††institutetext: Mathematical Institute, University of Oxford, Woodstock Road,
Oxford, OX2 6GG, UK

# Free field realisation of the chiral universal centraliser

Christopher Beem and Sujay Nair

###### Abstract

In the TQFT formalism of Moore–Tachikawa for describing Higgs branches of
theories of class $\mathcal{S}$, the space associated to the unpunctured
sphere in type $\mathfrak{g}$ is the universal centraliser $\mathfrak{Z}_{G}$,
where $\mathfrak{g}=Lie(G)$. In more physical terms, this space arises as the
Coulomb branch of pure $\mathcal{N}=4$ gauge theory in three dimensions with
gauge group $\check{G}$, the Langlands dual. In the analogous formalism for
describing chiral algebras of class $\mathcal{S}$, the vertex algebra
associated to the sphere has been dubbed the _chiral universal centraliser_.
In this paper, we construct an open, symplectic embedding from a cover of the
Kostant–Toda lattice of type $\mathfrak{g}$ to the universal centraliser of
$G$—extending a classic result of Kostant. Using this embedding and some
observations on the Poisson algebraic structure of $\mathfrak{Z}_{G}$, we
propose a free field realisation of the chiral universal centraliser for any
simple group $G$. We exploit this realisation to develop free field
realisations of chiral algebras of class $\mathcal{S}$ of type
$\mathfrak{a}_{1}$ for theories of genus zero with $n=1,\ldots,6$ punctures.
These realisations make generalised $S$-duality completely manifest, and the
generalisation to $n\geqslant 7$ punctures is conceptually clear, though
technically burdensome.

##  1 Introduction

The occurrence of vertex operator algebra (VOA) structures within the operator
algebras of supersymmetric quantum field theories in various dimensions (see,
e.g., Beem:2013sza ; Beem:2014rza ; Gaiotto:2017euk ; Costello:2018swh ;
Costello:2019 ; Costello:2020ndc ) has been a major theme in the study of such
systems in recent years, and has led to powerful new approaches to the
analysis of these (usually strongly coupled) theories. Furthermore, physical
intuition can potentially provide new insights into the study of VOAs arising
through these correspondences at a purely formal level, and indeed
developments in this area have led to a rich dialogue between physics and pure
mathematics (see, e.g., Arakawa:2018egx ; Creutzig:2017qyf ; Creutzig:2018lbc
; Costello:2018swh ; Xie:2019yds ; Xie:2019vzr ; Creutzig:2021ext ;
Beem:2022mde  for an extremeley incomplete selection).

A surprising property of many VOAs arising in a physical context is their
_geometric_ nature. This is perhaps most clearly exemplified through the Higgs
Branch Conjecture of Beem:2017ooy , which implies that the VOAs arising in
connection with four-dimensional $\mathcal{N}=2$ SCFTs are _chiral
quantisations_ of their Higgs branches (cf. Arakawa:2012c2 ). A related, but
less well formalised, idea is that the VOAs arising in this setting admit
certain (generalised) free field realisations (FFRs) the structure of which is
closely connected with the Poisson geometry of the Higgs branch Beem:2019tfp ;
Beem:2019snk ; Beem:2021jnm . Physically, these free field realisations should
be thought of as encoding the operator algebra of an SCFT in terms of degrees
of freedom that survive in the effective field theory on the moduli space
after (partial) Higgsing. This intuition (and the examples that bear it out)
has thus far been primarily restricted to the setting of VOAs associated to
four-dimensional $\mathcal{N}=2$ SCFTs, but it seems likely that the principle
holds somewhat more generally.

In this paper, we will be investigating the application of this free field
philosophy to a family of VOAs that are exotic from a four-dimensional
perspective, but potentially more conventional in the three-dimensional
setting. These are the _chiral universal centralisers_ defined in
Arakawa:2018egx , which were previously identified as the VOAs associated to
the _sphere_ in the setting of chiral algebras of class $\mathcal{S}$
Beem:2014rza . The associated variety of this VOA is the _universal
centraliser_ $\mathfrak{Z}_{G}$, the definition of which we recall in Section
2—this is the holomorphic-symplectic variety associated to the sphere by the
Moore–Tachikawa topological field theory for class $\mathcal{S}$ Higgs
branches Moore:2011ee . While in the context of class $\mathcal{S}$ the sphere
is a somewhat pathological case, the universal centraliser appears in a more
physically sensible way as the Coulomb branch of pure $\mathcal{N}=4$ gauge
theory in three dimensions. Thus the chiral quantisation of this space might
naturally be expected to appear as a boundary VOA in the sense of
Costello:2019  for pure gauge theory.

Our approach to constructing a candidate free field realisation for the chiral
universal centraliser will follow the general theme of Beem:2019tfp . We start
with a detailed understanding of the Poisson geometry of the universal
centraliser, an introduction to which we provice in Section 2. In Section 3,
we give an explicit characterisation of a map, $\varphi$, from the
Kostant–Toda lattice to a Zariski-open subset of $\mathfrak{Z}_{G}$. Such a
map was known to Kostant for the case when $G$ is of adjoint type, and in
Theorem 3.3 we construct a generalisation of (Kostant:1979195, , Theorem 2.6)
for any semi-simple $G$. This gives rise to Propositions 3.6 and 3.7, in which
we show that $\varphi$ is an open immersion and symplectic.

Altogether, this gives a finite-dimensional analogue of a free field
realisation for the Poisson algebra of functions
$\mathbb{C}[\mathfrak{Z}_{G}]$. In Section 4 we propose a chiralisation of
this construction in terms of a lattice vertex algebra that we call the chiral
Kostant–Toda lattice. In particular, we define a sub-VOA of this lattice VOA
(by specifying a weak generating set) that we conjecture to match the chiral
universal centraliser as defined by Arakawa. In defining this sub-VOA, we
encounter an (apparently novel) generalisation of the Miura transform, and in
Remark 4.1 we explain how that the usual Miura transform—realising principal
${\cal W}$-algebras in products of Heisenberg vertex algebras—can be
generalised to a family of such embeddings based on the lower central series
of the upper nilpotent subalgebra of $\mathfrak{g}$.

We verify that our construction reproduces the chiral universal centraliser
(defined _via_ Drinfel’d–Sokolov reduction) explicity for the case $G={\rm
SL}_{2}$, and we also study the $G={\rm SL}_{3}$ case in detail and confirm
that the VOA according to our prescription is indeed a chiral quantisation of
universal centraliser. Here a direct comparison to the definition in terms of
DS reduction would require a detailed analysis of said reduction, and this has
not appeared in the literature to date so far as the authors are aware. Some
interesting structure regarding gradings and simplicity of these VOAs become
apparent in our examples, and we elaborate on this in Section 4.4, where we
define a “more universal” centraliser that returns the universal centraliser
by central quotient.

An additional benefit of our free field construction is that for chiral
algebras of class $\mathcal{S}$ corresponding to physical theories (so with
three or more punctures), the sphere free field realisation can act as the
starting point for free field realisations that treat all punctures on equal
footing (and so manifest generalised $S$-duality at the level of FFRs). In
Section 5 we consider the case of $\mathfrak{a}_{1}$ theories with various
numbers of punctures. We give free field realisations for these vertex
algebras that incorporate the sphere FFR, which leads to a very uniform
construction. Analogous realisations for other class $\mathcal{S}$ families
should arise in much the same way. We conclude in Section 6 with several
comments and open questions. The proof of Theorem 2.2, regarding Poisson
generation in the ${\rm SL}_{N}$ case, is relegated to Appendix A.

The OPE computations in this paper were performed using the OPEdefs package,
which was developed by Kris Thielemans Thielemans:1994er , for Mathematica.

##  2 The universal centraliser

The universal centraliser admits several equivalent definitions, some of which
arise naturally by way of three- and four-dimensional physics. In this section
we will review these definitions and some important properties of the
universal centraliser, and also establish notation for what follows.

###  2.1 Background and definitions

Let $G$ be a simple algebraic Lie group and $\mathfrak{g}=\mathrm{Lie}(G)$.
Let $\mathfrak{g}^{\ast}$ be the linear dual of $\mathfrak{g}$, and using the
Killing form $(\cdot,\cdot)$ one may identify $\mathfrak{g}$ with
$\mathfrak{g}^{\ast}$ via the isomorphism $x\mapsto(x,\cdot)$ for
$x\in\mathfrak{g}$.

Furthermore, let $B\subset G$ be a Borel subgroup with splitting $B=N\rtimes
T$ for a maximal torus $T$ and unipotent group $N$. The Lie algebra
$\mathfrak{g}$ admits a Cartan decomposition
$\mathfrak{g}=\mathfrak{n_{+}}\oplus\mathfrak{t}\oplus\mathfrak{n_{-}}$, where
$\mathfrak{t}=\mathrm{Lie}~{}T$ and $\mathfrak{n}_{+}=\mathrm{Lie}~{}N$. (For
future reference, we also denote by $N_{-}$ the lower unipotent group with
$\mathfrak{n}_{-}=\mathrm{Lie}~{}N_{-}$.)

We denote adjoint and coadjoint actions of $G$ on $\mathfrak{g}$ by
$\mathrm{Ad}$ and $\mathrm{Ad}^{\ast}$, respectively. Similarly, $\mathrm{ad}$
and $\mathrm{ad}^{\ast}$ denote the respective $\mathfrak{g}$-actions. The
stabiliser, $G_{x}$, of an element $x\in\mathfrak{g}^{\ast}$ is the subgroup
of $G$ that leaves $x$ invariant under the coadjoint action,

$G_{x}=\\{g\in G|\,\mathrm{Ad}^{\ast}_{g}\,x=x\\}~{}.$ (1)

The Zariski open subset $\mathfrak{g}^{\ast}_{\rm
reg}\subset\mathfrak{g}^{\ast}$ is defined to be the set of all elements whose
centralisers have dimension equal to the rank of $\mathfrak{g}$. This is the
complement of the union of non-principal nilpotent orbits in
$\mathfrak{g}^{\ast}$, or alternatively, the union of all non-nilpotent orbits
along with the principal nilpotent orbit in $\mathfrak{g}^{\ast}$.

Let $T^{\ast}G$ be the total space of the cotangent bundle of $G$; one has the
trivialisation $T^{\ast}G\cong\mathfrak{g}^{\ast}\times G$ using the
realisation of $\mathfrak{g}$ by left-invariant vector fields on $G$. On the
cotangent bundle $T^{\ast}G$ there are two commuting, Hamiltonian $G$ actions,
with moment maps $\mu_{L}$ and $\mu_{R}$ given by

$\mu_{L}(g,x)=x~{},\qquad\mu_{R}(g,x)=\mathrm{Ad}_{g}^{*}\,x~{},$ (2)

for $(g,x)\in G\times\mathfrak{g}^{*}\cong T^{\ast}G$. Let $\mu_{\rm
diag}=\mu_{L}-\mu_{R}$ be the moment map for the diagonal $G$ action. The
universal centraliser is then defined as the following diagonal Hamiltonian
reduction Bezrukavnikov:2003 ,111Note that here our $\mathfrak{Z}_{G}$ is
denoted by $\mathfrak{Z}^{G}_{\mathfrak{g}}$.

$\mathfrak{Z}_{G}=\big{(}\mu_{\rm diag}^{-1}(0)\cap(\mathfrak{g}^{\ast}_{\rm
reg}\times G)\big{)}/\\!\\!/G_{\rm diag}~{}.$ (3)

This definition gives the universal centraliser the structure of a Poisson
variety (and in fact a symplectic variety). Furthermore, $\mathfrak{Z}_{G}$ is
naturally a scheme over the orbit space $\mathfrak{g}^{\ast}_{\rm
reg}/\\!\\!/G\cong\mathfrak{g}^{*}/\\!\\!/G$. We let
$(P_{i})_{i=1}^{\mathrm{rk}~{}\mathfrak{g}}$ denote the generators of the
$\mathbb{C}[\mathfrak{g}^{\ast}]^{G}$ (the _Harish-Chandra centre_)
corresponding to the fundamental invariants of $\mathfrak{g}$.

In terms of class $\SS$ physics, $\mathfrak{Z}_{G}$ can be formally identified
as the Higgs branch associated to an unpunctured sphere. While there is no
physical SCFT in four-dimensions for which this is the Higgs branch, one can
still interpret such an object in terms of, e.g., the Moore–Tachikawa
TQFT.222Or alternatively, in terms of compactification of the $(2,0)$ theory
on a Riemann surface of finite size, as in Gaiotto:2011xs . In that setting,
the operation of closing full punctures on the UV curve is implemented by
performing Kostant–Whittaker (KW) reduction on the associated symplectic
variety with respect to the Hamiltonian $G$-actions associated to the closed
punctures. The sphere can then be obtained, for instance, by closing two
punctures on the cylinder, which itself is associated to $T^{\ast}G$. This
leads to another definition of $\mathfrak{Z}_{G}$ as the two-sided KW
reduction of the cotangent bundle $T^{*}G$, as seen in Ginzburg:2018Nil . This
perspective will also have the most natural link to the chiralisation of
$\mathfrak{Z}_{G}$.

Let us make this characterisation more concrete. Let $\Delta$ denote the set
of (positive) simple roots of $\mathfrak{g}$, and let

$f=\sum_{\alpha\in\Delta}e_{-\alpha}~{},$ (4)

be the sum of the negative simple roots. By the Jacobson–Morozov theorem, such
an $f$ can be completed to an $\mathfrak{sl}_{2}$-triple $(e,h,f)$. Let
$\chi\in\mathfrak{g}^{*}$ be the (Killing) dual of $f$.

The moment maps on $T^{*}G$ can be restricted to $\mathfrak{n}_{-}^{*}$-valued
moment maps by means of the composition
$\bar{\mu}_{L,R}:T^{*}G\rightarrow\mathfrak{g}^{*}\twoheadrightarrow\mathfrak{g}^{*}/\mathfrak{b}_{+}^{*}\cong\mathfrak{n}^{*}_{-}$,
where $\mathfrak{b}_{+}=Lie(B)$ is our Borel subalgebra. The maximal (upper)
unipotent subgroups of the left and right $G$-actions are denoted by $N_{L}$
and $N_{R}$, respectively. The KW realisation of the universal centraliser
then takes the following form,

$\mathfrak{Z}_{G}\cong T^{*}G\overset{~{}\chi}{\,/\\!\\!/\,}(N_{L}\times
N_{R})=(\bar{\mu}^{-1}_{L}(\chi)\cap\bar{\mu}^{-1}_{R}(\chi))/\\!\\!/(N_{L}\times
N_{R})~{},$ (5)

which again encodes $\mathfrak{Z}_{G}$ as a symplectic variety.

Now let $\mathfrak{g}^{e}\subset\mathfrak{g}$ be the
$\mathrm{ad}_{e}$-invariant subspace. The _Slodowy slice_
$S_{f}\subset\mathfrak{g}^{\ast}_{\rm reg}\subset\mathfrak{g}^{\ast}$ is the
image of $f+\mathfrak{g}^{e}$ under the Killing isomorphism, or more
concretely,

$S_{f}=\chi_{f}+\\{(x,\cdot)\in\mathfrak{g}^{*}|\,x\in\mathfrak{g}^{e}\\}~{}.$
(6)

All $G$-orbits in $\mathfrak{g}^{\ast}_{\rm reg}$ intersect $S_{f}$
transversally Kostant:1978 , so in particular,

$\mathfrak{g}^{\ast}_{\rm reg}=G\,\cdot S_{f}~{}.$ (7)

This structure leads to Kostant’s isomorphism
$\mathfrak{g}^{*}\,/\\!\\!/\,G\cong S_{f}$. At the level of varieties
(schemes), performing KW reduction on a Hamiltonian $G$-space $X$ (with
respect to the principal nilpotent conjugacy class) is equivalent to taking
the fibre-product of $X$ with $S_{f}$ over $\mathfrak{g}^{*}$, see
Ginzburg:2008HC . We can perform two-sided KW reduction in stages, and after
taking just the reduction by the left action we have

$T^{*}G\overset{\chi}{\,/\\!\\!/\,}N_{L}\cong
T^{*}G\\!\\!\\!\underset{\mu_{L},\mathfrak{g}^{*}}{\times}\\!\\!\\!S_{f}=G\times
S_{f}~{}.$ (8)

The variety $G\times S_{f}$ (with the symplectic structure induced by KW
reduction) is also known as the equivariant Slodowy slice Losev1 ; Losev2 .
This inherits the right-action moment map, and performing the second reduction
then gives

$\begin{split}\mathfrak{Z}_{G}\cong(G\times
S_{f})\overset{\chi}{\,/\\!\\!/\,}N_{R}&\cong(G\times
S_{f})\underset{\mu_{R},\mathfrak{g}^{*}}{\times}S_{f}\\\ &\cong\\{(g,s)\in
G\times S_{f}\,|\,\mathrm{Ad}^{*}_{g}s\in S_{f}\\}~{}.\end{split}$ (9)

This gives a definition of $\mathfrak{Z}_{G}$ as the group scheme centralising
points in $S_{f}$, so in particular as a group scheme over
$S_{f}\cong\mathbb{C}^{\mathrm{rk}\,\mathfrak{g}}$. $S_{f}$ can then be
thought of as the base of an integrable system on $\mathfrak{Z}_{G}$ with
action variables given by the (restriction to $S_{f}$ of the) generators of
the Harish-Chandra centre.

The universal centraliser of type $G$ also arises in _three-dimensional_
physics as the Coulomb branch of pure ${\cal N}=4$ gauge theory with gauge
group $\check{G}$, the Langlands dual of $G$. (This is the three-dimensional
mirror for the circle compactification of the class $\SS$ theory associated to
the two-sphere.) The Coulomb branch of pure $\mathrm{SU}(2)$ gauge theory was
identified with the Atiyah–Hitchin manifold in Seiberg:1996nz , and based in
part upon this and further evidence from Argyres:1994xh ; Martinec:1995by ,
Teleman predicted in Teleman:2014jaa  that the Coulomb branch of pure
$\mathcal{N}=4$ gauge theory for any $\check{G}$ could be identified with the
(spectrum of the) Borel–Moore equivariant homology ring of the affine
Grassmannian $Gr_{\check{G}}$, which had been computed earlier by
Bezrukavnikov–Finkelberg–Mirković (BFM) Bezrukavnikov:2003 . This definition,
in terms of the affine Grassmannian, is confirmed by the construction of
three-dimensional Coulomb branches by Braverman–Finkelberg–Nakajima (BFN)
Nakajima:2015txa ; Braverman:2016wma ; Braverman:2017ofm ; Braverman:2016pwk
.333A physics-motivated computation of the Coulomb branches of ${\cal N}=4$
gauge theories by abelianisation techniques appeared contemporaneously in
Bullimore:2015lsa .

For completeness, we recall this definition here. Let $\check{G}({\cal
O})={\rm Hom}(\mathbb{D},G)$ and $\check{G}({\cal K})={\rm
Hom}(\mathbb{D}^{\times},G)$ be the spaces of homorphisms from the formal
disc, $\mathbb{D}=\mathrm{Spec}~{}\mathbb{C}[\\![t]\\!]$ and the formal
punctured disc, $\mathbb{D}^{\times}=\mathrm{Spec}~{}\mathbb{C}(\\!(t)\\!)$,
respectively, to $\check{G}$. The affine Grassmannian is the quotient,

$\mathrm{Gr}_{\check{G}}=\check{G}({\cal K})/\check{G}({\cal O})~{}.$ (10)

The universal centraliser is then identified with (the spectrum of) the
$\check{G}({\cal O})$-equivariant Borel–Moore homology,

$\mathfrak{Z}_{G}\cong\mathrm{Spec}~{}H_{\bullet}^{\check{G}({\cal
O})}(\mathrm{Gr}_{\check{G}})~{}.$ (11)

From this definition, realising the symplectic structure on $\mathfrak{Z}_{G}$
is a more elaborate procedure than in the previous classical definitions.

###  2.2 Examples in rank one

To make these spaces a bit more tangible, we consider in detail the two rank-
one examples: $G={\rm SL}_{2}$ and $G={\rm PGL}_{2}$.

By the Killing isomorphism, $S_{f}$ is identified with the elements of
$\mathfrak{sl}_{2}$ of the form

$\begin{pmatrix}\,0\,&\,S\,\\\ \,1\,&\,0\,\end{pmatrix}~{},\quad
S\in\mathbb{C}~{}.$ (12)

The most general element of ${\rm SL}_{2}$ that commutes with such a matrix is
of the form

$\sqrt{-1}\begin{pmatrix}Y&SX\\\ X&Y\end{pmatrix}~{},\quad
X,Y,S\in\mathbb{C}~{},$ (13)

where the overall factor of $\sqrt{-1}$ is purely conventional. Therefore,
$\mathfrak{Z}_{{\rm SL}_{2}}\subset\mathbb{A}^{3}$ can be realised as an
affine variety with coordinate ring

$I_{{\rm SL}_{2}}=\frac{\mathbb{C}[X,Y,S]}{\langle SX^{2}-Y^{2}-1\rangle}~{},$
(14)

which is the presentation appearing in Bezrukavnikov:2003 .

The Poisson bracket can be computed by carrying out the two sided KW reduction
of $T^{*}{{\rm SL}_{2}}$.444One can also obtain the Poisson brackets from the
blow-up computation of Bezrukavnikov:2003 .

$\\{S,X\\}=Y~{},\quad\\{S,Y\\}=SX~{},\quad\\{Y,X\\}=\frac{1}{2}X^{2}~{},$ (15)

with all other Poisson brackets vanishing. We observe that as a Poisson
algebra, $I_{{\rm SL}_{2}}$ is generated by just $S$ and $X$. This point will
arise again later when we discuss the general case for ${\rm SL}_{N}$.

Now for the adjoint case $\mathfrak{Z}_{{\rm PGL}_{2}}$, we have

$\mathfrak{Z}_{{\rm PGL}_{2}}\cong Z{({\rm
SL}_{2}})\backslash\mathfrak{Z}_{{\rm SL}_{2}}~{},$ (16)

where the left action of $Z({{\rm SL}_{2}})=\mathbb{Z}/2\mathbb{Z}$ is given
by left multiplication by $-\mathbbm{1}$ on the fibres. On functions this
action is given by

$X\mapsto-X~{},\quad Y\mapsto-Y~{}.$ (17)

Thus,

$I_{{\rm PGL}_{2}}\cong\frac{\mathbb{C}[S,X^{2},XY,Y^{2}]}{\langle
SX^{2}-Y^{2}-1\rangle}\cong\frac{\mathbb{C}[S,A,B,C]}{\langle
B^{2}-AC,SA-C-1\rangle}~{}.$ (18)

The Poisson structure can be obtained by pullback along the quotient map
$\mathfrak{Z}_{{\rm SL}_{2}}\rightarrow Z({{\rm
SL}_{2}})\backslash\mathfrak{Z}_{{\rm SL}_{2}}\cong\mathfrak{Z}_{{\rm
PGL}_{2}}$.

###  2.3 Example in rank two

It will be useful in what follows to also have an example in rank two at our
disposal. Here we choose the universal centraliser for $G={\rm SL}_{3}$.

Once again by using the Killing isomorphism, the Slodowy slice $S_{f}$ can be
identified as an affine subspaces in $\mathfrak{g}=\mathfrak{sl}_{3}$ of the
form

$\begin{pmatrix}0&\frac{1}{2}S&W\\\ 1&0&\frac{1}{2}S\\\
0&1&0\end{pmatrix}~{},\quad S,W\in\mathbb{C}~{}.$ (19)

Any element of ${\rm SL}_{3}$ that stabilises a point in the slice can be
written as

$\begin{pmatrix}Z&WX+\frac{1}{2}SY&\frac{1}{4}S^{2}X+WY\\\
Y&\frac{1}{2}SX+Z&WX+\frac{1}{2}SY\\\ X&Y&Z\end{pmatrix}~{},$ (20)

with $X$, $Y$, and $Z$ satisfying the following determinant relation,

$\begin{split}D_{{\rm
SL}_{3}}=&-\frac{1}{8}S^{3}X^{3}+W^{2}X^{3}+\frac{1}{2}SWX^{2}Y+\frac{1}{2}S^{2}XY^{2}+WY^{3}-\frac{1}{4}S^{2}X^{2}Z\\\
&-3WXYZ-SY^{2}Z+\frac{1}{2}SXZ^{2}+Z^{3}\overset{!}{=}1~{}.\end{split}$ (21)

Thus, we can identify $\mathfrak{Z}_{{\rm SL}_{3}}$ as an affine variety in
$\mathbb{A}^{5}$ with coordinate ring

$I_{{\rm SL}_{3}}\cong\frac{\mathbb{C}[S,W,X,Y,Z]}{\langle D_{{\rm
SL}_{3}}-1\rangle}~{}.$ (22)

The Poisson structure can again be computed from the two-sided KW reduction,
giving

$\displaystyle\\{S,X\\}$ $\displaystyle=-Y~{},$ $\displaystyle\quad\\{S,Y\\}$
$\displaystyle=-Z-\frac{1}{2}SX~{},$ $\displaystyle\quad\\{S,Z\\}$
$\displaystyle=-SY-WX~{},$ (23) $\displaystyle\\{W,X\\}$
$\displaystyle=-Z+\frac{1}{6}SX~{},$ $\displaystyle\quad\\{W,Y\\}$
$\displaystyle=-WX-\frac{1}{3}SY~{},$ $\displaystyle\quad\\{W,Z\\}$
$\displaystyle=-WY+\frac{1}{6}SZ-\frac{1}{4}S^{2}X~{},$ (24)
$\displaystyle\\{Y,X\\}$ $\displaystyle=\frac{2}{3}X^{2}~{},$
$\displaystyle\quad\\{Z,X\\}$ $\displaystyle=-\frac{5}{6}XY~{},$
$\displaystyle\quad\\{Z,Y\\}$
$\displaystyle=-\frac{1}{3}Y^{2}+\frac{1}{6}XZ-\frac{1}{4}SX^{2}~{}.$ (25)

Observe again that $I_{{\rm SL}_{3}}$ is generated, as a Poisson algebra, by
the generators $S,W$ of the Harish-Chandra centre and the coordinate $X$ that
is conjugate to the bottom left corner of the ${\rm SL}_{3}$ matrix. The
algebraic generator $Z$ is a shift of the Poisson bracket $\\{S,Y\\}$, but of
course one is free to use the generating set
$\big{(}S,W,X,\\{S,X\\},\\{S,\\{S,Y\\}\\}\big{)}$ instead.

###  2.4 Poisson algebraic structure for ${\rm SL}_{N}$

In this section, we will codify some patterns we have observed in the examples
of ${\rm SL}_{2}$ and ${\rm SL}_{3}$. We generalise these to statements about
${\rm SL}_{N}$ and discuss their further generalisation to arbitrary simple
algebraic Lie groups.

We realise ${\rm SL}_{N}$ as the closed subgroup of ${\rm GL}_{N}$ satisfying
the unit determinant condition. Let $(g_{ij})_{i,j=1}^{N}$ be the linear
coordinates on ${\rm GL}_{N}$ that are conjugate to the matrix entries. These
restrict to linear coordinates functions on ${\rm SL}_{N}$.

In both of the examples ${\rm SL}_{2}$ and ${\rm SL}_{3}$, the ring $I_{G}$
was generated by the Slodowy slice generators and the matrix entries in the
bottom row of the group element. This observation generalises as follows.

######  Proposition 2.1.

Let $s\in S_{f}$. An element, $g$, of $\,{\rm SL}_{N}$ that stabilises $s$ is
completely fixed by specifying the matrix entries of the bottom row of $g$.

In other words, if $g_{ij}$ are the coordinate functions conjugate to the
matrix entries, then $I_{{\rm SL}_{N}}$ is generated, as a ring, by the
restrictions of the fundamental invariants to $S_{f}$, $(P_{i})_{i=1}^{N}$,
and by $(g_{Ni})_{i=1}^{N}$.

###### Proof.

Let $g\in{\rm SL}_{N}$ and $s\in S_{f}$, then the condition
$\mathrm{Ad}^{\ast}_{g}s=s$ is equivalent to requiring $[g,s]=0$, where we
think of $g$ as a matrix of $\mathfrak{gl}_{N}$.

The kernel of $[-,s]$ in $\mathfrak{gl}_{N}$ is $N$-dimensional since
$S_{f}\subset\mathfrak{gl}_{N}$ is regular. The proposition will follow if we
can show that there is a basis, $(v_{i})_{i=1}^{N}$, of $\mathrm{ker}~{}[-,s]$
consisting of matrices $v_{i}$ such that $(v_{i})_{Nj}=\delta_{ij}$. In words,
$v_{i}$ is an element of $\mathrm{ker}~{}[-,s]$ where the $i$th entry along
the bottom row is equal to one and the rest of the bottom row entries vanish.

We can explicitly construct such a basis. First, let $v_{N}=\mathbbm{1}$,
i.e., the identity matrix. We also fix $v_{N-1}=s$. It is clear that both
$v_{N-1}$ and $v_{N}$ commute with $s$ and satisfy the required conditions on
the bottom row.

Now consider the matrix product, $s^{k}$, the entries $(s^{k})_{Ni}$ are zero
for $i<N-k$ and for $i=N-K+1$. The entry $(s^{k})_{N,N-k}=1$ and the entries
$s^{k}_{Ni}$ for $k>N-k$ are generically nonzero. By induction, we can take
linear combinations,

$s^{k}+\sum_{j=0}^{k-2}c_{j}s^{j}~{},$ (26)

for $c_{j}\in\mathbb{C}$ to set all the bottom entries to zero, except the
$(N,N-k)$ entry. We set $v_{N-k}$ equal to this linear combination.

This provides a linearly independent set $(v_{i})_{i=1}^{N}$ which all lie in
the kernel of $[-,s]$, since $s$ commutes with all powers $s^{k}$. Therefore
the $v_{i}$ form a basis of $\mathrm{ker}~{}[-,s]$ and we are done. ∎

Later on this Proposition will yield a significant simplification in the
procedure for computing the ring $I_{G}$ in terms of Kostant–Toda lattice
generators. There is also another simplification to be had.

######  Theorem 2.2.

The Poisson algebra $I_{{\rm SL}_{N}}$ is Poisson generated by
$(P_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ and the generator $X=g_{N1}$.

###### Proof.

The proof of this theorem relies upon machinery that will be introduced in
later sections. As such we relegate it to Appendix A. ∎

We expect a generalisation of these results to any simple algebraic Lie group
$G$ to hold as well. To justify this, let us characterise this preferred
element $X$ in more representation theoretic terms. By the Peter–Weyl
decomposition, the coordinate ring of $G$ decomposes into $G-G$ bi-modules
according to

$\mathbb{C}[G]\cong\bigoplus_{\lambda\in P^{+}}V_{\lambda}\otimes
V_{\lambda}^{*}~{},$ (27)

where $P^{+}$ is the set of integral dominant weights of $\mathfrak{g}$,
$V_{\lambda}$ is the finite-dimensional representation of $\mathfrak{g}$ with
highest weight $\lambda$ and $V_{\lambda}^{*}$ is the contragredient dual
representation. The vector space of functions on
$T^{*}G\cong\mathfrak{g}^{*}\times G$ therefore takes the form

$\mathbb{C}[T^{*}G]=\mathrm{Sym}~{}\mathfrak{g}~{}\otimes\bigoplus_{\lambda\in
P^{+}}V_{\lambda}\otimes V_{\lambda^{*}}~{}.$ (28)

Let $V_{F}$ be the lowest dimensional faithful representation of $G$. If $G$
is not of type $D_{2n}$ then $V_{F}$ is irreducible, so let us make this
assumption for the moment. Since $V_{F}$ is faithful, every irreducible
representation $V_{\lambda}$ will appear in the decomposition of the tensor
powers of $V_{F}$. In light of the Peter-Weyl decomposition, this means that
one can choose as generators of $\mathbb{C}[G]$ the functions transforming in
the representation $V_{F}\otimes V_{F}^{*}$ in (27). For example, if $G={\rm
SL}_{n}$ then $V_{F}$ is the defining representation. One usually picks the
linear coordinates $\\{g_{ij}\\}$—conjugate to the matrix entries—to be
generators of $\mathbb{C}[{\rm SL}_{n}]$, and indeed these constitute
precisely the term $V_{F}\otimes V_{F}^{*}$ in (27).

Let $v_{F}\in V_{F}$ be the highest-weight vector, then $V_{F}$ is generated,
as a $\mathfrak{g}$-module by $v_{F}$. Now, suppose $\widetilde{X}$ is the
function in $\mathbb{C}[G]\subset\mathbb{C}[T^{*}G]$ transforming in the
highest weight state $v_{F}\otimes v_{F}^{*}$. Then $\widetilde{X}$ is not in
the ideal generated by $\overline{\mu}_{L}$ and $\overline{\mu}_{R}$, and
moreover it is $N_{L}\times N_{R}$ invariant—since it is the highest-weight
state. As a result, $\widetilde{X}$ will descend unmodified to the two-sided
KW reduction of $T^{*}G$. For ${\rm SL}_{N}$, $\widetilde{X}$ is the linear
coordinate function conjugate to the matrix entry $g_{N1}$, and after KW
reduction it will become the function $X$ introduced in Theorem 2.2.

The bottom row $(g_{Ni})_{i=1}^{N}$ that algebraically generate (along with
the fundamental invariants) $I_{\rm SL_{N}}$ are identified with the vector
subspace $v_{F}\otimes V_{F}^{*}$ of $V_{F}\otimes V_{F}^{*}$. This provides a
more abstract characterisation of the algebraic generators of $I_{\rm
SL_{N}}$. For general $G$ these functions are $N_{L}$ invariant and can be
made $N_{R}$ invariant by corrections that are linear in the generators of
$v_{f}\otimes V_{F}^{*}$. We should therefore expect a correspondence between
(a subset) of the algebraic generators of $I_{G}$ and the weight vectors of
$v_{F}\otimes V_{F}^{*}$.

The Poisson algebra $\mathbb{C}[T^{*}G]$ has two Poisson-commuting
subalgebras, each isomorphic to $\mathrm{Sym}~{}\mathfrak{g}$, arising from
the moment maps for the left and right $G$-actions. The subspace $V_{F}\otimes
V_{F}^{*}\subset\mathbb{C}[T^{*}G]$ is generated by the (Poisson) action of
the left and right moment maps on $\widetilde{X}$, and since $V_{F}\otimes
V_{F}^{*}$ generates $\mathbb{C}[G]$, we can choose $\widetilde{X}$ and the
generators of the left and right moment map as Poisson generators of
$\mathbb{C}[T^{*}G]$. After two-sided KW reduction, what remains of the moment
maps are the generators of
$\mathbb{C}[\mathfrak{g}^{*}]^{G}\cong\mathbb{C}[S_{f}]$. It seems reasonable,
then, to conjecture that $X$ and $\mathbb{C}[S_{f}]$, which are in an
appropriate sense the reductions of the Poisson generators of
$\mathbb{C}[T^{*}G]$, will be Poisson generators of the reduced algebra.

Let us now return to the case where $G$ is of type $D_{2N}$. If $G$ is not
simply connected, then $Z(G)$ is cyclic and our previous description still
holds and $V_{F}$ is irreducible. If $G$ is of type $D_{2N}$ and simply
connected then $V_{F}$ is not irreducible—it is the Dirac spinor, $V_{\rm
s}\oplus V_{\rm c}$, where $V_{\rm s}$ and $V_{\rm c}$ are the smallest spin
representations of $\rm Spin(N)$. In this case, we must pick two such highest-
weight states, $X_{s}$ and $X_{c}$ in representations $V_{\rm s}\otimes V_{\rm
s}^{*}$ and $V_{\rm c}\otimes V_{\rm c}^{*}$ respectively. Similarly, the
$N_{L}$ invariant subspace is now $(v_{\rm s}\otimes V_{\rm
s}^{*})\oplus(v_{\rm c}\otimes V_{\rm c}^{*})$, where $v_{s}$ and $v_{c}$ are
highest weight states of $V_{\rm s}$ and $V_{\rm c}$ respectively.

Putting this all together, we have the following conjectures on the algebraic
and Poisson algebraic structures of $I_{G}$.

######  Conjecture 2.3.

If $G$ is not simply connected, or if $G$ is not of type $D_{2N}$, then
$I_{G}$ is generated, as a ring, by the fundamental invariants
$(P_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ and generators
$(g_{i})_{i=1}^{{\rm dim}\,V_{F}}$ which are in one-to-one correspondence with
the weight vectors of $V_{F}^{*}$.

If $G$ is simply connected and of type $D_{2N}$, $I_{G}$ is generated, as a
ring, by the fundamental invariants
$(P_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ and two sets of generators
$(g_{{\rm s}i})_{i=1}^{{\rm dim}\,V_{\rm s}}$, and $(g_{{\rm c}i})_{i=1}^{{\rm
dim}\,V_{\rm c}}$. The generators $g_{{\rm s}i}$ (repectively $g_{{\rm c}i}$)
are in one-to-one correspondence with the weight vectors of $V_{\rm s}^{*}$
(repectively $V_{\rm c}^{*}$).

######  Conjecture 2.4.

If $G$ is not simply connected, or if $G$ is not of type $D_{2N}$, then
$I_{G}$ is Poisson generated by the fundamental invariants
$\big{(}P_{i}\big{)}_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ and $X$.

If $G$ is simply connected and of type $D_{2N}$, i.e., if
$G=\mathrm{Spin}(2N)$, then $I_{G}$ is Poisson generated by the fundamental
invariants $\big{(}P_{i}\big{)}_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$, $X_{s}$,
and $X_{c}$.

##  3 The universal centraliser and the Kostant–Toda lattice

The geometric structure that apparently underlies the free field realisations
in Beem:2019tfp  is the existence of some particularly simple Zariski open
subset(s) $U$ of the Higgs branch $X$ in terms of which the global functions
on $X$ can be realised via the inclusion
$\mathbb{C}[X]\hookrightarrow\mathbb{C}[U]$. These subsets are generally
identified (including holomorphic symplectic structure) with cotangent bundles
of the form $(T^{\ast}\mathbb{C}^{\times})^{n}\times(T^{\ast}\mathbb{C})^{m}$
with canonical Poisson structures,555As in Beem:2019tfp , we take our
standard Poisson structure on $T^{\ast}\mathbb{C}^{\times}$ to be
$\\{p,e\\}=e$ where $p$ is the fibre coordinate and $e$ is the base
coordinate. so these can be thought of as Darboux (Zariski) neighbourhoods in
$X$. In practice, the expressions for the images of generators of
$\mathbb{C}[X]$ in terms of $\mathbb{C}[U]$ can often be inferred using
symmetry data, such as weights under the contracting $\mathbb{C}^{\ast}$
action on $X$ and the action of additional global symmetries. Such “finite-
dimensional free field realisations” form the foundation for a chiral
analogue, where the associated VOA is realised in terms of $\beta\gamma$
systems and isotropic lattice bosons that naturally chiralise $\mathbb{C}[U]$.
(We will say more about these isotropic lattice VOAs in Section 4.)

In previous examples in the literature, the open set $U$ has been defined with
reference to a (partial) Higgsing of some global symmetry acting on $X$.
However, it is not at all clear that there is anything fundamental about this
particular origin for such Darboux neighborhoods, and a more general story may
certainly exist. Indeed, for the case of the universal centraliser, there are
no global symmetries to speak of, so a new starting point is needed.

Fortunately, it has been known since work of Kostant (see Kostant:1979195 )
that the integrable system of the Kostant–Toda lattice of type $\mathfrak{g}$
and the universal centraliser $\mathfrak{Z}_{G}$ (for $G$ of adjoint type) are
intimately connected, with the former having precisely the Darboux structure
needed. In particular, the level sets of the Kostant–Toda lattice (as an
integrable system) can be identified with (open) subsets of stabilisers of
elements in $\mathfrak{g}^{\ast}_{\rm reg}$. Recently, Crooks has provided a
complex-analytic embedding of an open dense subset of the Kostant–Toda lattice
inside the universal centraliser Crooks:2020 , still with the caveat that $G$
be of adjoint type.

In this section, we will adapt this construction to the algebraic setting and
extend the domain of the morphism to the full Kostant–Toda lattice.
Generalising further to allow for any simple $G$ (including, e.g., both
adjoint and simply-connected forms of the group) will require us to replace
the Kostant–Toda lattice with a certain covering space that we define. Our
main result is the construction of an open immersion from the (cover of the)
Kostant–Toda lattice to the universal centraliser of $G$, and thus a finite-
dimensional free field realisation of the universal centraliser.

###  3.1 The Kostant–Toda lattice

Consider a dynamical system of $N$ particles on a line, with Hamiltonian

$\mathcal{H}_{KT}=\sum_{i=1}^{N}\frac{p_{i}^{2}}{2}+\sum_{i=1}^{l}e^{\psi_{i}(\mathbf{q})}~{},$
(29)

where $\psi_{i}(\mathbf{q})$ are linear functions of the canonical position
coordinates $\mathbf{q}=(q_{1},q_{2},\dots,q_{N})$. There is a natural inner
product on the vector space spanned by the $\psi_{i}$ Kostant:1978 . When this
inner product space agrees with the root system of some real Lie algebra
$\mathfrak{g}_{\mathbb{R}}$ endowed with the Killing form, the $\psi_{i}$
define a set of roots for $\mathfrak{g}_{\mathbb{R}}$, and in this case the
representation theory of $\mathfrak{g}_{\mathbb{R}}$ can be leveraged to
integrate Hamilton’s equations. Indeed, the system is integrable in the
classical sense.

The complexified phase space $\mathrm{KT}_{\mathfrak{g}}$ of the Kostant–Toda
Lattice of type $\mathfrak{g}_{\mathbb{R}}$ can be identified (adopting the
notation of the previous section) with an affine subspace of
$\mathfrak{g}^{\ast}\colonequals\mathfrak{g}_{\mathbb{C}}^{\ast}$,

$\mathrm{KT}_{\mathfrak{g}}=\chi+\mathfrak{t}^{*}+\sum_{\alpha\in\Delta}\mathfrak{g}^{*}_{\alpha}\backslash\\{0\\}~{}.$
(30)

Let us denote by $(\alpha_{i})_{i=1}^{\mathrm{rk}~{}\mathfrak{g}}$ the simple
roots of $\mathfrak{g}$, by $e_{\alpha_{i}}$ the corresponding simple root
elements of $\mathfrak{n}_{+}$, and by $h_{i}\in\mathfrak{t}$ the
corresponding Cartan elements. Let $h_{i}^{*}$ and $e_{\alpha_{i}}^{*}$ be the
Killing duals to $h_{i}$ and $e_{\alpha_{i}}$, respectively. An element
$y\in\mathrm{KT}_{\mathfrak{g}}$ is of the form

$y=\chi+\sum_{i=1}^{\mathrm{rk}~{}\mathfrak{g}}b_{i}(y)h_{i}^{*}+\sum_{i=1}^{\mathrm{rk}~{}\mathfrak{g}}\gamma_{i}(y)e_{\alpha_{i}}^{*}~{},$
(31)

so $b_{i}:\mathrm{KT}_{\mathfrak{g}}\rightarrow\mathbb{C}$ and
$\gamma_{i}:\mathrm{KT}_{\mathfrak{g}}\rightarrow\mathbb{C}^{\times}$ furnish
a basis of linear coordinates. The coordinate ring is then simply

$\mathbb{C}[\mathrm{KT}_{\mathfrak{g}}]\cong\mathbb{C}[b_{i},\gamma^{\pm
1}_{i}\,|\,i=1,\dots,\mathrm{rk}~{}\mathfrak{g}]~{}.$ (32)

The Poisson structure descends from the Kirillov–Kostant–Souriau bracket on
$\mathbb{C}[\mathfrak{g}^{*}]\cong\mathrm{Sym}\,\mathfrak{g}$ (see, e.g.,
Kostant:1979195 ), and we have

$\\{b_{i},\gamma_{j}^{\pm}\\}=\delta_{ij}\gamma_{i}^{\pm}~{},$ (33)

with all other brackets vanishing.666Note that the $b_{i}$, $\gamma_{i}$ are
linear duals of $h_{i}^{*}$ and $e_{\alpha_{i}}^{*}$, respectively. That is
$b_{i}(h^{\ast}_{j})=\delta_{ij}$ and
$\gamma_{i}(e^{\ast}_{\alpha_{j}})=\delta_{ij}$, so in particular $b_{i}\neq
h_{i}$ under the canonical identification
$\mathfrak{g}=(\mathfrak{g}^{\ast})^{\ast}$. Thus we identify
$\mathrm{KT}_{\mathfrak{g}}\cong
T^{*}(\mathbb{C}^{\times})^{\mathrm{rk}~{}\mathfrak{g}}$ as symplectic
varieties. The structure of a complex integrable system on
$\mathrm{KT}_{\mathfrak{g}}$ is then manifested through the action variables
$P_{i}|_{\mathrm{KT}_{\mathfrak{g}}}$ for
$i=1,2,\dots,\mathrm{rk}~{}\mathfrak{g}$, which are the restrictions of the
generators of the Harish-Chandra centre, $\mathbb{C}[\mathfrak{g}^{*}]^{G}$ to
$\mathrm{KT}_{\mathfrak{g}}$.

In the construction of Ginzburg:2018 —and its chiralisation in Arakawa:2018egx
—the structure of Higgs branches of class $\SS$ as $S_{f}$-schemes plays an
integral role. Here, the structure morphism of $\mathrm{KT}_{\mathfrak{g}}$ as
an $S_{f}$-scheme is the composition

$\begin{split}\mathrm{KT}_{\mathfrak{g}}&\xrightarrow{P}\mathfrak{g}^{*}/\\!\\!/G\xrightarrow{\sim}S_{f}~{},\\\
y&\mapsto(P_{1}(y),P_{2}(y),\dots
P_{\mathrm{rk}\,\mathfrak{g}}(y))~{},\end{split}$ (34)

where the second morphism is Kostant’s isomorphism
$\mathfrak{g}^{*}/\\!\\!/G\cong S_{f}$. This structure morphism is thus
exactly the map to the base of the aforementioned complex integrable system,
up to the identification of the base with $S_{f}$.

We have deliberately labelled the Kostant–Toda lattice with the Lie algebra
and not the group, whereas the universal centraliser depends on the choice of
group $G$. In Kostant:1979195 ; Crooks:2020 , the embedding from this lattice
targets the universal centraliser of the adjoint group. However, for the
purposes of class $\SS$ constructions we expect to be mainly interested in the
simply connected group $G$ with $\mathrm{Lie}~{}G=\mathfrak{g}$. To generalise
the embedding appropriately, we then need to replace
$\mathrm{KT}_{\mathfrak{g}}$ with a suitable cover.

The root lattice of the Lie group $G$ always contains the root lattice of the
Lie algebra $\mathfrak{g}$ as a sublattice. Let $\Delta_{G}$ denote the simple
roots of the Lie algebra $\mathfrak{g}$ embedded into the root lattice of the
Lie group $G$. Such a simple root, $\alpha\in\Delta_{G}$, is then a character
$\alpha:T\rightarrow\mathbb{C}^{\times}$, defined by its action on the simple
root generator $e_{\alpha}\in\mathfrak{g}$ as

$\mathrm{Ad}_{t}e_{\alpha}=\alpha(t)\,e_{\alpha}~{},\qquad\forall t\in T~{}.$
(35)

Let
$Z_{G}:T\rightarrow\big{(}\mathbb{C}^{\times}\big{)}^{\mathrm{rk}\,\mathfrak{g}}$
be defined by

$\begin{split}t\mapsto(\alpha(t))_{\alpha\in\Delta_{G}}~{},\qquad\forall t\in
T~{},\\\ \end{split}$ (36)

This map is an isomorphism for adjoint $G$, but if $G$ has a nontrivial
centre, $Z(G)$, then it is only surjective. For any $G$, $Z_{G}$ is étale and
the fibre above a point has cardinality $|Z(G)|$.

Now consider the map
$\mathrm{KT}_{\mathfrak{g}}\twoheadrightarrow\big{(}\mathbb{C}^{\times}\big{)}^{\mathrm{rk}\,\mathfrak{g}}$
that projects to the base of the cotangent bundle in
$\mathrm{KT}_{\mathfrak{g}}\cong
T^{*}(\mathbb{C}^{\times})^{\mathrm{rk}~{}\mathfrak{g}}$. We define
$\mathrm{KT}_{G}$ to be the pullback of this projection and the map $T$ given
above,

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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(37)

The structure sheaf of $\mathrm{KT}_{G}$ has the structure of an ${\cal
O}_{(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}}$-module given by

${\cal O}_{\mathrm{KT}_{G}}={\cal
O}_{\mathrm{KT}_{\mathfrak{g}}}\otimes_{{\cal
O}_{(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}}}{\cal O}_{T}~{},$ (38)

with the ${\cal O}_{(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}}$-module
structure on ${\cal O}_{T}$ coming from restriction of scalars along
$Z_{G}^{\\#}:{\cal
O}_{(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}}\rightarrow{\cal
O}_{T}$. Notice that we have the identification

$\begin{split}\mathrm{KT}_{G}&=\\{(y,t)\,|\,y\in\mathrm{KT}_{\mathfrak{g}}~{},~{}t\in
T~{},~{}(\gamma_{1}(y),\gamma_{2}(y),\dots,\gamma_{\mathrm{rk}\,\mathfrak{g}}(y))=Z_{G}(t)\\}~{},\\\
&\cong\\{(b_{1}(y),b_{2}(y),\dots,b_{\mathrm{rk}\,\mathfrak{g}}(y),t)\in(\mathbb{C}\times\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}\,\\}~{},\end{split}$
(39)

and so $\mathrm{KT}_{G}\cong\mathrm{KT}_{\mathfrak{g}}$ as varieties. (In
fact, by a linear redefinition of the coordinate functions $b$, this
isomorphism can be realised as one of symplectic varieties as well.)

However, the canonical projection
$\pi_{G}:\mathrm{KT}_{G}\rightarrow\mathrm{KT}_{\mathfrak{g}}$, coming from
the pullback (37) is not an isomorphism. Indeed, by standard properties of the
pullback $\pi_{G}$ is étale, since $Z_{G}$ is étale. We should therefore treat
$\mathrm{KT}_{G}$ as a covering space of $\mathrm{KT}_{\mathfrak{g}}$ by way
of $\pi_{G}$.

From its definition (37), we also have a canonical projection

$\pi_{T}:\mathrm{KT}_{G}\rightarrow T~{}.$ (40)

Since $\pi_{T}$ is a base change of the surjective morphism
$\mathrm{KT}_{\mathfrak{g}}\twoheadrightarrow(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}$
(and self-evidently from (39)), it too is surjective.

By construction, the fibre of $\pi_{G}$ above a point in
$\mathrm{KT}_{\mathfrak{g}}$ will contain $|Z(G)|$ many points. These fibres
admit a $Z(G)$ action which makes
$\pi_{G}:\mathrm{KT}_{G}\rightarrow\mathrm{KT}_{\mathfrak{g}}$ into a
$Z(G)$-torsor. In particular, let $y\in\mathrm{KT}_{\mathfrak{g}}$. The points
in the fibre $\pi_{G}^{-1}(y)$ are mapped bijectively by $\pi_{T}$ to a $Z(G)$
orbit in $T$. Let $\tilde{y}\in\pi_{G}^{-1}(y)$ and let $z\in Z(G)$. The
$Z(G)$ action is defined uniquely by specifying

$z\cdot\tilde{y}\coloneqq\tilde{y}^{\prime}~{},\quad\text{such
that}\quad\pi_{T}(\tilde{y}^{\prime})=z\cdot\pi_{T}(\tilde{y})~{},\quad\pi_{G}(\tilde{y})=\pi_{G}(\tilde{y}^{\prime})~{}.$
(41)

By construction, this gives the following:

######  Lemma 3.1.

The map $\pi_{T}:\mathrm{KT}_{G}\rightarrow T$ is $Z(G)$-equivariant.

In light of this, the projection
$\pi_{G}:\mathrm{KT}_{G}\rightarrow\mathrm{KT}_{\mathfrak{g}}$ is just the
natural projection to $Z(G)$ orbits,
$\mathrm{KT}_{G}\rightarrow\mathrm{KT}_{G}/\\!\\!/Z(G)$, where
$\mathrm{KT}_{\mathfrak{g}}$ is identified with the quotient space
$\mathrm{KT}_{G}/\\!\\!/Z(G)$. Letting $(y,t)\in\mathrm{KT}_{G}$, with
$y\in\mathrm{KT}_{\mathfrak{g}}$ and $t\in T$

$\pi_{G}(y,t)=\chi+\sum^{\mathrm{rk}\,\mathfrak{g}}_{i}b_{i}(y)h^{*}_{i}+\sum_{\alpha\in\Delta}\alpha(t)e_{\alpha}^{*}~{}.$
(42)

Since the map $\pi_{G}$ is étale, we can pullback the symplectic form on
$\mathrm{KT}_{\mathfrak{g}}$ to a symplectic form on $\mathrm{KT}_{G}$,
endowing its coordinate ring in particular with a Poisson bracket.
Furthermore, $\mathrm{KT}_{G}$ can be made into an $S_{f}$-scheme with
structure morphism

$\mathrm{KT}_{G}\xrightarrow{\pi_{G}}\mathrm{KT}_{\mathfrak{g}}\xrightarrow{P}\mathfrak{g}^{*}/\\!\\!/G\xrightarrow{\sim}S_{f}~{}.$
(43)

We are primarily interested in the case when $G$ is simply-connected; in this
case the functions on $\mathrm{KT}_{G}$ have a more concrete presentation. Let
$\big{(}\alpha_{i}\big{)}_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ be the simple
roots of $G$ and $a_{ij}$ be the matrix elements of the Cartan matrix of
$\mathfrak{g}$. Let $\widetilde{a}_{ij}$ be the matrix elements of the inverse
Cartan matrix. Then ${\cal
O}_{\mathrm{KT}_{\mathfrak{g}}}(\mathrm{KT}_{\mathfrak{g}})\cong\mathbb{C}[b_{\alpha_{i}},\gamma_{\alpha_{i}}\,|\,i=1,\dots,\mathrm{rk}\,\mathfrak{g}]$
and

${\cal
O}_{\mathrm{KT}_{G}}(\mathrm{KT}_{G})\cong\mathbb{C}[b_{\alpha_{i}},\widetilde{\gamma}_{\alpha_{i}}|i=1,\dots,\mathrm{rk}\,\mathfrak{g}]~{},$
(44)

where

$\widetilde{\gamma}_{\alpha_{i}}=\prod_{j=1}^{\mathrm{rk}\,\mathfrak{g}}\gamma^{\widetilde{a}_{ij}}_{\alpha_{j}}~{}.$
(45)

The map $\pi_{G}$, at the level of rings is,

$b_{\alpha_{i}}\mapsto
b_{\alpha_{i}}~{},\quad\gamma_{\alpha_{i}}\mapsto\prod_{i=1}^{\mathrm{rk}\,\mathfrak{g}}\widetilde{\gamma}_{j}^{a_{ij}}~{},\qquad
i=1,\dots,\mathrm{rk}~{}\mathfrak{g}~{}.$ (46)

The Poisson structure inherited by $\mathrm{KT}_{G}$ is just the natural
extension of the Poisson bracket on $\mathrm{KT}_{\mathfrak{g}}$ to the
$\widetilde{\gamma}_{\alpha_{i}}$ by imposing the chain rule.

When $G$ is simply connected, we can also give a more concrete presentation of
$\pi_{T}$. Let $(t_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ be the generators
of the root lattice of $G$. Since the root lattice of $G$ is isomorphic to the
weight lattice of the Lie algebra, $\mathfrak{g}$, these generators can be
identified with the fundamental weights,
$(\omega_{i})^{\mathrm{rk}\,\mathfrak{g}}_{i=1}$, of $\mathfrak{g}$. Let
$V_{i}$ be the finite dimensional representation of $\mathfrak{g}$ with
highest weight $\omega_{i}$, and let $v_{i}$ be the highest weight vector.
Then, if $h\in T$

$hv_{i}=t_{i}(h)v_{i}~{}.$ (47)

The $t_{i}$ are good coordinates on the torus and the map $\pi_{T}$ is induced
by the ring homomorphism,

$\begin{split}\pi_{T}^{\\#}:\mathbb{C}[t_{1}^{\pm 1},\dots,t^{\pm
1}_{\mathrm{rk}\,\mathfrak{g}}]&\rightarrow\mathbb{C}[b_{\alpha_{i}},\widetilde{\gamma}^{\pm
1}_{\alpha_{j}}\,|\,i,j=1,\dots\mathrm{rk}\,\mathfrak{g}]\\\
t_{i}&\mapsto\widetilde{\gamma}_{\alpha_{i}}=\prod_{j=1}^{\mathrm{rk}~{}\mathfrak{g}}\gamma_{\alpha_{j}}^{\widetilde{a}_{ij}}\end{split}$
(48)

Suppose $G$ is neither simply-connected nor of adjoint type. Then $G$ is a
quotient of the simply-connected group, $G_{sc}$, by some central subgroup
$Z^{\prime}\subset Z(G_{sc})$ with $Z^{\prime}\neq Z(G_{sc})$. Coordinates on
$G$ are then realised by passing to $Z^{\prime}$ invariants of the coordinate
ring of the simply-connected group.

###  3.2 The embedding in rank one

To make this abstract construction a bit more concrete, we consider in detail
the two rank-one examples: $G={\rm SL}_{2}$ and $G={\rm PGL}_{2}$. We will
then see directly how these Kostant–Toda lattices embed into the corresponding
universal centralisers.

The $\mathfrak{sl}_{2}$ Kostant–Toda lattice, thought of as a subset of
$\mathfrak{sl}_{2}$ using the Killing form, consists of all elements of the
form

$y=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}+\begin{pmatrix}b(y)&0\\\
0&-b(y)\end{pmatrix}+\begin{pmatrix}0&\gamma(y)\\\ 0&0\end{pmatrix}~{},$ (49)

with $b(y)\in\mathbb{C}$ and $\gamma(y)\in\mathbb{C}^{\times}$. The coordinate
ring is $\mathbb{C}[b,\gamma^{\pm}]$, with non-vanishing Poisson brackets

$\\{b,\gamma^{\pm}\\}=\pm\gamma^{\pm}~{}.$ (50)

For $G={\rm PGL}_{2}$ the maximal torus is isomorphic to $\mathbb{C}^{\times}$
and the simple root,
$\alpha:\mathbb{C}^{\times}\rightarrow\mathbb{C}^{\times}$, is just the
identity morphism. Following the construction above, this means that the
structure sheaf ${\cal O}_{\mathrm{KT}_{{\rm PGL}_{2}}}$ is isomorphic to
${\cal O}_{\mathrm{KT}_{\mathfrak{sl}_{2}}}$ as ${\cal
O}_{\mathbb{C}^{\times}}$-modules. The global sections are just
$\mathbb{C}[b,\gamma^{\pm 1}]={\cal
O}_{\mathrm{KT}_{\mathfrak{sl}_{2}}}(\mathrm{KT}_{\mathfrak{sl}_{2}})$ and the
map $\pi_{PGL_{2}}:\mathrm{KT}_{\rm
PGL_{2}}\rightarrow\mathrm{KT}_{\mathfrak{sl}_{2}}$ is an isomorphism. Here we
see directly that with $\rm PGL_{2}$ being of adjoint type, the covering space
$\mathrm{KT}_{\rm PGL_{2}}$ is exactly identified with
$\mathrm{KT}_{\mathfrak{sl}_{2}}$. The structure morphism to $S_{f}$ is given
by $b^{2}+\gamma$.

Alternatively, for $G={\rm SL}_{2}$ the simple root $\alpha$ is given by
$\alpha:\mathbb{C}^{\times}\xrightarrow{z^{2}}\mathbb{C}^{\times}$. The
structure sheaf of $\mathrm{KT}_{G}$ is ${\cal O}_{\mathrm{KT}_{G}}={\cal
O}_{T}\otimes_{{\cal O}_{\mathbb{C}^{\times}}}{\cal
O}_{\mathrm{KT}_{\mathfrak{g}}}$, with global sections

${\cal O}_{\mathrm{KT}_{G}}(\mathrm{KT}_{G})\cong\frac{\mathbb{C}[h^{\pm
1}]\otimes\mathbb{C}[b,\gamma^{\pm 1}]}{\langle
h^{2}-\gamma\rangle}\cong\mathbb{C}[b,\gamma^{\pm\frac{1}{2}}].$ (51)

which has the obvious module structure over $\mathbb{C}[b,\gamma^{\pm 1}]$.
The map $\pi_{{\rm SL}_{2}}$ on rings is just the natural inclusion $\pi_{{\rm
SL}_{2}}^{\\#}:\mathbb{C}[b,\gamma^{\pm
1}]\hookrightarrow\mathbb{C}[b,\gamma^{\pm\frac{1}{2}}]$. The Poisson
structure, inherited from $\mathrm{KT}_{\mathfrak{g}}$ is

$\\{b,\gamma^{\pm\frac{1}{2}}\\}=\pm\frac{1}{2}\gamma^{\pm\frac{1}{2}}~{}.$
(52)

Clearly, with a rescaling of $b$ we have $\mathrm{KT}_{{\rm SL}_{2}}\cong
T^{*}\mathbb{C}^{\times}\cong\mathrm{KT}_{{\rm PGL}_{2}}$ including Poisson
structures. However, unlike the adjoint case, here our construction gives
$\mathrm{KT}_{{\rm SL}_{2}}$ as a $2$-cover of
$\mathrm{KT}_{\mathfrak{sl}_{2}}$ along $\pi_{{\rm SL}_{2}}$. The structure
morphism for $\mathrm{KT}_{{\rm SL}_{2}}$ is
$b^{2}+\gamma^{\frac{1}{2}}\gamma^{\frac{1}{2}}$.

Now we turn to the universal centralisers for these two groups. From Section
2.2 we know that $\mathfrak{Z}_{{\rm SL}_{2}}\subset\mathbb{A}^{3}$ can be
realised as an affine variety with coordinate ring

$I_{{\rm SL}_{2}}=\frac{\mathbb{C}[X,Y,S]}{\langle SX^{2}-Y^{2}-1\rangle}~{}.$
(53)

Similarly, the adjoint version, $\mathfrak{Z}_{\rm PGL_{2}}$, has coordinate
ring

$I_{\rm PGL_{2}}\cong\frac{\mathbb{C}[S,X^{2},XY,Y^{2}]}{\langle
SX^{2}-Y^{2}-1\rangle}\cong\frac{\mathbb{C}[S,A,B,C]}{\langle
B^{2}-AC,SA-C-1\rangle}~{}.$ (54)

Let $U_{X}\subset\mathfrak{Z}_{{\rm SL}_{2}}$ be the elementary open set
defined by localising on the ideal $\langle X\rangle$. In this open subset,
the relation $SX^{2}-Y^{2}-1$ can be solved for $S$ according to

$S=\frac{1}{X^{2}}+\frac{Y^{2}}{X^{2}}~{}.$ (55)

So ${\cal O}_{\mathfrak{Z}_{{\rm SL}_{2}}}(U_{X})\cong\mathbb{C}[Y,X^{\pm
1}]$.

Constructing an open immersion $\mathrm{KT}_{{\rm
S}_{2}}\rightarrow\mathfrak{Z}_{{\rm SL}_{2}}$ is straightforward in this
case. The open subset $U_{X}$ can be identified with $\mathrm{KT}_{G}$ as
Poisson schemes over $S_{f}$ via the map

$X\mapsto\gamma^{-\frac{1}{2}}~{},\quad Y\mapsto-b\gamma^{-\frac{1}{2}}~{}.$
(56)

The Slodowy slice coordinate function is then realised as follows,

$S\mapsto b^{2}+\gamma^{\frac{1}{2}}\gamma^{\frac{1}{2}}=b^{2}+\gamma~{}.$
(57)

This, of course, agrees with the structure morphism of $\mathrm{KT}_{{\rm
SL}_{2}}$, so this morphism is an open immersion of (Poisson) $S_{f}$-schemes.

Now let $U_{A}$ be the elementary open set obtained by localising $I_{\rm
PGL_{2}}$ at the ideal $\langle A\rangle$. In this open set, the relations
(54) can be solved according to

$C=\frac{B^{2}}{A}~{},\quad S=\frac{1}{A}+\frac{B^{2}}{A^{2}}~{}.$ (58)

Therefore, $U_{A}\cong\mathrm{Spec}~{}\mathbb{C}[B,A^{\pm 1}]\cong
T^{*}\mathbb{C}^{\times}$ as schemes. We are free to pick generators
$\frac{B}{A}$ and $A^{\pm 1}$ for the functions on $U_{A}$. The Poisson
bracket is then determined by

$\left\\{\frac{B}{A},A^{\pm}\right\\}=\pm A^{\pm}~{},$ (59)

so we can identify $U_{A}$ with $T^{*}\mathbb{C}^{\times}$ as Poisson schemes.
Now to embed $\mathrm{KT}_{\rm PGL_{2}}$, we make the following
identifications

$A\mapsto\gamma^{-1}~{},\quad\frac{B}{A}\mapsto b~{}.$ (60)

This extends to

$C\mapsto b^{2}\gamma^{-1}~{},\quad S\mapsto b^{2}+\gamma$ (61)

These identifications induce an open immersion of $S_{f}$ schemes
$\mathrm{KT}_{\rm PGL_{2}}\rightarrow\mathfrak{Z}_{\rm PGL_{2}}$.

Since the open subsets $U_{X}$ and $U_{A}$ are both isomorphic to
$T^{*}\mathbb{C}^{\times}$ as Poisson varieties, one might at first inquire as
to whether there is any reason to introduce a covering space after all.
Indeed, one may construct the open immersion $\mathrm{KT}_{\rm
PGL_{2}}\rightarrow\mathfrak{Z}_{{\rm SL}_{2}}$, with image $U_{X}$, by making
the identificationes

$X\mapsto\gamma^{-1}~{},\quad Y\mapsto-\frac{1}{2}b\gamma^{-1}~{},\quad
S\mapsto\frac{1}{4}b^{2}+\gamma^{2}~{}.$ (62)

This map is an open immersion of Poisson schemes, however it fails to be a
morphism of $S_{f}$-schemes. Namely, the image of $S$ is not $b^{2}+\gamma$.
Insisting that the open immersion must be a morphism of $S_{f}$-schemes is
natural—it preserves some Lie theoretic data, namely the $\gamma$ can be
thought of as linear function on $\mathfrak{g}^{*}$. Furthermore, this ensures
that the inclusion $I_{{\rm
SL}_{2}}\hookrightarrow\mathbb{C}[\mathrm{KT}_{{\rm SL}_{2}}]$ is a map of
$\mathbb{C}[\mathfrak{g}^{*}]^{G}$-modules; this structure appears to be of
general importance in the construction of Higgs branches for theories of class
$\SS$.

###  3.3 The embedding in the general case and some technicalities

For higher rank cases, the relationship between $\mathrm{KT}_{G}$ and
$\mathfrak{Z}_{G}$ is more intricate. In this section we provide an abstract
description of an open immersion
$\mathrm{KT}_{G}\xrightarrow{\varphi}\mathfrak{Z}_{G}$. For $G$ of adjoint
type, this map was described in less concrete terms in Kostant:1979195 , and
our analysis follows and extends that of Crooks:2020 .

We decompose $\varphi$ into morphisms $\beta:\mathrm{KT}_{G}\rightarrow S_{f}$
to the base and $\lambda:\mathrm{KT}_{G}\rightarrow G$ to the fibre, i.e., for
$y\in\mathrm{KT}_{G}$, $\lambda(y)\in G_{\beta(y)}$. Let $\mathfrak{b}^{*}$ be
the image of the Borel subalgebra $\mathfrak{b}=\mathfrak{t}+\mathfrak{n}_{+}$
under the Killing isomorphism. Kostant Kostant:1978  tells us that there is an
isomorphism of affine varieties $\psi:N\times
S_{f}\xrightarrow{\sim}\chi+\mathfrak{b}^{*}$ realised as

$(n,s)\mapsto\mathrm{Ad}^{*}_{n}s~{}.$ (63)

Additionally, we have canonical projections $\pi_{S_{f}}:N\times
S_{f}\rightarrow S_{f}$ and $\pi_{N}:N\times S_{f}\rightarrow N$. The
Kostant–Toda lattice is a subset of $\chi+\mathfrak{b}^{*}$, so we can define
a map

$\begin{split}\beta:\mathrm{KT}_{\mathfrak{g}}&\rightarrow S_{f}~{},\\\
y&\mapsto\pi_{S_{f}}(\psi^{-1}(y))~{}.\end{split}$ (64)

Recall the generators $P_{i}$ of $\mathbb{C}[\mathfrak{g}^{*}]^{G}$ whose
restriction to $\mathrm{KT}_{\mathfrak{g}}$ define the map
$P:\mathrm{KT}_{\mathfrak{g}}\rightarrow\mathfrak{g}^{*}/\\!\\!/G$. By
(Kostant:1979195, , Theorem 2.6) this is surjective. The following gives an
important compatibility result between these various maps.

######  Proposition 3.2.

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},$ (65)

where the diagonal right arrow is the Kostant isomorphism. This diagram
commutes, so in particular $\beta$ is surjective.

The structure morphism for $\mathrm{KT}_{\mathfrak{g}}$ as an $S_{f}$-scheme
as introduced previously is the composition of the lower route around this
diagram, so can be identified with the map $\beta$.

The morphism $\beta:\mathrm{KT}_{\mathfrak{g}}\rightarrow S_{f}$ can be lifted
to a morphism $\mathrm{KT}_{G}\rightarrow S_{f}$ via the composition
$\mathrm{KT}_{G}\xrightarrow{\pi_{G}}\mathrm{KT}_{\mathfrak{g}}\xrightarrow{\beta}S_{f}$,
which is also surjective. In an abuse of notation, we denote this composition
as $\beta:\mathrm{KT}_{G}\rightarrow S_{f}$. This $\beta$ again agrees with
the structure morphism of $\mathrm{KT}_{G}$ as an $S_{f}$-scheme.

For the time being, let us assume $G$ is of adjoint type (subsequently we will
describe the generalisation of this story to non-adjoint $G$, and the more
general $\mathrm{KT}_{G}$ will make an appearance). Then the map $\beta$ gives
us a foliation of $\mathrm{KT}_{\mathfrak{g}}$ over the Slodowy slice, and the
fibre above a point $s\in S_{f}$ can be shown to be isomorphic to a Zariski
open subset of the $G$-stabiliser of $s$ according to (Kostant:1979195, ,
Theorem 2.6). These isomorphisms of fibres, taken collectively over $S_{f}$,
leads to an isomorphism of $S_{f}$-schemes between
$\mathrm{KT}_{\mathfrak{g}}$ and a Zariski open subset of $\mathfrak{Z}_{G}$
that can be made completely explicit. We now review and elaborate upon this
construction following Crooks:2020  with an eye towards the generalisation to
general $G$.

As we said before, the phase space $\mathrm{KT}_{\mathfrak{g}}$ has a
foliation

$\mathrm{KT}_{\mathfrak{g}}=\bigsqcup_{p\in\mathfrak{g}^{*}/\\!\\!/G}P^{-1}(p)~{}.$
(66)

Each leaf of this foliation can be related to the centraliser of a particular
element in $\chi+\mathfrak{t}^{*}$ as follows. Consider the diagram,

$\leavevmode\hbox to143.57pt{\vbox
to17.42pt{\pgfpicture\makeatletter\hbox{\hskip 71.78738pt\lower-8.75955pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-71.78738pt}{-8.65971pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
13.20555pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathrm{KT}_{\mathfrak{g}}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
13.20555pt\hfil&\hfil\hskip 42.8034pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-14.4979pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathfrak{g}^{*}/\\!\\!/G}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
18.80344pt\hfil&\hfil\hskip 39.7784pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\chi+\mathfrak{t}^{*}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-45.17628pt}{-6.15971pt}\pgfsys@lineto{-21.97627pt}{-6.15971pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.77629pt}{-6.15971pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.10945pt}{-3.80695pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{P}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{40.03053pt}{-6.15971pt}\pgfsys@lineto{16.83052pt}{-6.15971pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{18.07045pt}{-6.15971pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{16.63054pt}{-6.15971pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.50838pt}{-2.57678pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\sigma_{t}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},$ (67)

where $\sigma_{t}$ is the composition
$\chi+\mathfrak{t}^{*}\twoheadrightarrow\mathfrak{t}^{*}/\\!\\!/W\xrightarrow{\sim}\mathfrak{g}^{*}/\\!\\!/G$.
Let $y\in\mathrm{KT}_{\mathfrak{g}}$ and $p=P(y)$, then the fibre
$\sigma_{t}^{-1}(p)$ is nonempty. Let $\theta_{p}$ be the point in this fibre
such that $\theta_{p}-\chi$ lies in the fundamental Weyl chamber.777Note that
$y\mapsto\theta_{p}$ does not define an algebraic morphism between
$\mathrm{KT}_{\mathfrak{g}}$ and $\chi+\mathfrak{t}^{*}$, since the
fundamental domain is not algebraic. Nevertheless, the construction will
telescope and $\theta_{p}$ itself will not appear in the final definition of
the open immersion. Then $\theta_{p}$ and $y$ are $N$-conjugate, since
$P(y)=P(\theta_{p})$ and both elements are in $\chi+\mathfrak{b}^{*}$.
Furthermore, as $\beta(y)$ and $y$ are also $N$-conjugate, $\theta_{p}$ must
be $N$-conjugate to $\beta(y)$. Let $\delta_{p}\in N$ be the unique group
element such that $\mathrm{Ad}_{\delta(p)}^{*}\theta_{p}=\beta(y)$. This
element is unique since $\theta_{p}\in\chi+\mathfrak{b}^{*}\cong N\times
S_{f}$, and can be written as follows,

$\delta_{p}=\pi_{N}(\psi^{-1}(\theta_{p}))^{-1}~{}.$ (68)

Now let $w_{0}$ be the longest element of the Weyl group, i.e., the _Cartan
element_ of $W(G)$. There is a unique uplift of $w_{0}$ to a
$\widetilde{w}_{0}\in N_{G}(T)$, the normaliser of $T$, such that
$Ad_{\widetilde{w}_{0}}e_{\alpha}=e_{w_{0}\alpha}$ for any simple root
$\alpha$ and any associated generator $e_{\alpha}\in\mathfrak{n}_{+}$.
Consider the Zariski open subset $G^{*}\coloneqq N\widetilde{w}_{0}TN$, which
is just the big Bruhat cell in $G$. For $x\in\mathfrak{g}^{\ast}_{\rm reg}$,
let $G^{*}_{x}$ be the Zariski open subset of $G_{x}$ defined by the
intersection $G^{*}_{x}\coloneqq G^{*}\cap G_{x}$.

######  Theorem 3.3.

(Kostant:1979195, , Theorem 2.6) Let $p\in\mathfrak{g}^{*}/\\!\\!/G$ and let
$\theta_{p}$ be as above. The morphism

$\begin{split}K_{p}:G^{*}_{\theta_{p}}\cong P^{-1}(p)\\\
n^{\prime}\widetilde{w}_{0}hn\mapsto\mathrm{Ad}^{*}_{n}\theta_{p}~{},\end{split}$

where $n^{\prime},n\in N$ and $h\in T$, is an isomorphism of algebraic
varieties.

This theorem relates the fibres of $\mathrm{KT}_{\mathfrak{g}}$ over
$\mathfrak{g}^{*}/\\!\\!/G$ to a Zariski open subset of a stabiliser subgroup
of $G$; we next want to relate this stabiliser subgroup to a stabiliser
subgroup of the corresponding point $\beta(y)$ in the Slodowy slice.

For a given $p\in\mathfrak{g}^{*}/\\!\\!/G$, Kostant’s isomorphism gives a
unique $s_{p}\in S_{f}$, and once again the previously defined $\delta_{p}$ is
precisely such that $\mathrm{Ad}^{*}_{\delta_{p}}\theta_{p}=s_{p}$. Thus the
stabilisers of $s_{p}$ and $p$ are related by

$G^{*}_{s_{p}}=\delta_{p}G^{*}_{\theta_{p}}\delta_{p}^{-1}~{}.$ (69)

We can therefore define a Zariski open set,
$\mathfrak{Z}_{G}^{*}\coloneqq(G^{*}\times S_{f})\cap\mathfrak{Z}_{G}$, which
has foliation

$\mathfrak{Z}_{G}^{*}=\bigsqcup_{s\in S_{f}}G^{*}_{s}~{},$ (70)

with each fibre isomorphic to $G^{*}_{\theta_{s}}$ under conjugation by
$\delta_{s}$. By Theorem 3.3 we can therefore construct an open immersion

$\varphi:\mathrm{KT}_{\mathfrak{g}}\rightarrow\mathfrak{Z}_{G}~{},$ (71)

by sending each leaf of the foliation
$\mathrm{KT}_{\mathfrak{g}}\cong\bigsqcup_{p\in\mathfrak{g}^{*}/\\!\\!/G}P^{-1}(p)$
to the stabiliser $G^{*}_{\theta_{p}}$ and then conjugating by $\delta_{p}$.
Each step is an isomorphism, so $\varphi$ is an open immersion with image
$\mathfrak{Z}_{G}^{*}$.

Defining the morphism $\varphi$ requires an explicit realisation of the
inverse $K_{p}^{-1}:P^{-1}(p)\rightarrow G_{\theta_{p}}^{*}$. To construct the
inverse to $K_{p}$, we define a few more morphisms. Via the Kostant
isomorphism, we have a map $\rho_{N}:\mathrm{KT}_{\mathfrak{g}}\rightarrow N$
given by the composition

$\mathrm{KT}_{\mathfrak{g}}\xrightarrow{\psi^{-1}}N\times
S_{f}\xrightarrow{\pi_{N}}N~{}.$ (72)

Let $\widetilde{\rho}_{N}$ be the further composition

$\mathrm{KT}_{\mathfrak{g}}\xrightarrow{\mathrm{Ad}^{*}_{\widetilde{w}_{0}\pi_{T}(-)^{-1}}}\mathrm{KT}_{\mathfrak{g}}\xrightarrow{\psi^{-1}}N\times
S_{f}\xrightarrow{\pi_{N}}N~{},$ (73)

In other words,

$\begin{split}\widetilde{\rho}_{N}:\mathrm{KT}_{G}&\rightarrow N\\\
y&\mapsto\pi_{N}(\psi^{-1}(\mathrm{Ad}_{\widetilde{w}_{0}\pi_{T}(y)^{-1}}^{*}y))~{}.\end{split}$
(74)

Here, $\pi_{T}:\mathrm{KT}_{\mathfrak{g}}\rightarrow T$ arises from the
identification
$\pi_{G}:\mathrm{KT}_{G}\xrightarrow{\sim}\mathrm{KT}_{\mathfrak{g}}$, for
adjoint $G$. The inverse $K_{p}^{-1}:P^{-1}(p)\rightarrow G_{\theta_{p}}^{*}$
is given by

$\begin{split}K_{p}^{-1}(y)&=\bigg{(}(\widetilde{\rho}_{N}(y)\delta_{p})^{-1}\widetilde{w}_{0}(\pi_{T}(y))^{-1}\rho_{N}(y)\delta_{p}\bigg{)}~{},\\\
&=\delta_{p}^{-1}\widetilde{\rho}_{N}(y)^{-1}\widetilde{w}_{0}(\pi_{T}(y))^{-1}\rho_{N}\delta_{p}~{},\end{split}$
(75)

for any $y\in P^{-1}(p)$. As this expression is a bit opaque, let us try and
explain informally why it stabilises $\theta_{p}$. A more detailed analysis of
its origin can be found in the proof of (Crooks:2020, , Lemma 19).

Given $y\in P^{-1}(p)$ we want an element of $G^{*}$ that stabilises
$\theta_{p}$. Every such element has the form $n^{\prime}\widetilde{w}_{0}hn$
for some $n,n^{\prime}\in N$ and $h\in T$. In terms of the ingredients
appearing in $K_{p}^{-1}$, we have

$\begin{split}n^{\prime}&=\big{(}\widetilde{\rho}_{N}(y)\delta_{p}\big{)}^{-1}~{},\\\
h&=\big{(}\pi_{T}(y)\big{)}^{-1}~{},\\\
n&=\rho_{N}(y)\delta_{p}~{}.\end{split}$ (76)

The composite expression for $K_{p}^{-1}(y)$ acts on $\theta_{p}$ in the
following way. First $\delta_{p}$ takes $\theta_{p}$ to $s_{p}$, then
$\rho_{N}(y)$ takes $s_{p}$ to $y$—this follows from the definitions of
$\rho_{N}(y)$ and $\delta_{p}$. Note that this is precisely the image of the
map $K_{p}$.

Next the element $y$ gets mapped to another $y^{\prime}\in P^{-1}(p)$ by
$\widetilde{w}_{0}h$, where the form of $h$ is dictated by the requirement
that this indeed lands in $\mathrm{KT}_{\mathfrak{g}}$. Finally,
$\widetilde{\rho}_{N}(y)^{-1}$ is the unique element of $N$ that takes this
$y^{\prime}$ back to $s_{p}$, and $\delta_{p}^{-1}$ will conjugate $s_{p}$
back to $\theta_{p}$—stabilising $\theta_{p}$ as a result.

Now, for any $y\in\mathrm{KT}_{\mathfrak{g}}$, the morphism $\varphi$ is given
by

$\begin{split}\varphi(y)&=\big{(}\delta_{p}K_{p}^{-1}(y)\delta_{p}^{-1},\beta(y)\big{)}~{},\\\
&=\big{(}\widetilde{\rho}_{N}(y)\widetilde{w}_{0}(\pi_{T}(y))^{-1}\rho_{N}(y),\beta(y)\big{)}~{},\end{split}$
(77)

where $p=P(y)$. Note that the definition has telescoped, so that the map,
$\lambda:\mathrm{KT}_{\mathfrak{g}}\rightarrow G^{*}$, which identifies leaves
of $\mathrm{KT}_{\mathfrak{g}}$ and $\mathfrak{Z}_{G}$, does not depend
explicitly on $p$, $\theta_{p}$ or $\delta_{p}$. Indeed, the map (77) is
manifestly algebraic.

Now let us return to the case where $G$ is not necessarily of adjoint type.
Then $p\in\mathfrak{g}^{*}/\\!\\!/G$ and $\theta_{p}$ can be defined as
before, while the longest element of the Weyl group, $w_{0}\in W$, no longer
has a unique uplift to $N_{G}(T)$—there is now a $Z(G)$-orbit worth of
uplifts. We fix $\widetilde{w}_{0}\in N_{G}(T)$ to be the unique uplift of
$w_{0}$ that lies in the connected component of the identity. The Zariski open
$G^{*}\subset G$ is defined, as before, as the big Bruhat cell
$N\widetilde{w}_{0}TN$ for this choice of $\widetilde{w}_{0}$, and if
$x\in\mathfrak{g}^{\ast}_{\rm reg}$, $G^{*}_{x}\coloneqq G^{*}\cap G_{x}$.

The map $K_{p}:G^{*}_{\theta_{p}}\rightarrow P^{-1}(p)$ is no longer an
isomorphism; the fibres of the map have cardinality $|Z(G)|$. Therefore, it
cannot be inverted and the previous definition of $\lambda$ is not fit for
purpose. To rectify the wituation we will pass to the cover $\mathrm{KT}_{G}$.
We define the morphism
$\overline{P}:\mathrm{KT}_{G}\rightarrow\mathfrak{g}^{*}/\\!\\!/G$ as the
composite

$\mathrm{KT}_{G}\xrightarrow{\pi_{G}}\mathrm{KT}_{\mathfrak{g}}\xrightarrow{P}\mathfrak{g}^{*}/\\!\\!/G~{},$
(78)

and for $p\in\mathfrak{g}^{*}/\\!\\!/G$, $\overline{P}^{-1}(p)$ is a cover of
$P^{-1}(p)$ and we will want to define a lift
$\overline{K}_{p}:G^{*}_{\theta_{p}}\rightarrow\overline{P}^{-1}(p)$ that is
an isomorphism. For the purposes of constructing $\varphi$, we are actually
interested in the putative inverse
$\overline{K}_{p}^{-1}:\overline{P}^{-1}(p)\rightarrow G^{*}_{\theta_{p}}$, so
in what follows we will construct $\overline{K}_{p}^{-1}$ and show that it is
an isomorphism.

The fibres $\overline{P}^{-1}(p)$ have $|Z(G)|$ many disconnected components,
all of which are isomorphic to $P^{-1}(p)$ as varieties. The strategy will be
to identify each component of $\overline{P}^{-1}(p)$ with a component of
$G^{*}_{\theta_{p}}$ in a $Z(G)$-equivariant way.

To define our inverse morphism, we need to lift $\rho_{N}$ and
$\widetilde{\rho}_{N}$ to the cover, which can be done by precomposition. For
convenience, we continue to use $\rho_{N}$ to denote the composition
$\mathrm{KT}_{G}\xrightarrow{\pi_{G}}\mathrm{KT}_{\mathfrak{g}}\xrightarrow{\rho_{N}}N$,
and $\widetilde{\rho}_{N}$ for the composition

$\mathrm{KT}_{G}\xrightarrow{\pi_{G}}\mathrm{KT}_{\mathfrak{g}}\xrightarrow{\mathrm{Ad}^{*}_{\widetilde{w}_{0}\pi_{T}(-)^{-1}}}\mathrm{KT}_{\mathfrak{g}}\xrightarrow{\psi^{-1}}N\times
S_{f}\xrightarrow{\pi_{N}}N~{}.$ (79)

In other words,

$\begin{split}\widetilde{\rho}_{N}:\mathrm{KT}_{G}&\rightarrow N\\\
y&\mapsto\pi_{N}(\psi^{-1}(\mathrm{Ad}_{\widetilde{w}_{0}\pi_{T}(y)^{-1}}^{*}\pi_{G}(y)))~{}.\end{split}$
(80)

Note that now the morphisms $\rho_{N}$ and $\widetilde{\rho}_{N}$ are both
invariant on the $Z(G)$ orbits of $\mathrm{KT}_{G}$. For $\rho_{N}$ this is
clear from the definition. For $\widetilde{\rho_{N}}$, note that the twisting
by $\mathrm{Ad}^{*}_{\widetilde{w}_{0}(\pi_{T}(y))^{-1}}$ that enters the
definition is insensitive to the centre, since the coadjoint action is blind
to it.

We now define $\overline{K}_{p}^{-1}:\overline{P}^{-1}(p)\rightarrow
G_{\theta_{p}}^{*}$, by the same expression as before,

$\begin{split}\overline{K}_{p}^{-1}(y)&=\bigg{(}(\widetilde{\rho}_{N}(y)\delta_{p})^{-1}\widetilde{w}_{0}(\pi_{T}(y))^{-1}\rho_{N}(y)\delta_{p}\bigg{)}~{},\\\
&=\delta_{p}^{-1}\widetilde{\rho}_{N}(y)^{-1}\widetilde{w}_{0}(\pi_{T}(y))^{-1}\rho_{N}\delta_{p}~{},\end{split}$
(81)

where now the definition applies for any $y\in\overline{P}^{-1}(p)$. For
adjoint $G$, of course, this recovers the inverse $\overline{K}^{-1}_{p}$.
Notice that the map $\pi_{T}$ _is sensitive_ to which connected component of
$\mathrm{KT}_{G}$ $y$ lies in, and moreover, it is the only part of the map
which is sensitive to this. Recalling the equivariance properties of Lemma 3.1
we can anticipate that this map will target every connected component of
$G^{*}_{\theta_{p}}$. We have the following.

######  Theorem 3.4.

Let $p\in\mathfrak{g}^{*}/\\!\\!/G$, then
$\overline{K}_{p}^{-1}:\overline{P}^{-1}(p)\rightarrow G^{*}_{\theta_{p}}$ is
an isomorphism.

###### Proof.

The following is based on the proof of (Crooks:2020, , Lemma 19), modified to
account for the subtleties of $G$ not being of adjoint type.

First, we establish surjectvity. Let $n^{\prime}\widetilde{w}_{0}hn\in
G_{\theta_{p}}^{*}$ with $n^{\prime},n\in N$ and $h\in T$. Then
$y=Ad^{*}_{n}\theta_{p}$ is some element in $P^{-1}(p)$ by Theorem 3.3. Since
$n^{\prime}\widetilde{w}_{0}hn$ stabilises $\theta_{p}$, we have

$Ad^{*}_{n^{\prime}\widetilde{w}_{0}hn}\theta_{p}=Ad^{*}_{n^{\prime}\widetilde{w}_{0}h}y\overset{!}{=}\theta_{p}~{}.$
(82)

Concretely $\theta_{p}=\chi+\tau^{*}$ for some $\tau^{*}$ in the fundamental
Weyl chamber and
$y=\chi+\tau^{\prime}+\sum_{\alpha\in\Delta}\gamma_{\alpha}e_{\alpha}^{*}$.
Therefore

$\begin{split}\mathrm{Ad}^{*}_{n^{\prime}\widetilde{w}_{0}h}y&=\mathrm{Ad}^{*}_{n^{\prime}\widetilde{w}_{0}h}\big{(}\chi+\tau^{\prime}+\sum_{\alpha\in\Delta}\gamma_{\alpha}(y)e^{*}_{\alpha}\big{)}~{},\\\
&=c_{+}+\sum_{\alpha\in\Delta}\alpha(h)\gamma_{\alpha}(y)e_{w_{0}\alpha}^{*}~{},\end{split}$
(83)

with $c_{+}\in\mathfrak{b}^{*}$. Since this must be equal to
$\theta_{p}=\chi+\tau$, we must have that

$\chi=\sum_{\alpha\in\Delta}\alpha(h)\gamma_{\alpha}(y)e^{*}_{w_{0}\alpha}~{},$
(84)

which implies

$\alpha(h)=\gamma_{\alpha}^{-1}(y)~{}.$ (85)

This means that $h$ can be written as $\big{(}\pi_{T}(\tilde{y})\big{)}^{-1}$
for some $\tilde{y}\in\pi^{-1}_{G}(y)$. Moreover, this choice of $\tilde{y}$
must be unique, since $\pi_{T}$ is $Z(G)$ equivariant from Lemma 3.1. Since
$\theta_{p}$ and $y$ are in the same $N$-orbit of a point on the Slodowy
slice, we must have that $n=\rho_{N}(\tilde{y})\delta_{p}$ for every
$\tilde{y}\in\pi_{G}^{-1}(y)$.

Next, to fix $n^{\prime}$ we can follow, with slight modifications for non-
adjoint $G$, the proof of (Crooks:2020, , Lemma 19) to establish that

$n^{\prime}=\big{(}\widetilde{\rho}_{N}(\tilde{y})\delta_{p}\big{)}^{-1}~{}.$
(86)

Thus, every element of $G^{*}_{\theta_{p}}$ looks like
$\overline{K}_{p}^{-1}(\tilde{y})$ for some
$\tilde{y}\in\overline{P}^{-1}(p)$, and $K^{-1}_{p}$ is surjective.

Now to establish injectivity, let $x,y\in\overline{P}^{-1}(p)$ such that
$\overline{K}^{-1}_{p}(x)=\overline{K}^{-1}_{p}(y)=n^{\prime}hn\in
G^{*}_{\theta_{p}}$. We see that $Ad^{*}_{n}\theta_{p}=\pi_{G}(x)=\pi_{G}(y)$
and so $x,y$ are both in the fibre of $\pi_{G}$ above a common point on
$\mathrm{KT}_{\mathfrak{g}}$. But by Lemma 3.1, this means $\pi_{T}(x)$ and
$\pi_{T}(y)$ will differ by an action of $Z(G)$, and so will
$\overline{K}^{-1}_{p}(x)$ and $\overline{K}^{-1}_{p}(y)$. Thus, we conclude
that $x\overset{!}{=}y$ if they map to the same point under
$\overline{K}_{p}^{-1}$. ∎

To summarise, the map $\lambda:\mathrm{KT}_{G}\rightarrow G^{*}$ is defined by

$\lambda(y)\coloneqq\delta_{p}\overline{K}^{-1}_{p}(y)\delta^{-1}_{p}~{},$
(87)

for $y\in\mathrm{KT}_{G}$ and $p=\overline{P}(y)$. Unravelling the definitions
again gives the more compact and algebraic form,

$\begin{split}\lambda:~{}&\mathrm{KT}_{G}\rightarrow G^{*}\\\
&y\mapsto\widetilde{\rho}_{N}(y)^{-1}\widetilde{w_{0}}\big{(}\pi_{T}(y)\big{)}^{-1}\rho_{N}(y)~{}.\end{split}$
(88)

For any $y\in\mathrm{KT}_{G}$, we have that $\lambda(y)\in G^{*}_{\beta(y)}$,
and as an immediate corollary of Theorem 3.4, we have the following:

######  Corollary 3.5.

Let $p\in\mathfrak{g}^{*}/\\!\\!/G$ and let $s_{p}\in S_{f}$ be the
corresponding Slodowy slice element under Kostant’s isomorphism. Then, the
restriction of $\lambda$ to $\overline{P}^{-1}(p)$ gives an isomorphism

$\lambda:\overline{P}^{-1}(p)\xrightarrow{\sim}G_{s_{p}}^{*}~{}.$ (89)

###### Proof.

Follows immediately from Theorem 3.4 and the definition of $\lambda$. ∎

Thus, we have the morphism

$\begin{split}\varphi:\mathrm{KT}_{G}&\rightarrow\mathfrak{Z}_{G}~{},\\\
x&\mapsto(\lambda(x),\beta(x))~{}.\end{split}$ (90)

Let $\mathfrak{Z}^{*}_{G}\subset G^{*}\times S_{f}$ be the closed subscheme,

$\mathfrak{Z}^{*}_{G}=\\{(g,s)\in G^{*}\times
S_{f}|\mathrm{Ad}^{*}_{g}s=s\\}~{}.$ (91)

Note that $\mathfrak{Z}_{G}^{*}=(G^{*}\times S_{f})\cap\mathfrak{Z}_{G}$
inside $G\times S_{f}$, and so is Zariski open in $\mathfrak{Z}_{G}$. The open
subset $\mathfrak{Z}^{*}_{G}$ can alternatively be characterised as the
Hamiltonian reduction

$\mathfrak{Z}^{*}_{G}\cong\big{(}\mu_{diag}^{-1}(0)\cap(G^{*}\times\mathfrak{g}^{\ast}_{\rm
reg})\big{)}/\\!\\!/G_{diag}~{}.$ (92)

######  Proposition 3.6.

The morphism $\varphi:\mathrm{KT}_{G}\rightarrow\mathfrak{Z}_{G}$ is an open
immersion of $S_{f}$-schemes, with image equal to $\mathfrak{Z}^{*}_{G}$

###### Proof.

We have a foliation of $\mathrm{KT}_{G}$,

$\mathrm{KT}_{G}=\bigsqcup_{p\in\mathfrak{g}^{*}/\\!\\!/G}\overline{P}^{-1}(p)$
(93)

From Proposition 3.2, the morphism $\beta$ can be used to rewrite the above as
a foliation over $S_{f}$,

$\mathrm{KT}_{G}=\bigsqcup_{s\in S_{f}}\beta^{-1}(s)~{}.$ (94)

The leaf $\beta^{-1}(s)$ can be identifed with the leaf $\overline{P}^{-1}(p)$
when $s=s_{p}$. By Corollary 3.5, the morphism $\lambda$ gives an isomorphism
between each leaf of this foliation and $G^{*}_{s}$. Thus,

$\mathrm{KT}_{G}\cong\bigsqcup_{s\in S_{f}}G^{*}_{s}=\mathfrak{Z}_{G}^{*}~{}.$
(95)

By construction, $\varphi$ sends the leaf of $\mathrm{KT}_{G}$ over $s\in
S_{f}$ to the leaf of $\mathfrak{Z}_{G}^{*}$ over the same point. Therefore
the diagram,

${\mathrm{KT}_{G}}$${\mathfrak{Z}_{G}}$${S_{f}}$$\scriptstyle{\beta}$$\scriptstyle{\varphi}$
(96)

commutes and so $\varphi$ is a morphism of $S_{f}$-schemes.

∎

We would like $\varphi$ to not just be an open immersion, but also a
symplectic morphism.

######  Proposition 3.7.

For any simple, algebraic Lie group $G$,
$\varphi:\mathrm{KT}_{G}\rightarrow\mathfrak{Z}_{G}$ is symplectic.

###### Proof.

Let $G_{ad}$ be a simple algebraic Lie group of adjoint type. From
Bezrukavnikov:2003 , it is known that

$\varphi_{ad}:\mathrm{KT}_{G_{ad}}\rightarrow\mathfrak{Z}_{G_{ad}}$ (97)

is symplectic.

Now let $G$ be a simple algebraic Lie group such that
$\mathrm{Lie}~{}G=\mathrm{Lie}~{}G_{ad}$. The universal centralisers are
related by

$\mathfrak{Z}_{G_{ad}}\cong Z(G)\backslash\mathfrak{Z}_{G}~{},$ (98)

where the centre $Z(G)$ acts by left multiplication on the fibres of
$\mathfrak{Z}_{G}$. Let $\pi_{\mathfrak{Z}}$ be the projection
$\mathfrak{Z}_{G}\rightarrow
Z(G)\backslash\mathfrak{Z}_{G}\cong\mathfrak{Z}_{G_{ad}}$, which is
symplectic.

Consider the diagram,

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}\pgfsys<EMAIL_ADDRESS>(99)

Since $\varphi$ is $Z(G)$ equivariant, the square commutes. Let
$\omega_{\mathfrak{Z}_{ad}}$ be the symplectic form on
$\mathfrak{Z}_{G_{ad}}$, then

$\varphi^{*}_{ad}(\omega_{\mathfrak{Z}_{ad}})=\omega_{\mathrm{KT}_{\mathfrak{g}}}~{},$
(100)

where $\omega_{\mathrm{KT}_{\mathfrak{g}}}$ is the symplectic form on
$\mathrm{KT}_{\mathfrak{g}}$. Since the symplectic form on $\mathrm{KT}_{G}$,
$\omega_{\mathrm{KT}_{G}}$, is defined by pullback along $\pi_{G}$, we have
that

$(\pi_{G}^{*}\circ\varphi_{ad}^{*})(\omega_{\mathfrak{Z}_{ad}})=\omega_{\mathrm{KT}_{G}}~{},$
(101)

and $\varphi_{ad}\circ\pi_{G}$ is symplectic. As the square commutes,

$(\varphi^{*}\circ\pi_{\mathfrak{Z}}^{*})(\omega_{\mathfrak{Z}_{ad}})=(\pi_{G}^{*}\circ\varphi_{ad}^{*})(\omega_{\mathfrak{Z}_{ad}})=\omega_{\mathrm{KT}_{G}}~{}.$
(102)

But
$\pi_{\mathfrak{Z}}^{*}(\omega_{\mathfrak{Z}_{ad}})=\omega_{\mathfrak{Z}_{G}}$,
where $\omega_{\mathfrak{Z}_{G}}$ is the symplectic form on
$\mathfrak{Z}_{G}$. Therefore,

$(\varphi^{*}\circ\pi_{\mathfrak{Z}}^{*})(\omega_{\mathfrak{Z}_{ad}})=\varphi^{*}(\omega_{\mathfrak{Z}_{G}})=\omega_{\mathrm{KT}_{G}}~{},$
(103)

and $\varphi$ is symplectic. ∎

###  3.4 An algorithmic construction for $\varphi$

With the morphism $\varphi:\mathrm{KT}_{G}\rightarrow\mathfrak{Z}_{G}$
established, we turn to the more practical problem of finding explicit
expressions for the components of this map. Here we will restrict our
attention to the case $G={\rm SL}_{N}$, whose structure as an algebraic group
is particularly transparent. In the case of other algebraic Lie groups, this
level of explicit detail would entail more work, though the story should be
essentially analogous.

The embedding $\varphi$ consists of two parts, $\lambda$ and $\beta$. Let us
first discuss $\beta$. An element $y\in\mathrm{KT}_{\mathfrak{g}}$ has form

$y=\begin{pmatrix}b_{1}(y)&\gamma_{1}(y)&0&\dots\\\
1&b_{2}(y)-b_{1}(y)&\gamma_{2}(y)&\dots\\\ 0&1&b_{3}(y)-b_{2}(y)&\dots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}~{},$ (104)

when considered as an element of $\mathfrak{g}$.888Throughout this section we
make liberal use of the Killing isomorphism to represent elements of
$\mathfrak{g}^{*}$ as elements in $\mathfrak{g}$ without further comment. The
fibre of the map $\pi_{G}$ at $y$ consists of elements
$\\{\tilde{y}_{k},~{}k=0,\ldots,N-1\\}$ that are permuted cyclically under the
action of $Z(G)$.

For $x\in\mathfrak{sl}_{n}$ realised as a matrix in
$\mathrm{End}(\mathbb{C}^{n})$, the fundamental invariants (up to conventions
regarding overall scaling) can be computed as

$P_{j}(x)=\frac{1}{j+1}\mathrm{Tr}\big{(}x^{j+1}\big{)}~{},\qquad\text{for
}n=1,2,\dots,\mathrm{rk}~{}\mathfrak{g}~{}.$ (105)

The components of $\beta(y)$ are given by $P_{1}(y),P_{2}(y),\ldots
P_{\mathrm{rk}_{\mathfrak{g}}}(y)$ and for a given value of $N$ can be easily
computed by the trace formula above. To compute $\beta(\tilde{y}_{k})$ we
simply project to $y$ by $\pi_{G}$ before applying the expression above.

Next let us consider the more elaborate morphism $\lambda$. The definition
(88) of $\lambda$ applied to a point in $\mathrm{KT}_{G}$ takes the form

$\lambda(\tilde{y}_{k})=\widetilde{\rho}_{N}(\tilde{y}_{k})^{-1}\widetilde{w}_{0}\big{(}\pi_{T}(\tilde{y}_{k})\big{)}^{-1}\rho_{N}(\tilde{y}_{k})~{}.$
(106)

From the expression in (48), the map $\pi_{T}$ is given by

$\pi_{T}=\begin{pmatrix}\widetilde{\gamma}_{1}&0&0&0&\dots\\\
0&\frac{\widetilde{\gamma}_{2}}{\widetilde{\gamma}_{1}}&0&0&\dots\\\
0&0&\ddots&0&\dots\\\
\dots&0&0&\frac{\widetilde{\gamma}_{N}}{\widetilde{\gamma}_{N-1}}&0\\\
\dots&0&0&0&\widetilde{\gamma}_{N}^{-1}\end{pmatrix}~{}.$ (107)

Like $\beta$, the maps $\rho_{N}$ and $\widetilde{\rho}_{N}$ only depend on
the image of $\tilde{y}_{k}$ under $\pi_{G}$, whereas $\pi_{T}$ satisfies

$\pi_{T}(\tilde{y}_{k})=\xi^{k}\pi_{T}(\tilde{y}_{0})~{}.$ (108)

A key input into the maps $\rho_{N}$ and $\widetilde{\rho}_{N}$ is the inverse
map $\psi^{-1}:\chi+\mathfrak{b}^{*}\rightarrow N\times S_{f}$ of the Kostant
isomorphism. An element of $N\times S_{f}$ can be written as $(n,s)$, where we
have

$n=\begin{pmatrix}1&n_{1,1}&n_{1,2}&n_{1,3}&\dots\\\
0&1&n_{2,1}&n_{2,2}&\dots\\\ 0&0&1&n_{3,1}&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}~{},\qquad
s=\begin{pmatrix}0&p_{1}&p_{2}&p_{3}&\dots\\\ 1&0&p_{1}&p_{2}&\dots\\\
0&1&0&p_{1}&\dots\\\ \vdots&\vdots&\vdots&\ddots\end{pmatrix}~{}.$ (109)

All entries are $\mathbb{C}$-valued. Under $\psi$ the element $(n,s)$ is taken
to the element of $\mathrm{KT}_{\mathfrak{g}}$ whose matrix entries are

$\begin{split}Ad^{\ast}_{n}s=nsn^{-1}&=\begin{pmatrix}n_{1,1}&p_{1}-(n_{1,1})^{2}+n_{1,2}&\ast&\dots\\\
1&n_{2,1}-n_{1,1}&p_{1}-(n_{2,1})^{2}+n_{2,1}n_{1,1}+n_{2,2}-n_{2,1}&\dots\\\
0&1&n_{3,1}-n_{2,1}&\dots\\\ \vdots&\vdots&\vdots&\ddots\end{pmatrix}\\\
&\overset{!}{=}\begin{pmatrix}b_{1}&\gamma_{1}&0&\dots\\\
1&b_{2}-b_{1}&\gamma_{2}&\dots\\\ 0&1&b_{3}-b_{2}&\dots\\\
\vdots&\vdots&\vdots&\vdots\end{pmatrix}=y~{}.\end{split}$ (110)

The inverse morphism $\psi^{-1}$ is determined by systematically solving this
equation for the coordinates $n_{i,j}$ and $p_{i}$ on $N\times S_{f}$. In
practice, equality of diagonals sets $n_{i,1}=b_{i}$, and qquating the first
superdiagonals then fixes the values of $n_{i,2}$ and $p_{1}$. The next
superdiagonal fixes $n_{i,3}$ and $p_{2}$, and so on continuing to higher
superdiagonals. At each step, the new unfixed coordinates appear only
linearly. The value of $\rho_{N}(\tilde{y}_{k})$ is then just $n$, re-
expressed in terms of the $b$ and $\tilde{\gamma}$ variables according to this
procedure.

The map $\widetilde{\rho}_{N}$ can be determined in the same manner. Note that
the conjugate $\mathrm{Ad}_{\widetilde{w}_{0}(\pi_{T}(\tilde{y}_{0}))^{-1}}y$
will be the reflection of $y$ along the antidiagonal. To see this, observe
that the element $\pi_{T}(\tilde{y}_{0})^{-1}$ leaves the $b_{i}$ invariant
and contracts each $\gamma_{i}$ entry to $1$ while dilating $\chi$ to
$\sum_{i=1}^{\mathrm{rk}\,\mathfrak{g}}\gamma_{i}e_{-\alpha_{i}}$ (as the
negative simple roots transform with opposite weights). In effect, this
reflects $y$ across the diagonal. On the other hand, the element
$\widetilde{w}_{0}$, acts by sequential reflection across the diagonal and the
antidiagonal, so in combination the effect is that
$\mathrm{Ad}_{\widetilde{w}_{0}(\pi_{T}(\tilde{y}_{0}))^{-1}}$ is the
antidiagonal reflection. Consequently, in practice the expression for
$\widetilde{\rho}_{N}(y)$ can simply be obtained from the $n_{i,j}$ above by
making the replacement
$(b_{i},\gamma_{i})\leftrightarrow(-b_{N-1-i},\gamma_{N-1-i})$.

For the purpose of realising the universal centraliser as a symplectic
variety, this procedure can be expedited as a result of Proposition 2.1 and
Theorem 2.2. These imply that rather than constructing $\lambda$ in its
entirety, it is sufficient to determine just the lower left-hand corner entry.

By definition, both $\rho_{N}(\tilde{y}_{k})$ and
$\widetilde{\rho}_{N}(\tilde{y}_{k})$ are upper unitriangular matrices, while
$\widetilde{w}_{0}\big{(}\pi_{T}(\tilde{y})_{k}\big{)}^{-1}$ is supported
along the antidiagonal as in (107). The bottom row of the stabilising group
element is therefore completely fixed by the first row of the map $\rho_{N}$,
and what’s more, the lower-left corner entry is just equal to the reciprocal
of the upper-left corner entry of $\pi_{T}$, so in particular we have

$X\coloneqq\widetilde{\gamma}^{-1}_{1}=\prod_{j=1}^{\mathrm{rk}~{}\mathfrak{g}}\gamma_{j}^{-\widetilde{a}_{1j}}~{}.$
(111)

From this and the expression for the $P_{j}$ functions, Theorem 2.2 allows us
to generate all of $I_{{\rm SL}_{N}}$.

For $G\neq\rm SL_{N}$ we have a conjectural expression for $X$, coming from
the discussion in Section 2.4. Suppose that $G$ is not $\mathrm{Spin}(4N)$,
then the lowest dimensional faithful representation, $V_{F}$ is irreducible.
Let $(a_{1},a_{2},\dots_{,}a_{\mathrm{rk}\,\mathfrak{g}})$ be the components
of the highest weight of $V_{F}$ in the simple root basis. We conjecture that

$X=\prod_{j=1}^{\mathrm{rk}\,\mathfrak{g}}\gamma_{j}^{-a_{j}}~{}.$ (112)

For $G={\rm SL}_{N}$ this specialises to
$X=\prod_{j=1}^{\mathrm{rk}\,\mathfrak{g}}\gamma_{j}^{-\widetilde{a}_{1j}}$,
where $\widetilde{a}_{ij}$ are again the entries of the inverse Cartan matrix.
Similarly, if $G$ is ${\rm Spin}(4N)$ we define $X_{\rm s}$ and $X_{\rm c}$
analogously to (112) using the highest weights of $V_{\rm s}$ and $V_{\rm c}$
respectively.

###  3.5 The embedding for ${\rm SL}_{3}$

The ring of functions on the Kostant–Toda lattice for $\mathfrak{sl}_{3}$ is
given by

$\mathbb{C}[\mathrm{KT}_{\mathfrak{g}}]=\mathbb{C}[b_{1},b_{2},\gamma_{1}^{\pm
1},\gamma_{2}^{\pm 1}]~{}.$ (113)

Following the general prescription, for $G={\rm SL}_{3}$ the covering space
$\mathrm{KT}_{G}$ comes with coordinate ring

$\mathbb{C}[\mathrm{KT}_{G}]=\mathbb{C}[b_{1},b_{2},\widetilde{\gamma}^{\pm
1}_{1},\widetilde{\gamma}^{\pm 1}_{2}]~{},$ (114)

where
$\widetilde{\gamma}_{1}=\gamma^{-\frac{2}{3}}_{1}\gamma_{2}^{-\frac{1}{3}}$
and
$\widetilde{\gamma}_{2}=\gamma^{-\frac{2}{3}}_{2}\gamma_{1}^{-\frac{1}{2}}$,
and the Poisson bracket follows accordingly. We can make explicit the maps
involved in the definition of $\varphi$. For
$\pi_{T}:\mathrm{KT}_{G}\rightarrow T$ we have (at the level of rings of
functions)

$t_{1}\mapsto\widetilde{\gamma}_{1}~{},\quad
t_{2}\mapsto\widetilde{\gamma}_{2}~{}.$ (115)

The longest element of the Weyl group of $A_{2}$ can be lifted to $N_{{\rm
SL}_{3}}(T)$ as

$\widetilde{w}_{0}=\begin{pmatrix}0&0&-1\\\ 0&-1&0\\\ -1&0&0\end{pmatrix}~{}.$
(116)

The coordinate ring of the Slodowy slice, $S_{f}$, is embedded (up to
normalisation) as

$\begin{split}S&=\frac{1}{2}\mathop{\mathrm{Tr}}\nolimits(y^{2})=b_{1}^{2}+b_{2}^{2}-b_{1}b_{2}+\gamma_{1}+\gamma_{2}~{},\\\
W&=\frac{1}{3}\mathop{\mathrm{Tr}}\nolimits(y^{3})=b_{1}^{2}b_{2}-b_{1}b_{2}^{2}+\gamma_{1}b_{2}-\gamma_{2}b_{1}~{}.\end{split}$
(117)

The image of $\lambda$ is realised as follows. Solving for the matrix
$\rho_{N}(y)$ as described above, one finds

$\rho_{N}(y)=\begin{pmatrix}~{}~{}1~{}~{}&~{}~{}b_{1}~{}~{}&\frac{1}{2}(b_{1}^{2}+b_{1}b_{2}-b_{2}^{2}+\gamma_{1}-\gamma_{2})\\\
0&1&b_{2}\\\ 0&0&1\end{pmatrix}~{}.$ (118)

The group element $\tilde{w}_{0}(\pi_{T}(y))^{-1}$ is given as follows,

$\tilde{w}_{0}(\pi_{T}(y))^{-1}=\begin{pmatrix}0&0&-1\\\ 0&-1&0\\\
-1&0&0\end{pmatrix}\begin{pmatrix}\tilde{\gamma}_{1}&0&0\\\
0&\frac{\tilde{\gamma}_{2}}{\tilde{\gamma}_{1}}&0\\\
0&0&\frac{1}{\tilde{\gamma}_{2}}\end{pmatrix}^{-1}=\begin{pmatrix}0&0&-\gamma_{1}^{\frac{1}{3}}\gamma_{2}^{\frac{2}{3}}\\\
0&-\frac{\gamma_{2}^{\frac{1}{3}}}{\gamma_{1}^{\frac{1}{3}}}&0\\\
-\frac{1}{\gamma_{1}^{\frac{2}{3}}\gamma_{2}^{\frac{1}{3}}}&0&0\end{pmatrix}~{}.$
(119)

Solving for the matrix $\widetilde{\rho_{N}}(y)$ then produces the result

$\widetilde{\rho_{N}}(y)=\begin{pmatrix}~{}~{}1~{}~{}&~{}~{}-b_{2}~{}~{}&\frac{1}{2}(-b_{1}^{2}+b_{1}b_{2}+b_{2}^{2}-\gamma_{1}+\gamma_{2})\\\
0&1&-b_{1}\\\ 0&0&1\end{pmatrix}~{}.$ (120)

Putting everything together, this gives the following parameterisation of (an
open subset of) stabiliser of the point $\beta(y)\in S_{f}$,

$\lambda(y)=-\frac{1}{\gamma_{1}^{\frac{2}{3}}\gamma_{2}^{\frac{1}{3}}}\begin{pmatrix}b_{1}^{2}+\gamma_{1}-\frac{S}{2}&W+\frac{b_{1}S}{2}&b_{1}W+\frac{S^{2}}{4}\\\
b_{1}&b_{1}^{2}+\gamma_{1}&W+\frac{b_{1}S}{2}\\\
1~{}&~{}b_{1}~{}&~{}b_{1}^{2}+\gamma_{1}-\frac{S}{2}\end{pmatrix}~{},$ (121)

where to simplify the expression we have used $S$ and $W$ as defined in (117).
We define the additional algebraic generators of the coordinate ring as the
entries on the bottom row of $\lambda(y)$,

$\begin{split}X&\coloneqq-\gamma_{1}^{-\frac{2}{3}}\gamma_{2}^{-\frac{1}{3}}~{},\\\
Y&\coloneqq-\\-\gamma_{1}^{-\frac{2}{3}}\gamma_{2}^{-\frac{1}{3}}b_{1}~{},\\\
Z&\coloneqq-\frac{1}{2}\gamma_{1}^{-\frac{2}{3}}\gamma_{2}^{-\frac{1}{3}}(\gamma_{1}-\gamma_{2}+b_{1}^{2}-b_{2}^{2}+b_{1}b_{2})~{}.\end{split}$
(122)

By Proposition 2.1, only the bottom row is needed to uniquely specify
$\lambda(y)$, and indeed, the remaining rows can be expressed in terms of the
$S,W,X,Y,Z$ (and could be determined directly by solving the
$[\lambda(y),\beta(y)]=0$). Altogether, we have for the embedding
$\varphi(y)$,

$\varphi(y)=\left(\begin{pmatrix}Z~{}&~{}WX+\frac{1}{2}SY~{}&~{}\frac{1}{4}S^{2}X+WY\\\
Y&\frac{1}{2}SX+Z&WX+\frac{1}{2}SY\\\
X&Y&Z\end{pmatrix}~{},~{}\begin{pmatrix}0&\frac{1}{2}S&W\\\
1&0&\frac{1}{2}S\\\ 0&1&0\end{pmatrix}\right)~{}.$ (123)

Note that this agrees with the presentation of Section 2.3. The determinantal
relation

$\begin{split}D_{{\rm
SL}_{3}}=&-\frac{1}{8}S^{3}X^{3}+W^{2}X^{3}+\frac{1}{2}SWX^{2}Y+\frac{1}{2}S^{2}XY^{2}+WY^{3}-\frac{1}{4}S^{2}X^{2}Z\\\
&-3WXYZ-SY^{2}Z+\frac{1}{2}SXZ^{2}+Z^{3}\overset{!}{=}1~{},\end{split}$ (124)

is satisfied identically by the expressions for $S,W,X,Y,Z$ in terms of
$\mathrm{KT}_{G}$ variables. The Poisson structure is inherited from the
structure on $\mathrm{KT}_{G}$, and one can verify that the Poisson brackets
also agree with (23).

##  4 Chiralising the universal centraliser

In Arakawa:2018egx , the (critical level) chiral universal centraliser
($\mathbf{I}_{G}$, denoted in _loc._ cit. by $\mathbf{I}_{G}$), was defined to
be the two-sided quantum Drinfel’d–Sokolov reduction of critical level chiral
differential operators (CDOs) on $G$
($\mathcal{D}^{ch}_{G,\kappa_{c}}$).999The chiral universal centraliser
$\mathbf{I}_{{\rm SL}_{2}}$ at _non-critical level_ appeared previously in
Frenkel:200657  as the regular representation of the Virasoro algebra. The
operation of Drinfel’d–Sokolov reduction is a chiral analogue of
Kostant–Whittaker reduction, and $\mathcal{D}^{ch}_{G,\kappa_{c}}$ represents
a chiral quantisation (defined in the following section) of $T^{*}G$, so
$\mathbf{I}_{G}$ is indeed a natural chiral analogue of $\mathfrak{Z}_{G}$. By
virtue of its definition in terms of DS reduction, it should be identified
with the VOA assigned to the sphere of type $G$ in class $\SS$ Beem:2014rza .

In the previous section, we presented a method for realising
$I_{G}=\mathbb{C}[\mathfrak{Z}_{G}]$ by way of the Kostant–Toda lattice
$\mathrm{KT}_{G}$. The open immersion
$\varphi:\mathrm{KT}_{G}\rightarrow\mathfrak{Z}_{G}$ gives a realisation of
$I_{G}$ in terms of functions on an algebraically trivial space, which we
think of as a (finite-dimensional, classical) “free field realisation” of
$I_{G}$. In this section we describe a chiral analogue of this construction,
and propose a strategy for producing (chiral, quantum) free field realisations
for $\mathbf{I}_{G}$ for general $G$. We make this completely explicit for the
cases $G={\rm SL}_{2}$ and $G={\rm SL}_{3}$.

In this section $G$ will be assumed to be simply connected unless otherwise
stated.

###  4.1 Chiralising the Kostant–Toda lattice

The Kostant–Toda lattice is isomorphic as a symplectic variety to
$(T^{\ast}\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}$. The standard
chiral quantisation(s) of this space are given by CDOs on
$(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}$. In Beem:2019tfp , these
CDOs were presented in terms of a lattice boson system. We will review this
construction here in connection with the Kostant–Toda lattice and its
refinement.

Let chiral bosons $\phi_{i}$, $\delta_{j}$ for
$i,j=1,2,\dots,\mathrm{rk}\,\mathfrak{g}$ be normalised according to

$\phi_{i}(z)\phi_{j}(w)\sim-\delta_{ij}\log(z-w)~{},\quad\delta_{i}(z)\delta_{j}(w)\sim\delta_{ij}\log(z-w)~{}.$
(125)

Let $\Pi_{\mathfrak{g}}$ be the isotropic lattice vertex algebra generated by
$\partial\phi_{i}$, $\partial\delta_{i}$, and vertex operators,
$e^{n(\phi_{i}+\delta_{i})}$, for $n\in\mathbb{Z}$. Since the
$\phi_{i}+\delta_{i}$ operator has regular self-OPE, the exponential vertex
operators form a commutative vertex algebra. This is a “half-lattice vertex
algebra” in the sense of half_lattice .

In fact, $\Pi_{\mathfrak{g}}$ is strongly generated by the operators

$\gamma_{i}^{\pm 1}=e^{\pm(\phi_{i}+\delta_{i})}~{},\quad
b_{i}=\frac{1}{2}\partial(\delta_{i}-\phi_{i})~{},$ (126)

whose only nonvanishing singular OPEs are given by

$b_{i}(z)\gamma_{j}^{\pm 1}(w)\sim\frac{\pm\delta_{ij}\gamma_{j}^{\pm
1}(w)}{z-w}~{},$ (127)

and which also satisfy the suggested relation
$(\gamma_{i}^{-1}\gamma_{i}^{+1})=1$. This presentation is the one relevant to
description of this vertex algebra as CDOs on the multiplicative group
$(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}$, which itself can be
thought of as the localisation of $\mathrm{rk}\,\mathfrak{g}$ many
$\beta\gamma$ systems (i.e., CDOs on $\mathbb{C}^{\mathrm{rk}\,\mathfrak{g}}$
Malikov:1999 ). The lattice representation then arises via
Friedan–Martinec–Shenker bosonisation of the $\beta\gamma$ system.

This presentation in terms of lattice bosons makes the following
generalisation natural. We pass to a refined lattice with rational spacings,
specifically with denominator $|Z(G)|$ for $G$ simply connected, and define
the vertex subalgebra $\Pi_{G}$ to be strongly generated by

$\widetilde{\gamma}^{\pm
1}_{i}=\prod_{i=1}^{\mathrm{rk}{\mathfrak{g}}}e^{\pm\tilde{a}_{ij}(\phi_{j}+\delta_{j})}~{},\quad
b_{i}=\frac{1}{2}\partial(\delta_{i}-\phi_{i})~{},$ (128)

where, as previously, the $\tilde{a}_{ij}$ are matrix elements of the inverse
Cartan matrix of $\mathfrak{g}$. Note that the $b_{i}$ are unchanged. For
$\mathfrak{g}=\mathfrak{sl}_{2}$, this is precisely the lattice
$\Pi_{\frac{1}{2}}$ appearing in the free field constructions of Beem:2019tfp
(and previously in Adamovic2019 ). The vertex algebra $\Pi_{G}$ admits a
natural action of $Z(G)$ according to the prescription in (41), extended to be
$\mathbb{C}[\partial]$-linear.

Now suppose $G^{\prime}$ is a non-simply connected algebraic Lie group with
the same Lie algebra as $G$. Then $G^{\prime}\cong Z^{\prime}(G)\backslash G$
for some central subgroup $Z^{\prime}(G)\subset Z(G)$. Let
$\gamma_{i}^{\prime}$ be the generators of the ring
$\mathbb{C}[\widetilde{\gamma}_{i}]^{Z^{\prime}(G)}$. Then we define a lattice
vertex subalgebra $\Pi_{G^{\prime}}\subset\Pi_{G}$ that is strongly generated
by the $b_{i}$ and the $Z(G)$ invariants, $(\gamma_{i}^{\prime})^{\pm 1}$.

There is an obvious inclusion of vertex algebras
$\Pi_{\mathfrak{g}}\hookrightarrow\Pi_{G}$, given by

$\gamma_{i}^{\pm
1}\mapsto\prod_{j=1}^{\mathrm{rk}\,\mathfrak{g}}(\widetilde{\gamma}_{j}^{\pm
1})^{a_{ij}}~{},\qquad b_{i}\mapsto b_{i}~{}.$ (129)

The image of $\Pi_{\mathfrak{g}}$ inside $\Pi_{G}$ is precisely the $Z(G)$
invariant subalgebra of $\Pi_{G}$—indeed, for the case that $G^{\prime}$ is
the adjoint group, we have $\Pi_{G^{\prime}}=\Pi_{\mathfrak{g}}$.

Our vertex algebras $\Pi_{G}$ and the Kostant–Toda lattices are related in the
sense that the former is a chiral quantisation of the latter. The relationship
between vertex algebras and geometric spaces were explored by Arakawa–Moreau
Arakawa:2012c2 ; Arakawa2017introduction ; Arakawa_Moreau:2018 ;
Arakawa_Moreau:2017157 ; Arakawa_Moreau:2018542 . We shall recall some of the
relevant definitions now. For a more pedagogical introduction, we recommend
the monograph Arakawa_Moreau:jet .

For a scheme $X$ over $\mathbb{C}$, we denote its arc space by
$J_{\infty}X\coloneqq\mathrm{Hom}(\mathbb{D},X)$, where
$\mathbb{D}=\mathrm{Spec}~{}\mathbb{C}[\\![t]\\!]$ is the formal disc. For $X$
Poisson, functions on the arc space ${\cal O}(J_{\infty}X)$ has a canonical
vertex Poisson algebra structure Arakawa:2012c2 .

Every vertex algebra has a canonical decreasing filtration, $F^{\bullet}$,
known as the Li filtration Li:2005 . Specifically, for $V$ a vertex algebra
with vacuum $|0\rangle$, we have

$F^{i}V=\bigg{\\{}v_{1,(-n_{1}-1)}v_{2,(-n_{2}-1)}\dots
v_{k,(-n_{k}-1)}|0\rangle\,\bigg{|}\,k\in\mathbb{N}\,,~{}~{}v_{1},v_{2},\dots,v_{k}\in
V\,,~{}~{}\sum_{j=1}^{k}n_{j}\geqslant i\,\bigg{\\}}~{}.$ (130)

$V$ is separated if $\bigcap_{i=1}F^{i}V=0$ and if $V$ is a separated vertex
algebra, then the vector space $R_{V}\coloneqq V/F^{1}(V)$ is a Poisson
algebra, with multiplication arising from the normally ordered product and the
Poisson bracket coming from simple poles in the OPE. This Poisson algebra is
called the Zhu’s $C_{2}$ algebra of $V$, and
$X_{V}\coloneqq\mathrm{Specm}\,R_{V}$ is the _associated variety_ of $V$.
$\overline{X}_{V}\coloneqq\mathrm{Spec}\,R_{V}$ is then the _associated
scheme_ of $V$.

Now let $X$ be an affine, Poisson scheme; a vertex algebra $V$ is said to be a
chiral quantisation of $X$ Arakawa2017introduction  if, as Poisson varieties,

$X_{V}\cong X_{red}~{}.$ (131)

It is a strict chiral quantisation of $X$ if furthermore,

$\overline{X}_{V}\cong
X\qquad\mathrm{and}\qquad\mathrm{Spec}(\mathrm{gr}_{F}\,V)\cong
J_{\infty}X~{}.$ (132)

A vertex algebra satisfying $\mathrm{Spec}~{}(\mathrm{gr}_{F}V)\cong
J_{\infty}\mathrm{Spec}~{}\overline{X}_{V}$ is also known as classically free.

The vertex algebra $\Pi_{\mathfrak{g}}$ is a strict chiral quantisation of
$T^{\ast}\big{(}\mathbb{C}^{\times}\big{)}^{\mathrm{rk}\,\mathfrak{g}}$, and
given its definition in terms of strong generators, it is straightforward to
see that
$R_{\Pi_{G}}\cong\mathbb{C}[b_{i},\widetilde{\gamma}_{j}\,|\,i,j=1,\dots,\mathrm{rk}\,\mathfrak{g}]$
with the expected Poisson brackets. Therefore, $\overline{X}_{\Pi_{G}}$ is
reduced and can be identified with $\mathrm{KT}_{G}$ as a symplectic variety.
Moreover, $\Pi_{G}$ is classically free, so $\Pi_{G}$ is a strict chiral
quantisation of $\mathrm{KT}_{G}$ for any $G$. Moreover, the inclusion
$\Pi_{\mathfrak{g}}\hookrightarrow\Pi_{G}$ as the $Z(G)$ invariant subalgebra
descends, at the level of associated schemes, to the covering map
$\pi_{G}^{\\#}:{\cal O}(\mathrm{KT}_{\mathfrak{g}})\hookrightarrow{\cal
O}(\mathrm{KT}_{G})$.

Finally, let us briefly remark on the conformal structure of these vertex
algebras. Chiral bosons admit a well-known family of conformal vectors, which
in our case amount to a $\mathbb{C}^{2\mathrm{rk}\,\mathfrak{g}}$ parameter
space of conformal vectors of the form

$T=\sum_{i}\frac{1}{2}\bigg{(}\partial\delta_{i}\partial\delta_{i}-\partial\phi_{i}\partial\phi_{i}+\alpha_{i}\partial^{2}\phi_{i}+\beta_{i}\partial^{2}\delta\bigg{)}~{},$
(133)

where $\alpha_{i},\beta_{i}\in\mathbb{C}$ are background charges for
$\phi_{i}$ and $\delta_{i}$, respectively. Tuning the values of $\alpha_{i}$
and $\beta_{i}$ changes the conformal grading on $\Pi_{G}$, and in particular
changes the weight of the $\gamma_{i}$ operators. To match with the
$\mathbb{C}^{\ast}$ grading on $\mathrm{KT}_{G}$, we require the weights
$\Delta_{\gamma_{i}}=2$ and $\Delta_{b_{i}}=1$, which is accomplished by
setting $\alpha_{i}+\beta_{i}=-2$. With no further input, this leaves
$\mathrm{rk}\,\mathfrak{g}$ many unconstrained background charges, one
combination of which could further be constrained by demanding that the
central charge match the expected value for the chiral universal centraliser,

$c_{\mathfrak{g}}=\mathrm{rk}\,\mathfrak{g}+48(\rho_{\mathfrak{g}},\rho_{\mathfrak{g}}^{\vee})~{},$
(134)

where $\rho_{\mathfrak{g}}$ is the half-sum of positive roots and
$\rho_{\mathfrak{g}}^{\vee}$ is the half-sum of positive coroots. For simply
laced $\mathfrak{g}$, we can apply the strange formula of Freudenthal and de
Vries to simplify this expression to

$c_{\mathfrak{g}}=\mathrm{rk}\,\mathfrak{g}+2h_{\mathfrak{g}}^{\vee}\,{\rm
dim}\,\mathfrak{g}~{},$ (135)

where $h_{\mathfrak{g}}^{\vee}$ is the dual Coxeter number.

###  4.2 A chiral embedding and free field realisation

The aim in this section is to identify a candidate for a free field
realisation of the chiral universal centraliser by constructing a chiral
version of the morphism $\varphi:\mathrm{KT}_{G}\rightarrow\mathfrak{Z}_{G}$.
Recall that $\varphi$ decomposes into a map $\beta$ to the Slodowy slice
$S_{f}$ and a map $\lambda$ to the centraliser in $G$ of $\beta(y)$. We will
treat the chiral version in a similar matter.

The chiral analogue of $\mathbb{C}[S_{f}]$ is the Feigin–Frenkel centre