stable.

The action of ${\it Shifts}$, and thus the action of $\widehat{DLoc}$, on
${\it Loc}_{W}$ is only partially defined, as we cannot shift the base point
to $0$. We will refer to orbits of this partial action with the obvious
meaning: the intersection of an orbit in ${\it Loc}_{W_{\geq-1}}$ with
$W^{*}$.

######  Corollary 4.19.

Let $\mathfrak{g}=W$ or $W_{\geq-1}$. If $n$ is even, then ${\it
Loc}_{\mathfrak{g}}^{n}$ is a single $\widehat{{\it DLoc}}^{\leq n}$-orbit. If
$n$ is odd, there is a polynomial
$F\in{\Bbbk}[x,\beta_{0},\dots,\beta_{n},\beta_{n}^{-1}]$ (if
$\mathfrak{g}=W_{\geq-1})$ or in
${\Bbbk}[x,x^{-1},\beta_{0},\dots,\beta_{n},\beta_{n}^{-1}]$ (if
$\mathfrak{g}=W$) so that the $\widehat{{\it DLoc}}^{\leq n}$-orbits are
precisely the fibres of the morphism $F:{\it
Loc}_{\mathfrak{g}}^{n}\to\mathbb{A}^{1}.$

###### Proof.

For $\mathfrak{g}=W_{\geq-1}$ the result follows directly from Corollary 4.16.
If $n$ is odd, note that by definition, applying an element of ${\it Shifts}$
changes $x$ but not any of the $\beta_{i}$. Thus we may take $F$ to be the
polynomial defined in (4.17), regarded as an element of
${\Bbbk}[x,\beta_{0},\dots,\beta_{n},\beta_{n}^{-1}]$. The proof for
$\mathfrak{g}=W$ is similar. ∎

######  Remark 4.20.

The dimension of a $\widehat{DLoc}^{\leq n}$-orbit in Corollary 4.19 is

$2\lfloor\frac{n+2}{2}\rfloor=\begin{cases}n+2&\mbox{ $n$ even}\\\ n+1&\mbox{
$n$ odd}\end{cases}.$

###  4.3. Implicit description of pseudo-orbits

Let $\mathfrak{g}=W$, $W_{\geq-1}$, or $W_{\geq 1}$. We now consider arbitrary
local functions on $\mathfrak{g}$. We will provide every pair of local
functions $\chi,\nu\in\mathfrak{g}^{*}$ with an affine variety $X$ acted on by
an algebraic group $H$, an open subset $U\subseteq X$, a Poisson map
$\phi:U\to\mathfrak{g}^{*}$, and a pair of points $x,y\in X$ which will
satisfy the conditions of Proposition 2.13. Thus we will reduce the mysterious
infinite-dimensional case to the more easily comprehensible action of an
algebraic group on an affine variety. We use this to determine the pseudo-
orbits in $\operatorname{S}(\mathfrak{g})$.

######  Theorem 4.21.

Let $\mathfrak{g}=V\\!ir$ or $\mathfrak{g}=W$ or $W_{\geq-1}$. Let
$\chi^{{I}}$ and $\chi^{{II}}$ be local functions on $\mathfrak{g}^{*}$
represented by sums

$\chi^{{I}}=\sum_{i=1}^{\ell}\chi_{i}^{{I}},\hskip
10.0pt\chi^{{II}}=\sum_{i=1}^{\ell}\chi_{i}^{{II}},\hskip
10.0pt\chi_{i}^{{I}}\in{\it Loc}_{x_{i}^{{I}}},\chi_{i}^{{II}}\in{\it
Loc}_{x_{i}^{{II}}},$

such that $x^{{I}}_{i}\neq x^{{I}}_{j}$ and $x^{{II}}_{i}\neq x^{{II}}_{j}$
for $i\neq j$. Then $\chi^{{I}}$ and $\chi^{{II}}$ are in the same pseudo-
orbit if and only if it is possible to reorder $x^{{I}}_{i}$ and
$x^{{II}}_{i}$ in such a way that

(4.22) ${\it DLoc}_{1}({\it Shift}_{1-x_{i}^{{I}}}\chi^{{I}}_{i})={\it
DLoc}_{1}({\it Shift}_{1-x_{i}^{{II}}}\chi^{{II}}_{i})\mbox{ for all $i$;}$

that is,

$\widehat{DLoc}(\chi^{{I}}_{i})=\widehat{{\it DLoc}}(\chi^{{II}}_{i})\mbox{
for all $i$.}$

In particular, if $\mathbb{O}(\chi^{{I}})=\mathbb{O}(\chi^{{II}})$ then
$\lambda(\chi^{{I}})=\lambda(\chi^{{II}})$ and hence the supports of
$\chi^{{I}}$ and $\chi^{{II}}$ have the same cardinality.

######  Remark 4.23.

Let $\mathfrak{g}=W$ or $W_{\geq-1}$. Fix a partition
$\lambda=(m_{1},\dots,m_{r})$ and consider the set
$V^{\lambda}_{\mathfrak{g}}$ of all local functions
$\chi\in{\mathfrak{g}}^{*}$ with $\lambda(\chi)=\lambda$. Let
$\chi=\chi_{1}+\dots+\chi_{r},\nu=\nu_{1}+\dots+\nu_{r}\in
V^{\lambda}_{\mathfrak{g}}$, where each $\chi_{i}$ (respectively, $\nu_{i}$)
is a one-point local function of order $m_{i}$. By Theorem 4.21,
$\mathbb{O}(\chi)=\mathbb{O}(\nu)$ $\iff$
$\mathbb{O}(\chi_{i})=\mathbb{O}(\nu_{i})$ for $1\leq i\leq r$. Combining
Corollary 4.19 and Theorem 4.21 applied to one-point local functions, if
$m_{i}$ is even, then ${\it Loc}^{m_{i}}_{\mathfrak{g}}$ is a single pseudo-
orbit. On the other hand, if $m_{i}$ is odd then the pseudo-orbits in ${\it
Loc}_{\mathfrak{g}}^{m_{i}}$ are parameterised by $\mathbb{A}^{1}$. Thus
pseudo-orbits in $V^{\lambda}_{\mathfrak{g}}$ (and the corresponding Poisson
primitive ideals of $\operatorname{S}({\mathfrak{g}})$) are parameterised by a
symmetric product of $k$ copies of $\mathbb{A}^{1}$, where $k$ is the number
of odd $m_{i}$. More specifically, pseudo-orbits in
$V^{\lambda}_{\mathfrak{g}}$ are parameterised by $(\mathbb{A}^{1})^{\times
k}/\mathfrak{S}_{\lambda}\cong\mathbb{A}^{k}$, where
$\mathfrak{S}_{\lambda}=\\{\sigma\in\mathfrak{S}_{k}\ |\
m_{i}=m_{\sigma(i)}\mbox{ for all }i\\}$. In Subsection 5.2 we will refine
this parameterisation to apply to arbitrary prime Poisson ideals of
$\operatorname{S}(\mathfrak{g})$.

To prove Theorem 4.21 we need two preparatory results.

######  Lemma 4.24.

Let $\chi\in W^{*}$ be a local function with $\chi=\chi_{1}+\ldots+\chi_{s}$
and $\chi_{i}\in{\it Loc}_{x_{i}}$ such that $x_{i}\neq x_{j}$ if $i\neq j$.
Then $W\cdot\chi=\bigoplus_{i}W\cdot\chi_{i}$. Likewise, if $\chi\in
W_{\geq-1}^{*}$, then
$W_{\geq-1}\cdot\chi=\bigoplus_{i}W_{\geq-1}\cdot\chi_{i}$.

###### Proof.

This statement is basically implied by the Chinese remainder theorem, but we
give the details for $W$. It is easy to verify that
$W\cdot\chi=\\{B_{\chi}(u,\cdot)\mid u\in W\\}$. Since $\chi$ is local, there
are $d_{1},\ldots,d_{s}\in\mathbb{Z}_{\geq 1}$ so that
$\chi|_{W(\prod_{i}(t-x_{i})^{d_{i}})}=0$. Let
$f:=\prod_{i}(t-x_{i})^{d_{i}+1}$. Then $B_{\chi}|_{W(f)}=0$. By the Chinese
remainder theorem,

(4.25)
${\Bbbk}[t,t^{-1}]/(\prod(t-x_{i})^{d_{i}+1})\cong\bigoplus_{i}{\Bbbk}[t]/(t-x_{i})^{d_{i}+1}.$

This leads to the decomposition

$W/W(f)\cong\bigoplus_{i}[W(f_{i})/W(f)],\mbox{ where
}f_{i}:=\frac{f}{(t-x_{i})^{d_{i}+1}}.$

It is straightforward to verify that the partial restriction
$B_{\chi_{i}}(W(f_{j}),\cdot)$ is nonzero only if $i=j$. Hence

$W\cdot\chi=\\{B_{\chi}(u,\cdot)\mid u\in
W\\}=\bigoplus_{i}\\{B_{\chi}(u_{i},\cdot)\mid u_{i}\in W(f_{i})\\}.$

This implies the desired result. ∎

We next apply the general results from Section 2 and the group action from
Subsection 4.1 to the Poisson cores of local functions and deduce some
consequences.

######  Proposition 4.26.

  1. (a)

Let $\chi\in W^{*}$ be a local function. Then

$\operatorname{P}(\chi)\cap\operatorname{S}(W_{\geq-1})=\operatorname{P}(\chi|_{W_{\geq-1}}).$

  2. (b)

Let $\mathfrak{g}$ be $W$ or $W_{\geq-1}$ or $W_{\geq 1}$ and pick
$\chi\in\mathfrak{g}^{*}$. Then

$\dim\mathbb{O}(\chi)=\operatorname{GKdim}\operatorname{S}(\mathfrak{g})/\operatorname{P}(\chi)=\dim\mathfrak{g}\cdot\chi.$

We alert the reader that the proof of Proposition 4.26 establishes notation
that is also used in the proof of Theorem 4.21. (This notation generalises
that used in the proof of the last part of Corollary 4.16.)

###### Proof.

(a). Write $\chi=\sum_{i=1}^{\ell}\chi_{i}$ with $\chi_{i}\in{\it
Loc}_{x_{i}}$ and $x_{i}\neq x_{j}$ if $i\neq j$; further assume that no
$\chi_{i}=0$. Let $n$ be the order of $\chi$.

Let $X:=\Bigl{(}\widetilde{{\it Loc}}^{\leq
n}_{W_{\geq-1}}\Bigr{)}^{\times\ell}$, let $Y:=\Bigl{(}\widetilde{{\it
Loc}}^{\leq n}_{W}\Bigr{)}^{\times\ell}$, and let $U\subseteq Y$ be the
complement of all the diagonals $\Delta_{ij}$ in $Y$, where for $i\neq j$ we
define

$\Delta_{ij}=\\{(x^{1},\underline{\alpha}^{1},\dots,x^{\ell},\underline{\alpha}^{\ell})\in
Y\ |\ x^{i}=x^{j}\\},$

where $\underline{\alpha}^{i}$ stands for the sequence
$\underline{\alpha}^{i}_{0},\underline{\alpha}^{i}_{1},\ldots$. That is, $U$
gives representations of local functions as sums of one-point local functions
based at distinct points. Note that $U,Y$ are open subsets of $X$, and
$H:=\Bigl{(}\widehat{{\it DLoc}}^{\leq n}\Bigr{)}^{\times\ell}$ acts on $X$.
Let $\Sigma_{W}:(W^{*})^{\times\ell}\to W^{*}$ be the summation map, and
likewise define $\Sigma_{W_{\geq-1}}$. Let $\phi_{W}:U\to W^{*}$ be
$\Sigma_{W}\circ(\phi_{W}^{\leq n})^{\times\ell}$ and let
$\phi_{W_{\geq-1}}:X\to W_{\geq-1}^{*}$ be
$\Sigma_{W_{\geq-1}}\circ(\phi_{W_{\geq-1}}^{\leq n})^{\times\ell}$. There is
a commutative diagram

(4.27)
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{W_{\geq-1}}}$$\textstyle{W_{\geq-1}^{*}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{W}}$$\scriptstyle{\subseteq}$$\textstyle{W^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the right vertical arrow is restriction. This induces maps

(4.28)
$\textstyle{{\Bbbk}[X]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{\operatorname{S}(W_{\geq-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{W_{\geq-1}}^{*}}$$\scriptstyle{\subseteq}$$\textstyle{{\Bbbk}[U]}$$\textstyle{\operatorname{S}(W).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{W}^{*}}$

By Lemma 4.24, $\dim W\cdot\chi=\oplus_{i}\dim W\cdot\chi_{i}$, which is
$\dim\Bigl{(}(\widehat{\operatorname{lie}}^{\leq
n})^{\times\ell}\Bigr{)}\cdot\chi$ by Lemma 4.7. Let $y\in\phi^{-1}(\chi)$. By
Proposition 2.13, $\operatorname{P}(\chi)$ is the full preimage under
$\phi_{W}^{*}$ of $I^{U}(Hy)$, and likewise,
$\operatorname{P}(\chi|_{W_{\geq-1}})$ is the full preimage under
$\phi_{W_{\geq-1}}^{*}$ of $I^{U}(Hy)$. By commutativity of (4.28), this is
$\operatorname{P}(\chi)\cap\operatorname{S}(W_{\geq-1})$.

(b) If $\dim\mathfrak{g}\cdot\chi=\infty$ then the result follows from Lemma
2.5. Thus we can assume $\mathfrak{g}\cdot\chi<\infty$. For $\mathfrak{g}=W$
or $W_{\geq-1}$ this follows from the discussion above and Proposition 2.13;
see in particular (2.15). The proof for $\mathfrak{g}=W_{\geq 1}$ is similar.
∎

We now give the proof of Theorem 4.21.

###### Proof of Theorem 4.21.

We give the proof for the case $\mathfrak{g}=W$.

By definition, $\mathbb{O}(\chi^{{I}})=\mathbb{O}(\chi^{{II}})$ if and only if
$\operatorname{P}(\chi^{{I}})=\operatorname{P}(\chi^{{II}})$. So we compute
the Poisson cores of $\chi^{{I}}$ and $\chi^{{II}}$.

First, by changing $\ell$ and switching $\chi^{{I}}$ and $\chi^{{II}}$ if
necessary, we may assume that none of the $\chi_{i}^{{I}}$ are zero. Let $n$
be the maximum of the orders of $\chi^{{I}}$ and of $\chi^{{II}}$; thus both
are in the image of $({\it Loc}_{W}^{\leq n})^{\times\ell}$.

Define $U,V,X,H$ as in the proof of Proposition 4.26, and set up the maps in
(4.27) and (4.28). Let $\phi=\phi_{W}$. Then $\chi^{{I}}=\phi(y^{{I}})$,
$\chi^{{II}}=\phi(y^{{II}})$ for some $y^{{I}},y^{{II}}\in U$. As in the proof
of Proposition 4.26, $\operatorname{P}(\chi^{{I}})$ is the full preimage under
${\phi}^{*}$ of the defining ideal of $I^{U}(Hy^{{I}})$. Likewise,
$\operatorname{P}(\chi^{{II}})$ is the full preimage under ${\phi}^{*}$ of the
defining ideal of $I^{U}(Hy^{{II}})$.

Note that $\phi$ factors through the dominant map $U\to
U/{\mathfrak{S}}_{\ell}$, where $\mathfrak{S}_{\ell}$ is the symmetric group.
It is clear that if we can reorder the $\chi_{i}^{{I}}$ and $\chi_{i}^{{II}}$
as described then
$\operatorname{P}(\chi^{{I}})=\operatorname{P}(\chi^{{II}})$.

Suppose now that $\operatorname{P}(\chi^{{I}})=\operatorname{P}(\chi^{{II}})$.
Then by Proposition 2.13(c), there are open sets $U^{{I}}\subseteq
Hy^{{I}}\cap U$ and $U^{{II}}\subseteq Hy^{{II}}\cap U$ such that their images
in $W^{*}$ are the same. In particular, there are
$h_{1}^{{I}},h_{1}^{{II}},\dots,h_{\ell}^{{I}},h_{\ell}^{{II}}\in\widehat{{\it
DLoc}}^{\leq n}$, with
$(h_{1}^{{I}}\chi_{1}^{{I}},\ldots,h_{\ell}^{{I}}\chi_{\ell}^{{I}})\in U$, and
so that $\sum h_{i}^{{I}}\chi_{i}^{{I}}-\sum h_{i}^{{II}}\chi_{i}^{{II}}=0$.
Now, a set of one-point local functions with distinct base points is linearly
independent. Since by definition of $U$ the base points of the one-point local
functions $h_{i}^{{I}}\chi_{i}^{{I}}$ are distinct, the only option is that
the supports of $\sum h_{i}^{{I}}\chi_{i}^{{I}}$ and $\sum
h_{i}^{{II}}\chi_{i}^{{II}}$ are equal, and we can reorder
$\chi_{1}^{{II}},\dots,\chi_{\ell}^{{II}}$ so that
$h_{i}^{{I}}\chi_{i}^{{I}}=h_{i}^{{II}}\chi_{i}^{{II}}$ for all $i$. This is
clearly equivalent to (4.22). ∎

######  Remark 4.29.

Let $\mathfrak{g}=V\\!ir$ or $W$ or $W_{\geq-1}$ and let
$\chi\in\mathfrak{g}^{*}$. If $\chi$ is not local then
$\operatorname{P}(\chi)=(0)$ (if $\mathfrak{g}=W$ or $W_{\geq-1}$) or
$\operatorname{P}(\chi)=(z-\chi(z))$ (if $\mathfrak{g}=V\\!ir$). If $\chi$ is
local, then by Lemma 2.4(a) we may construct
$\operatorname{P}(\chi)=\bigcap\\{\mathfrak{m}_{\nu}\ |\
\nu\in\mathbb{O}(\chi)\\}$ from Theorem 4.21.

Further combining Corollary 4.19 with Theorem 4.21 one may in principle
compute all the Poisson primitive ideals of $\operatorname{S}(W)$ or
$\operatorname{S}(W_{\geq-1})$; the function $F$ whose fibres by Corollary
4.19 give the pseudo-orbits of a one-point local function of order $2k+1$ may
be worked out using Lemma 4.8 for any $k$. There is an example of this in
Section 7, although we have not given a fully general formula.

Note also that we have not studied generators (in any sense) for the Poisson
prime ideals $\operatorname{P}(\chi)$ — this can be an interesting subject of
research.

A version of Theorem 4.21 holds for $W_{\geq 1}$ and $W_{\geq 0}$ with almost
the same proof. We state it as follows:

######  Theorem 4.30.

Let $\mathfrak{g}=W_{\geq 0}$ or $W_{\geq 1}$. Let
$\chi^{{I}},\chi^{{II}}\in\mathfrak{g}^{*}$ be local functions. Then there are
unique local functions $\chi_{W}^{{I}},\chi_{W}^{{II}}\in W^{*}$ and
$\chi^{{I}}_{[0]},\chi^{{II}}_{[0]}\in{\it Loc}_{0}$ such that

$\chi^{{I}}=\chi^{{I}}_{W}|_{\mathfrak{g}}+\chi^{{I}}_{[0]},\hskip
10.0pt\chi^{{II}}=\chi^{{II}}_{W}|_{\mathfrak{g}}+\chi^{{II}}_{[0]}.$

Moreover $\operatorname{P}(\chi^{{I}})=\operatorname{P}(\chi^{{II}})$ if and
only if $\operatorname{P}(\chi^{{I}}_{W})=\operatorname{P}(\chi^{{II}}_{W})$
and ${\it DLoc}_{0}(\chi^{{I}}_{[0]})={\it DLoc}_{0}(\chi^{{II}}_{[0]})$. ∎

The description of pseudo-orbits in Theorem 4.21 has several immediate
applications. We start by comparing pseudo-orbits in $W^{*}$ and
$W^{*}_{\geq-1}$.

######  Corollary 4.31.

Let $\chi^{{I}},\chi^{{II}}\in W^{*}$ be local functions. Then
$\mathbb{O}(\chi^{{I}})=\mathbb{O}(\chi^{{II}})$ iff
$\mathbb{O}(\chi^{{I}}|_{W_{\geq-1}})=\mathbb{O}(\chi^{{II}}|_{W_{\geq-1}})$.
Further, restriction induces a bijection

(4.32) $\operatorname{PSpec}_{\rm
prim}\operatorname{S}(W)\to\operatorname{PSpec}_{\rm
prim}\operatorname{S}(W_{\geq-1}).$

In Corollary 5.16 we will show the bijection (4.32) is actually a
homeomorphism.

###### Proof.

The first statement is immediate from Theorem 4.21. By that result, if
$\chi\in W_{\geq-1}^{*}$ is a local function then there is
$\nu\in\mathbb{O}(\chi)$ so that $0$ is not in the support of $\nu$. Thus
restriction of local functions induces a bijection on pseudo-orbits.

By Lemma 2.4 there are bijections

$\mbox{nonzero Poisson primitive ideals of
}\operatorname{S}(W)\leftrightarrow\mbox{ pseudo-orbits in }W^{*}\\\
\leftrightarrow\mbox{ pseudo-orbits in }W_{\geq-1}^{*}\leftrightarrow\mbox{
nonzero Poisson primitive ideals of }\operatorname{S}(W_{\geq-1}),$

where the middle bijection comes from restriction of local functions as in the
first paragraph of the proof. By Proposition 4.26(a)
$\operatorname{P}(\chi)\cap\operatorname{S}(W_{\geq-1})=\operatorname{P}(\chi|_{W_{\geq-1}})$.
Thus the bijection from left to right above agrees with restriction of ideals
from $\operatorname{S}(W)$ to $\operatorname{S}(W_{\geq-1})$. ∎

######  Remark 4.33.

Let $P=\operatorname{P}(\chi)$ be a Poisson primitive ideal of
$\operatorname{S}(W_{\geq-1})$, where $\chi\in(W_{\geq-1})^{*}$. As we may
apply elements of ${\it Shifts}$ to $\chi$ without affecting
$\operatorname{P}(\chi)$, we may assume that $\chi$ does not involve any
element of ${\it Loc}_{0}$. Then $\chi$ extends uniquely to a local function
$\overline{\chi}$ on $W$, and we have seen that
$\operatorname{P}(\chi)=\operatorname{P}(\overline{\chi})\cap\operatorname{S}(W_{\geq-1})$.

Let $\mathfrak{g}$ be a Lie algebra and let $\chi\in\mathfrak{g}^{*}$. The
closure of $\mathbb{O}(\chi)$ is, by definition, $V(\operatorname{P}(\chi))$.
We call $V(\operatorname{P}(\chi))=\overline{\mathbb{O}(\chi)}$ the orbit
closure of $\chi$. (Technically, this term should probably be “pseudo-orbit
closure”; we have used wording which we find more pleasant at the cost of a
slight abuse of terminology.) Note that $\mu$ is in the orbit closure of
$\chi$ if and only if $\operatorname{P}(\mu)\supseteq\operatorname{P}(\chi)$.

Our final corollary is a partial description of orbit closures of arbitrary
local functions.

######  Corollary 4.34.

Let $\mathfrak{g}=V\\!ir$ or $W$ or $W_{\geq-1}$ or $W_{\geq 1}$. Let
$\mu,\chi\in\mathfrak{g}^{*}$ be local functions with disjoint support, and
let $\nu$ be in the orbit closure of $\mu$. Then $\nu+\chi$ is in the orbit
closure of $\mu+\chi$. In particular, $\mu$ and $\chi$ are in the orbit
closure of $\mu+\chi$.

###### Proof.

By Theorem 4.21, the pseudo-orbit of $\mu+\chi$ consists of all local
functions of the form $\mu^{\prime}+\chi^{\prime}$ where
$\mu^{\prime}\in\mathbb{O}(\mu)$, $\chi^{\prime}\in\mathbb{O}(\chi)$, and the
supports of $\mu^{\prime}$ and $\chi^{\prime}$ are disjoint. A modification of
Corollary 4.16 for the multipoint case implies that the orbit closure
$\overline{\mathbb{O}(\mu+\chi)}$ contains $\nu+\chi$.

The final statement follows as $0$ is in every orbit closure (equivalently,
$\mathfrak{m}_{0}=\operatorname{P}(0)$ contains all nontrivial Poisson ideals
of $\operatorname{S}(\mathfrak{g})$). ∎

######  Remark 4.35.

The orbit closure relations for arbitrary local functions are not completely
given by Corollary 4.34. If ${\Bbbk}=\mathbb{C}$, let $s\in{\Bbbk}^{\times}$
and consider the local function

$\chi_{s}:=\chi_{s;1/s}+\chi_{0;-1/s}\in W_{\geq-1}^{*}.$

By Theorems 4.21 and 4.9, the $\chi_{s}$ are all in
$\mathbb{O}(\chi_{0;1}+\chi_{1;1})$. For $f\in{\Bbbk}[t]$, we have

$\lim_{s\to 0}\chi_{s}(f\partial)=\lim_{s\to
0}\frac{f(s)-f(0)}{s}=f^{\prime}(0).$

As Zariski closed sets are closed in the complex analytic topology,
$\lim_{s\to
0}\chi_{s}=\chi_{0;0,1}\in\overline{\mathbb{O}(\chi_{0;1}+\chi_{1;1})}$. The
same statement holds for arbitrary ${\Bbbk}$ by the Lefschetz principle.

##  5\. Prime Poisson ideals

In this section we apply our results on Poisson primitive ideals to the
structure of arbitrary Poisson prime ideals of
$\operatorname{S}(\mathfrak{g})$, where $\mathfrak{g}$ is one of our Lie
algebras of interest. After a preliminary subsection where we develop a
version of Chevalley’s theorem on the image of a morphism of varieties for our
context, we show that the partition data associated to local functions may be
used to parameterise Poisson primes of $\operatorname{S}(W)$ and
$\operatorname{S}(W_{\geq 1})$. This will allow us to show that the bijection
$\operatorname{PSpec}_{\rm
prim}\operatorname{S}(W)\to\operatorname{PSpec}_{\rm
prim}\operatorname{S}(W_{\geq-1})$ of Corollary 4.31 is a homeomorphism. We
then investigate when the Poisson analogue of the Dixmier-Moeglin equivalence
holds for $\operatorname{S}(\mathfrak{g})$ and, in particular, when Poisson
primitive ideals are locally closed in
$\operatorname{PSpec}\operatorname{S}(\mathfrak{g})$; we will see that this is
almost always, but not always, the case. Finally, we answer a question of León
Sánchez and the second author [LS] on heights of Poisson prime ideals, and in
the process show that $\operatorname{PSpec}\operatorname{S}(W)$ has the
somewhat counterintuitive property that every proper radical Poisson ideal
contains a proper Poisson primitive ideal.

###  5.1. Chevalley’s theorem

A fundamental result of Chevalley [G, IV.1.8.4] says that if $\phi:X\to Y$ is
a morphism of algebraic varieties, then $\phi(X)$ is a constructible subset of
$Y$: that is, a finite union of locally closed subsets. We will need a version
of this theorem which works if $Y=\mathfrak{g}^{*}$ in order to prove the main
results of this section. We will prove our nonnoetherian, and somewhat weaker,
version of Chevalley’s theorem following [H, Ex. II.3.19]. At the first step
we require a technical lemma. All of this is standard, but we must verify that
it works in our non-affine context.

######  Lemma 5.1.

Let $A$, $B$ be ${\Bbbk}$-algebras which are domains, and let $\phi:A\to B$ be
an injective homomorphism. Assume also that $B$ is affine (we make no
additional assumption on $A$). Let $b\in B\setminus\\{0\\}$. There is $a\in
A\setminus\\{0\\}$ so that if $\chi:A\to{\Bbbk}$ is a homomorphism with
$\chi(a)\neq 0$, then there is some $\chi^{\prime}:B\to{\Bbbk}$ with
$\chi^{\prime}\phi=\chi$ and $\chi^{\prime}(b)\neq 0$.

###### Proof.

Without loss of generality $A\subseteq B$ and $\phi$ is the inclusion.

Since $B$ is affine, it is finitely generated over $A$, say by
$b_{1},\dots,b_{n}$. We may assume that $b_{n}=b$.

First assume that $n=1$. If $b=b_{1}$ is transcendental over $A$ then simply
set $a=1$: any nonzero $\chi:A\to{\Bbbk}$ extends to
$\chi^{\prime}:B\to{\Bbbk}$ for any choice of $\chi^{\prime}(b)$, so we may
certainly choose $\chi^{\prime}(b)\neq 0$. If $b$ is algebraic over $A$ then
we have $\sum_{i=0}^{m}a_{i}b^{i}=0$ for some $a_{i}\in A$; without loss of
generality, $a_{0}\neq 0$ and $m$ is minimal. Let $a=a_{0}$. Given $\chi$ with
$\chi(a_{0})\neq 0$, define $\chi^{\prime}$ by setting $\chi^{\prime}(b)$ to
be any root of the nonzero polynomial $\sum\chi(a_{i})x^{i}$; there is such a
root as ${\Bbbk}$ is algebraically closed. Then
$\chi(a_{0})=-\chi^{\prime}(b)\sum_{i=1}^{m}\chi(a_{i})\chi^{\prime}(b)^{i-1}$
and so $\chi^{\prime}(b)\neq 0$.

If $n>1$, first, by the $n=1$ case, there is $0\neq c\in
A[b_{1},\dots,b_{n-1}]$ so that any
$\chi^{\prime}:A[b_{1},\dots,b_{n-1}]\to{\Bbbk}$ with $\chi^{\prime}(c)\neq 0$
extends to $\chi^{\prime\prime}:B\to{\Bbbk}$ with $\chi^{\prime\prime}(b)\neq
0$. Now by induction on $n$, there is $a\in A\setminus\\{0\\}$ so that any
$\chi:A\to{\Bbbk}$ with $\chi(a)\neq 0$ extends to
$\chi^{\prime}:A[b_{1},\dots,b_{n-1}]\to{\Bbbk}$ with $\chi^{\prime}(c)\neq
0$. Then $\chi^{\prime}$ extends to $\chi^{\prime\prime}:B\to{\Bbbk}$ with
$\chi^{\prime\prime}(b)\neq 0$, as needed. ∎

######  Corollary 5.2.

Let $B$ be an affine ${\Bbbk}$-algebra which is a domain, and let $A$ be a
(not necessarily noetherian) ${\Bbbk}$-algebra which is a domain with
$\dim_{\Bbbk}A<|{\Bbbk}|$. Let $X=\operatorname{MSpec}B$ and let
$Y=\operatorname{MSpec}A$, and let $\psi:X\to Y$ be an algebraic map. Then
$\psi(X)$ contains a nonempty open subset of $\overline{\psi(X)}$.

###### Proof.

Replacing $Y$ by $\overline{\psi(X)}$ and $A$ by $A/I(\psi(X))$ we may assume
that $\psi(X)$ is dense in $Y$ and that $\psi^{*}:A\to B$ is injective. Use
Lemma 5.1 with $b=1$ to choose $a\in A$ satisfying the conclusion of the
Lemma. Let $\mathfrak{m}$ be a maximal ideal of $A$ with
$a\not\in\mathfrak{m}$; by the Nullstellensatz there is a homomorphism
$\chi:A\to{\Bbbk}$ with $\mathfrak{m}=\ker\chi$. By the Lemma, there is a
nonzero homomorphism $\chi^{\prime}:B\to{\Bbbk}$ with
$\chi^{\prime}\psi^{*}=\chi$. As $\chi^{\prime}\neq 0$, thus
$\ker\chi^{\prime}$ is a maximal ideal of $B$, and
$\mathfrak{m}=(\psi^{*})^{-1}(\ker\chi^{\prime})$ is the image of
$\ker\chi^{\prime}\in X$ under $\psi$. Thus $\psi(X)\supseteq Y\setminus
V(a)$. ∎

The final result of this subsection allows us to conclude that injective
algebraic maps $X\to Y$ are birational, even in a nonnoetherian context.

######  Corollary 5.3.

Let $A$, $B$ be ${\Bbbk}$-algebras which are domains, and let $\phi:A\to B$ be
an injective homomorphism. Assume also that $B$ is affine, (we make no
additional assumption on $A$) and that the map
$\operatorname{MSpec}B\to\operatorname{MSpec}A$ is also injective. Then there
is $f\in A\setminus\\{0\\}$ so that the natural map $A[f^{-1}]\to B[f^{-1}]$
is an isomorphism.

###### Proof.

Since $\dim_{\Bbbk}A\leq\dim_{\Bbbk}B\leq\aleph_{0}<|{\Bbbk}|$, $A$ is a
Jacobson ring and the Nullstellensatz applies. In particular, if
$\mathfrak{m}\in\operatorname{MSpec}A$ then the natural map ${\Bbbk}\to
A/\mathfrak{m}$ is an isomorphism.

Without loss of generality $A\subseteq B$ and $\phi$ is the inclusion. Let
$Q(A)$ and $Q(B)$ be the quotient fields of $A$ and $B$ respectively. Assume
that $Q(B)$ is not algebraic over $Q(A)$. Then there exists $x\in B$ so that
$x$ is not algebraic over $Q(A)$, i.e. the natural map $A[t]\to B,t\to x$ from
the polynomial algebra $A[t]$ is injective; we denote the image of this map by
$A[x]$.

Choose $p\in A[x]\setminus\\{0\\}$ to be the element defined by Lemma 5.1 for
$A=A[x],B=B,b=1$. Let $p_{0}\in A$ be a nonzero coefficient of $p$. Then, for
every $\mathfrak{m}\in\operatorname{MSpec}A$ with $p_{0}\not\in\mathfrak{m}$
there are at most finitely many $x_{0}\in{\Bbbk}$ such that
$p(x_{0})\in\mathfrak{m}$. Every other $x_{0}$ provides a maximal ideal
$(\mathfrak{m},x-x_{0})$ of $A[x]$ which extends to a maximal ideal of $B$
thanks to Lemma 5.1. Thus every such $\mathfrak{m}$ has uncountably many
preimages in $\operatorname{MSpec}B$, contradicting our assumption.

Thus we can assume that $Q(B)$ is algebraic over $Q(A)$. Let $B$ be generated
as a ${\Bbbk}$-algebra by $b_{1},\dots,b_{k}$. If $Q(A)=Q(B)$ then every
$b_{i}$ can be written $\frac{a^{+}_{i}}{a^{-}_{i}}$ with $a_{i}^{\pm}\in
A\setminus\\{0\\}$; further we have that $A[f^{-1}]=B[f^{-1}]$ for
$f=a_{1}^{-}\dots a_{k}^{-}$. Thus we can assume that $Q(A)\neq Q(B)$ and
hence there exists $x\in B\setminus Q(A)$.

As $x$ is algebraic over $Q(A)$ there are $a_{0},a_{1},\ldots,a_{n}\in A$ with
$a_{0}\neq 0,a_{n}\neq 0$ such that

$F(x):=a_{0}x^{0}+a_{1}x^{1}+a_{2}x^{2}+\ldots+a_{n}x^{n}=0$

with $n\geq 2$. Let $p=\frac{p^{+}}{a_{n}^{d}}$ with $p^{+}\in
A\setminus\\{0\\},d\in\mathbb{Z}_{\geq 0}$ be the resultant of $F$. Consider
$\mathfrak{m}\in\operatorname{MSpec}A$ with $a_{n},p^{+}\not\in\mathfrak{m}$.
Then it is well-known that there are exactly $n$ maximal ideals
$\mathfrak{m}^{\prime}$ of $A[x]$ with $\mathfrak{m}=\mathfrak{m}^{\prime}\cap
A$. Pick $y\in A[x]$ to be the element defined by Lemma 5.1 for
$A=A[x],B=B,b=1$. This element $y$ is algebraic over $A$ and thus there are
$a_{0}^{\prime},a_{1}^{\prime},\ldots\in A$ with $a_{0}^{\prime}\neq 0$ such
that

$a_{0}^{\prime}y^{0}+a_{1}^{\prime}y^{1}+a_{2}^{\prime}y^{2}+\ldots=0.$

If $\mathfrak{m}^{\prime}$ is a maximal ideal of $A[x]$ such that
$a_{0}^{\prime}\not\in\mathfrak{m}=\mathfrak{m}^{\prime}\cap A$ then we have
$y\notin\mathfrak{m}^{\prime}$.

Pick $\mathfrak{m}\in\operatorname{MSpec}A$ such that
$a_{n}p^{+}a_{0}^{\prime}\notin\mathfrak{m}$. Then there are at least $n$
maximal ideals $\mathfrak{m}^{\prime\prime}$ of $B$ such that
$\mathfrak{m}^{\prime\prime}\cap A=\mathfrak{m}$. This contradicts the
injectivity of $\operatorname{MSpec}B\to\operatorname{MSpec}A$. ∎

###  5.2. Parameterising Poisson primes

Throughout this subsection, let $\mathfrak{g}=W$ or $W_{\geq-1}$. We will
refine Remark 4.23 to a parameterisation of Poisson primes of
$\operatorname{S}(\mathfrak{g})$; we will use this to show in Corollary 5.16
that the bijection of Corollary 4.31 is a homeomorphism.

Given a local function $\chi$, recall the definition of the partition
$\lambda(\chi)$ from Section 3. By Proposition 4.26 and Remark 4.20,

(5.4) $\dim\mathbb{O}(\chi)=2\sum\lfloor\frac{m_{i}+2}{2}\rfloor,$

and in particular $\dim\mathbb{O}(\chi)$ depends only on $\lambda(\chi)$.

Fix a partition $\lambda=(m_{1}\geq m_{2}\geq\cdots\geq m_{r})$, where
$m_{i}\in\mathbb{Z}_{\geq 1}$, and let

$V^{\lambda}_{\mathfrak{g}}:=\\{\chi\in\mathfrak{g}^{*}\ |\
\lambda(\chi)=\lambda\\}.$

Let

${\it Loc}^{\lambda}_{\mathfrak{g}}:=\prod_{i=1}^{r}{\it
Loc}^{m_{i}}_{\mathfrak{g}},$

where recall that ${\it
Loc}^{m}_{\mathfrak{g}}=\\{\chi_{x;\alpha_{0},\dots,\alpha_{m}}\in\mathfrak{g}^{*}\
|\ \alpha_{m}\neq 0\\}.$ There is a natural summation map
$\Sigma^{\lambda}_{\mathfrak{g}}:{\it
Loc}^{\lambda}_{\mathfrak{g}}\to\mathfrak{g}^{*}$ defined by
$\Sigma^{\lambda}_{\mathfrak{g}}(\chi_{1},\dots,\chi_{r})=\sum\chi_{i}$. As in
the proof of Proposition 4.26, let

$U^{\lambda}_{\mathfrak{g}}:=\\{(\chi_{x_{1};\underline{\alpha}^{1}},\dots,\chi_{x_{r};\underline{\alpha}^{r}})\in{\it
Loc}^{\lambda}_{\mathfrak{g}}\ |\ \mbox{the $x_{i}$ are all distinct }\\}$

(here $\underline{\alpha}^{i}$ stands for a sequence
$\underline{\alpha}^{i}_{0},\underline{\alpha}^{i}_{1},\ldots$). Note that
$\Sigma^{\lambda}_{\mathfrak{g}}(U^{\lambda}_{\mathfrak{g}})$ is precisely
$V^{\lambda}_{\mathfrak{g}}$, whereas $\Sigma^{\lambda}_{\mathfrak{g}}({\it
Loc}^{\lambda}_{\mathfrak{g}})$ usually includes other local functions.

Let the partition

$\lambda^{\rm o}:=(m_{i}\ |\ \mbox{$m_{i}$ is odd})$

consist of the odd integers occurring in $\lambda$ and let
$k:=\operatorname{len}(\lambda^{\rm o})$. For any $m_{i}\in\lambda^{\rm o}$
let $\pi_{i}:{\it Loc}^{m_{i}}_{\mathfrak{g}}\to\mathbb{A}^{1}$ be the
morphism defined in Corollary 4.19, whose fibres are pseudo-orbits (that is,
$\widehat{{\it DLoc}}^{\leq m_{i}}$-orbits) in ${\it
Loc}^{m_{i}}_{\mathfrak{g}}$.

Let

$\pi^{\lambda}_{\mathfrak{g}}:{\it
Loc}^{\lambda}_{\mathfrak{g}}\to\mathbb{A}^{k}$

be the composition

$\textstyle{{\it
Loc}^{\lambda}_{\mathfrak{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rm
pr}$$\textstyle{{\it Loc}^{\lambda^{\rm
o}}_{\mathfrak{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod\pi_{i}}$$\textstyle{\mathbb{A}^{1}\times\dots\times\mathbb{A}^{1}=\mathbb{A}^{k},}$

where ${\rm pr}$ denotes projection.

Now, $\widehat{{\it DLoc}}^{\leq m_{i}}$-orbits in ${\it
Loc}^{m_{i}}_{\mathfrak{g}}$ are nonsingular and all are of dimension
$2\lfloor\frac{m_{i}+2}{2}\rfloor$. Further, $\pi_{i}$ is flat: any torsion-
free module over ${\Bbbk}[x]$ is flat. Thus, by [H, Theorem III.10.2],
$\pi_{i}$ is smooth, and so $\pi^{\lambda}_{\mathfrak{g}}$ is also smooth.

Given a partition $\lambda=(m_{i})$ as above, we also write

$\lambda=(\underbrace{n_{1},\ldots,n_{1}}_{j_{1}},\underbrace{n_{2},\ldots,n_{2}}_{j_{2}},\dots,\underbrace{n_{\ell},\ldots,n_{\ell}}_{j_{\ell}}),$

where $n_{1}\geq n_{2}\geq\dots\geq n_{\ell}$. Note that
$m_{1}=\dots=m_{j_{1}}=n_{1}$,
$m_{j_{1}+1}=m_{j_{1}+2}=\dots=m_{j_{1}+j_{2}}=n_{2}$, etc., and
$r=\operatorname{len}(\lambda)=\sum_{i=1}^{\ell}j_{i}$. The multisymmetric
group
$\mathfrak{S}_{\lambda}:=\mathfrak{S}_{j_{1}}\times\cdots\times\mathfrak{S}_{j_{\ell}}$
acts on ${\it Loc}^{\lambda}_{\mathfrak{g}}$ and on
$U^{\lambda}_{\mathfrak{g}}$ by permuting the factors. Further,
$\mathfrak{S}_{\lambda}$ also acts on $\mathbb{A}^{k}$ by permutations (if
$n_{i}$ is even, the factor $\mathfrak{S}_{j_{i}}$ acts trivially). We will
name the action map

(5.5)
$\mu_{\lambda}:\mathfrak{S}_{\lambda}\times\mathbb{A}^{k}\to\mathbb{A}^{k}$

since we will need to refer to it later.

Note that $\pi^{\lambda}_{\mathfrak{g}}$ is
$\mathfrak{S}_{\lambda}$-equivariant. There is thus an induced map

$\overline{\pi^{\lambda}_{\mathfrak{g}}}:U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}\to\mathbb{A}^{k}/\mathfrak{S}_{\lambda}.$

We remark that $U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$ and
$\mathbb{A}^{k}/\mathfrak{S}_{\lambda}$ are affine. Further,
$\mathbb{A}^{k}/\mathfrak{S}_{\lambda}$ is nonsingular — in fact, isomorphic
to $\mathbb{A}^{k}$ — since $\mathfrak{S}_{\lambda}$ is a reflection group.
The fibres of $\overline{\pi^{\lambda}_{\mathfrak{g}}}$ are isomorphic to
fibres of the smooth morphism $\pi^{\lambda}_{\mathfrak{g}}$ and are thus
nonsingular. Therefore $U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$ is
also nonsingular.

The summation map $\Sigma^{\lambda}_{\mathfrak{g}}$ factors through ${\it
Loc}^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$. This induces a morphism
of varieties

$\psi^{\lambda}_{\mathfrak{g}}:U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}\to\mathfrak{g}^{*},$

which is injective, as any collection of one-point functions with distinct
supports is linearly independent. The image of $\psi^{\lambda}_{\mathfrak{g}}$
is precisely the set $V^{\lambda}_{\mathfrak{g}}$ of local functions with
order partition $\lambda$. Further note that $\psi_{\mathfrak{g}}^{\lambda}$
maps the tangent space to a $\widehat{{\it DLoc}}$-orbit of
$\chi\in\mathfrak{g}^{*}$ at $\chi^{*}$ to
$\mathfrak{g}_{\psi^{\lambda}_{\mathfrak{g}}(\chi)}$ (see Lemma 4.7 and 4.24),
i.e. it satisfies the conditions of Propositions 2.11 and 2.13.

######  Remark 5.6.

We may identify $V^{\lambda}_{\mathfrak{g}}$ with
$U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$ as a set. This
construction allows us to write the set of local functions on $\mathfrak{g}$
as a countable union of the affine varieties
$U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$, further justifying the
statement in Remark 3.3 that most elements of $\mathfrak{g}^{*}$ are not
local.

We abuse notation slightly, and identify ${\it Loc}^{\lambda}_{W}$, via
restriction, with the subset of ${\it Loc}^{\lambda}_{W_{\geq-1}}$ consisting
of functions whose support does not contain $0$. The diagram

(5.7)
$\textstyle{W^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rm
res}$$\textstyle{W_{\geq-1}^{*}}$$\textstyle{U^{\lambda}_{W}/\mathfrak{S}_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\lambda}_{W}}$$\scriptstyle{\subset}$$\scriptstyle{\overline{\pi^{\lambda}_{W}}}$$\textstyle{U^{\lambda}_{W_{\geq-1}}/\mathfrak{S}_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{\lambda}_{W_{\geq-1}}}$$\scriptstyle{\overline{\pi^{\lambda}_{W_{\geq-1}}}}$$\textstyle{\mathbb{A}^{k}/\mathfrak{S}_{\lambda}}$

commutes, and both maps to $\mathbb{A}^{k}/\mathfrak{S}_{\lambda}$ are
surjective.

Let $\widehat{{\it DLoc}}^{\lambda}:=\widehat{{\it DLoc}}^{\leq
m_{1}}\times\cdots\times\widehat{{\it DLoc}}^{\leq m_{r}}$. The group
$\widehat{{\it DLoc}}^{\lambda}$ acts on ${\it Loc}^{\lambda}_{W_{\geq-1}}$,
and there are partial $\widehat{{\it DLoc}}^{\lambda}$-actions on ${\it
Loc}^{\lambda}_{W}$, on $U^{\lambda}_{W}$, and on $U^{\lambda}_{W_{\geq-1}}$;
as usual, we speak of $\widehat{{\it DLoc}}^{\lambda}$-orbits on these
varieties to mean the intersections with full orbits in ${\it
Loc}^{\lambda}_{W_{\geq-1}}$. This (partial) action does not commute with the
$\mathfrak{S}_{\lambda}$ action. Nevertheless, it is compatible with the
natural action of $\mathfrak{S}_{\lambda}$ on $\widehat{{\it DLoc}}^{\lambda}$
and altogether this provides an action of the semidirect product
$\widehat{{\it DLoc}}^{\lambda}\rtimes\mathfrak{S}_{\lambda}$. Further by
Corollary 4.19 the fibres of $\overline{\pi^{\lambda}_{\mathfrak{g}}}$ are
precisely the images in $U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$ of
$\widehat{{\it DLoc}}^{\lambda}\rtimes\mathfrak{S}_{\lambda}$-orbits in
$U^{\lambda}_{\mathfrak{g}}$. It follows from Theorem 4.21 that these fibres
correspond under $\psi^{\lambda}_{\mathfrak{g}}$ precisely with pseudo-orbits
of local functions with partition $\lambda$. We will refer to the image in
$U^{\lambda}/\mathfrak{S}_{\lambda}$ of a $\widehat{{\it
DLoc}}^{\lambda}\rtimes\mathfrak{S}_{\lambda}$-orbit in
$U^{\lambda}_{\mathfrak{g}}$ as simply a $\widehat{{\it
DLoc}}^{\lambda}$-orbit quotient. We abuse terminology and say a subset or
subvariety of $U^{\lambda}/\mathfrak{S}_{\lambda}$ is $\widehat{{\it
DLoc}}^{\lambda}$-invariant if it is a union of $\widehat{{\it
DLoc}}^{\lambda}$-orbit quotients.

We can use the maps $\psi^{\lambda}_{\mathfrak{g}}$ defined above to
strengthen Proposition 2.11 in our setting.

######  Proposition 5.8.

Let $\mathfrak{g}=W$ or $\mathfrak{g}=W_{\geq-1}$. Let $Q$ be a radical ideal
of $\operatorname{S}(\mathfrak{g})$ and let $Z:=V(Q)$. Suppose also that for
some $\lambda$ the set $V^{\lambda}_{\mathfrak{g}}\cap Z$ is (Zariski) dense
in $Z$. Then $Q$ is Poisson if and only if
$(\psi^{\lambda}_{\mathfrak{g}})^{-1}(Z\cap V^{\lambda}_{\mathfrak{g}})$ is
$\widehat{{\it DLoc}}^{\lambda}$-invariant.

###### Proof.

Let $\psi:=\psi^{\lambda}_{\mathfrak{g}}$, let
$U:=U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$, let
$X:=\overline{\psi^{-1}(Z)}\subseteq U$, and let $\widetilde{X}$ be the
preimage of $X$ in $U^{\lambda}_{W}$. As $Z\cap V^{\lambda}$ is dense in $Z$,
thus $Q$ is the kernel of the composition

$\textstyle{\operatorname{S}(\mathfrak{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{*}}$$\textstyle{{\Bbbk}[U]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Bbbk}[X]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Bbbk}[\widetilde{X}]}$

and by Proposition 2.11, $Q$ is Poisson if and only if
$\mathfrak{g}_{\psi(x)}\subseteq\psi_{x}(T_{x}X)$ for all $x\in X$.

Denote by $A$ the image of $\operatorname{S}(\mathfrak{g})$ in ${\Bbbk}[X]$.
The map $U\to\mathfrak{g}^{*}$ is injective and therefore the map
$X\to\operatorname{MSpec}A$ is also injective. Let $f\in A$ be the element
given by Corollary 5.3. Then the open subsets
$\operatorname{MSpec}(A[f^{-1}])$ and
$\operatorname{MSpec}({\Bbbk}[X][f^{-1}])$ of $\operatorname{MSpec}(A)$ and
$X$, respectively, are isomorphic to each other. This implies that $\psi_{x}$
identifies the tangent spaces $T_{x}X$ and $T_{\psi(x)}Z$ for every $x$ from
this open subset.

Moreover, $\widetilde{X}\to X$ is finite and so by generic smoothness there
are dense open affine subsets $X^{\prime}\subseteq X$,
$\widetilde{X^{\prime}}\subseteq\widetilde{X}$ so that
$\widetilde{X^{\prime}}\to X^{\prime}$ is étale and so identifies tangent
spaces.

Let $X^{\prime\prime}=X^{\prime}\cap\operatorname{MSpec}({\Bbbk}[X][f^{-1}])$
and let $\widetilde{X^{\prime\prime}}\subseteq\widetilde{X}$ be the preimage
of $X^{\prime\prime}$, which is also affine. For every $x\in X^{\prime\prime}$
and every preimage $\widetilde{x}\in\widetilde{X^{\prime\prime}}$ of $x$, the
map $\widetilde{X}\to\mathfrak{g}^{*}$ gives an isomorphism of the tangent
spaces $T_{\widetilde{x}}\widetilde{X}$ and $T_{\psi(x)}Z$.

Let $\mathfrak{h}$ be the Lie algebra of $\widehat{{\it DLoc}}^{\lambda}$. The
proofs of Lemma 4.7 and Lemma 4.24 show that
$\psi_{x}(\mathfrak{h}_{x})=\mathfrak{g}_{\psi(x)}$ for any $x\in U$. By
Proposition 2.11 applied to the morphism $X^{\prime\prime}\to\mathfrak{g}^{*}$
and the above discussion, the ideal $Q$ is Poisson if and only if
$\mathfrak{h}_{x}\subseteq T_{x}\widetilde{X}$ for all
$x\in\widetilde{X^{\prime\prime}}$. To finish the proof we note that the
following conditions are equivalent:

  * •

$\mathfrak{h}_{x}\subseteq T_{x}\widetilde{X}$ for all
$x\in\widetilde{X^{\prime\prime}}$ (that is, tangent spaces to $\widehat{{\it
DLoc}}^{\lambda}$-orbits of points of $\widetilde{X^{\prime\prime}}$ are
tangent to $\widetilde{X}$);

  * •

$\widetilde{X}=\overline{\widetilde{X^{\prime\prime}}}$ is $\widehat{{\it
DLoc}}^{\lambda}$-invariant: that is, a union of
$\widehat{DLoc}^{\lambda}$-orbits;

  * •

the defining ideal of $X=\overline{X^{\prime\prime}}$ is $\widehat{{\it
DLoc}}^{\lambda}$-invariant.

∎

Let $J^{\lambda}_{\mathfrak{g}}:=I(V^{\lambda}_{\mathfrak{g}})$. We
immediately obtain:

######  Corollary 5.9.

The ideal $J^{\lambda}_{\mathfrak{g}}$ is a prime Poisson ideal.

###### Proof.

That $J^{\lambda}_{\mathfrak{g}}$ is Poisson is an immediate application of
Proposition 5.8. To see that it is prime, note that
$U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}$ is irreducible and that
$J^{\lambda}_{\mathfrak{g}}$ is the kernel of
$(\psi^{\lambda}_{\mathfrak{g}})^{*}:\operatorname{S}(\mathfrak{g})\to{\Bbbk}[U^{\lambda}_{\mathfrak{g}}/\mathfrak{S}_{\lambda}]$.
∎

######  Remark 5.10.

By Corollary 5.2 $V^{\lambda}_{\mathfrak{g}}\subset\mathfrak{g}^{*}$ contains
a nonempty open subset of $\overline{V^{\lambda}_{\mathfrak{g}}}$. Let $U$ be
the union of all open subsets of $\overline{V^{\lambda}_{\mathfrak{g}}}$
contained in $V^{\lambda}_{\mathfrak{g}}$. It is surely true that $U$ is
Poisson, in the sense that the defining ideal of