The singular Hitchin fibration, cameral data, and representation theoryThanks: The author acknowledges support from the EPSRC grant EP/V520275/1, and the projects “Wobbly and Very stable Higgs bundles” (Proyectos de generación del conocimiento ref. PID2023-147785NA-I00) and “Real Calabi-Yau and Hitchin systems” (Proyectos de Consolidación ref. CNS2022-136042), funded by MICIU/AEI/10.13039/501100011033 and NextGenerationEU/PRTR

We give an outline of the framework of generalised Hitchin fibrations of Morrissey and Ngô (as surveyed in [72]) to motivate the statements of our results. For simplicity, throughout this work we will exclusively consider the case where the ground field is $\mathbb{C}$.

We first consider the usual Hitchin system. We denote by $\mathcal{M}$ the moduli stack of $G$-Higgs bundles on $\Sigma$, for a complex reductive group $G$ (with Lie algebra $\mathfrak{g}$) and a smooth projective curve $\Sigma$. In the formulation of [70], this is viewed as the stack of maps from the curve $\Sigma$ to a twist of the adjoint quotient stack $[\mathfrak{g}/G]$. The Hitchin map $h:\mathcal{M}\rightarrow\mathcal{A}$ is then viewed as the global version of the map $\chi_{G}:[\mathfrak{g}/G]\rightarrow\mathfrak{g}//G=\mathfrak{c}$ induced by the Chevalley restriction morphism. The abelianisation phenomenon for regular Higgs bundles can be seen as a consequence of the gerbe structure of $\chi_{G}$ when restricted to the regular locus $[\mathfrak{g}^{reg}/G]$, where $\mathfrak{g}^{reg}$ is the open subset of regular elements in $\mathfrak{g}$.

A number of the properties of the Hitchin fibration can be abstracted by replacing the adjoint action with an arbitrary action of $G$ on a normal affine variety $V$. In particular, we can define a stack of generalised Higgs bundles ${\bf M}(V)$, and a generalised Hitchin map ${\bf h}_{V}:{\bf M}(V)\rightarrow{\bf A}(V)$ as a global version of the affinisation map $\chi_{V,G}:[V/G]\rightarrow V//G$. We denote by $V^{reg}$ the locus of regular elements, i.e. those with minimal centraliser dimension. Unlike the special case of the adjoint action, the restriction $\chi_{V,G}:[V^{reg}/G]\rightarrow V//G$ may fail to be a gerbe. However, given a flat open subgroup scheme $\mathcal{I}^{fl}$ of the group scheme $\mathcal{I}^{reg}$ of centralisers on $V^{reg}$, one can construct a Deligne-Mumford stack $\mathcal{B}$, the regular quotient, and a factorisation of $\chi_{V,G}$ through a gerbe $\rho_{V,G}:[V^{reg}/G]\rightarrow\mathcal{B}$. As a consequence, the fibres of $h_{V}$ on the regular locus ${\bf M}^{reg}(V)$ can be described using stacks of torsors on $\Sigma$ whose structure group is determined by $\mathcal{I}^{fl}$.

One of the primary questions for generalised Hitchin fibrations is whether they can be described by cameral data. We sketch a simplified version of this for the usual Hitchin fibration. For any point $a\in\mathcal{A}$ in the Hitchin base, there is a finite cover $\check{\Sigma}_{a}$ of the curve $\Sigma$, the cameral curve associated to $a\in\mathcal{A}$, which admits an action of the Weyl group $W$ over $\Sigma$. We fix a Cartan subgroup $T$ for $G$, and denote by $\hat{\mathcal{P}}_{a}$ the stack of strongly $W$-equivariant $T$-bundles $\mathcal{T}$ with the following property:

• $(*)$

  for any ramification point $x\in\check{\Sigma}_{a}$ of the cover $\check{\Sigma}_{a}\rightarrow\Sigma$, there is an isomorphism $\mathcal{T}_{x}\cong T$ of $T$-torsors which is equivariant with respect to the action of $Stab_{W}(x)$.

Then, for a generic choice of $a\in\mathcal{A}$, the Hitchin fibre $h^{-1}(a)$ can be identified with $\hat{\mathcal{P}}_{a}$ up to isogeny (note that a substantially more precise statement is possible [27, Theorem 6.4]).

This can be derived as a consequence of the structure of the centraliser group scheme $\mathcal{I}^{reg}$. Let $\mathfrak{t}$ be the Lie algebra of $T$, and recall that $\mathfrak{t}/W$ can be identified with the Chevalley base $\mathfrak{c}$. The Weil restriction $\Pi$ of the torus $T$ under the induced map $\mathfrak{t}\rightarrow\mathfrak{c}$ inherits a $W$-action, and the $W$-fixed points define a group scheme $\hat{\mathcal{J}}$ on $\mathfrak{c}$. Then, there is a canonical open embedding of group schemes

which is generically an isomorphism [27, Proposition 12.5] (again, a stronger statement describing the image of the map $\kappa$ is possible [27, Theorem 11.6]).

For any $d\in\mathbb{N}$, we now consider the substack $\mathcal{M}^{d}$ of $\mathcal{M}$ of $G$-Higgs bundles on $\Sigma$ whose Higgs field has centraliser of dimension $d$ at every point of $\Sigma$. The locus $\mathfrak{g}_{d}\subseteq\mathfrak{g}$ of elements with centraliser dimension $d$ is not necessarily irreducible; its irreducible components are known as sheets. The decomposition

defines a decomposition of $\mathcal{M}^{d}$ into closed substacks $\mathcal{M}_{S}$. It is more natural to describe the restriction of the Hitchin fibration to each of the substacks $\mathcal{M}_{S}$ than to directly study the fibration on $\mathcal{M}^{d}$; in particular $\mathcal{M}_{S}$ can be viewed as the regular locus for a generalised Hitchin system if the sheet $S$ is normal. Throughout, we will make the simplifying assumption that $S$ is in fact non-singular; this is not a particularly restrictive condition, as it includes all sheets in classical Lie algebras [50], and most of the sheets in the exceptional Lie algebras [17].

The geometry of sheets and their quotients under the restriction of the adjoint action is already well-studied. Their initial motivation came from the study of the primitive ideals of the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ [26], [8], [10], [9]. More recently, there has been renewed interest in sheets, and the related notion of birational sheets, for their connection to the representation theory of finite $W$-algebras and their role in the orbit method [78], [76], [66], [87].

The centraliser group scheme $\mathcal{I}_{S}$ on $S$ is non-abelian whenever $S$ is not the regular sheet, and can fail to be flat as an $S$-scheme (see Example 2.25 and Corollary 3.19). Nonetheless, we show that $\mathcal{I}_{S}$ contains a maximal smooth normal subgroup scheme $\mathcal{I}_{S}^{sm}$ of finite index. This determines a canonical choice for the regular quotient $\mathcal{B}$ of [72], and thus a factorisation of the Chevalley map on $S$ through a morphism $\rho_{S}:S\rightarrow\mathcal{B}$, which we call the $S$-Chevalley map.

The regular quotient $\mathcal{B}$ is a smooth Deligne-Mumford stack whose coarse moduli space is the geometric quotient $\mathfrak{c}_{S}$ for the $G$-action on the sheet $S$. Moreover, it can be described explicitly as a stack quotient of an affine space $\tilde{\mathfrak{c}}_{S}$ by a certain finite group $F$ associated to $S$. The group $F$, which we call the Katsylo group, also plays a role in other aspects of the geometry of the sheet, and we give a number of different descriptions of it in this paper (see e.g. Definition 3.2, Remark 8.6, and Theorem 8.14).

The map $\rho_{S}$ induces a factorisation of the Hitchin map on $\mathcal{M}_{S}$ as

Here, $\mathcal{A}_{S}$ is a smooth Deligne-Mumford stack; moreover, $\mathcal{A}_{S}$ contains a distinguished connected component $\mathcal{A}^{0}_{S}$ which is a stack quotient of an affine space $\tilde{\mathcal{A}}_{S}^{0}$ by an action of the Katsylo group $F$. The map $\tilde{\mu}_{S}$ is quasi-finite, and is generically injective on $\mathcal{A}^{0}_{S}$.

For any Higgs bundle $(E,\Phi)$ representing a $\mathbb{C}$-point of $\mathcal{M}_{S}$, the group scheme $\mathcal{I}_{(E,\Phi)}$ of local automorphisms of $(E,\Phi)$ over $\Sigma$ contains a smooth normal subgroup scheme $\mathcal{I}^{sm}_{(E,\Phi)}$ induced by the subgroup scheme $\mathcal{I}_{S}^{sm}$ of $\mathcal{I}_{S}$.

We also give a global version of this statement over the component $\mathcal{A}^{0}_{S}$ by constructing a generalisation of the Hitchin section (Theorem 5.20).

In order to generalise the cameral description from the regular case, we first consider how to generalise the map $\kappa$ of (1.1). We consider this only when $S$ is a Dixmier sheet; i.e. $S$ contains a dense locus of semisimple elements. In this case the centraliser of any semisimple element in $S$ is conjugate to some Levi subgroup $L$ of $G$. If $\mathfrak{z}$ denotes the centre of $Lie(L)$, and $W_{L}=N_{G}(L)/L$ denotes the relative Weyl group, the geometric quotient $\mathfrak{c}_{S}$ of $S$ by the $G$-action can be identified with $\mathfrak{z}/W_{L}$.

There are obstacles to a cameral description for the group scheme $\mathcal{I}^{sm}_{S}$ in general. In particular, even for $G=GL_{n}$, the obvious generalisation of $\hat{\mathcal{J}}$ will not always be suitable, as there are examples of Levi subgroups for which $W_{L}$ is trivial (this can be compared with [42, Example 4.2]). Despite this, it is possible to produce a generalised version of the map $\kappa$ defined in (1.1) by regarding it instead as an abelianisation map; this is natural from its construction which factors through the abelianisation of the universal Borel subgroup. We construct our abelianisation map $\kappa_{S}$, which we call the cameral homomorphism, over the sheet $S$ instead of the regular quotient $\mathcal{B}$, using the generalised Grothendieck-Springer theory of sheets [7], [16]. While it no longer makes sense to ask whether $\kappa_{S}$ is an open embedding in general, we show that $\kappa_{S}$ is smooth in the case that $G$ is a classical group.

We now assume that the abelianisation map $\kappa_{S}$ is smooth. Then, there is a factorisation of the $S$-Hitchin map as

We consider $h_{S}^{ab}:\mathcal{M}_{S}^{ab}\rightarrow\mathcal{A}_{S}$ to be the abelianisation of the $S$-Hitchin map. For any $\mathbb{C}$-point $\tau$ of $\mathcal{A}_{S}$, we can define a commutative group stack $\mathcal{P}_{S,\tau}$ over $\mathbb{C}$ using the morphism $\kappa_{S}$; and if we restrict to $\mathcal{A}_{S}^{0}$ this defines a commutative group stack $\mathcal{P}_{S}^{0}\rightarrow\mathcal{A}_{S}^{0}$ by varying $\tau$ over $\mathcal{A}_{S}^{0}$. We denote by $\mathcal{M}_{S}^{ab,0}$ the restriction of $\mathcal{M}_{S}^{ab}$ to $\mathcal{A}_{S}^{0}$.

For any $\mathbb{C}$-point $\tau$ of $\mathcal{A}_{S}$, we can define a cover $\hat{\Sigma}_{\tau}\rightarrow\Sigma$ which we call the $S$-cameral curve; generically, this is the normalisation of the reduced subscheme of the usual cameral curve $\check{\Sigma}_{\tilde{\mu}_{S}(\tau)}$. We denote by $\bar{Z}$ the abelianisation of the Levi subgroup $L$, and let $\hat{\mathcal{P}}_{S,\tau}$ be the stack of $W_{L}$-equivariant $\bar{Z}$-bundles $\mathcal{Z}$ satisfying the analogue of the property $(*)$ noted above for the regular case (see Theorem 5.31 for the precise statement).

We make these constructions and results explicit in the examples $G=GL_{n}$ (for arbitrary $n$) and $G=Sp_{4}$. In each case we sketch a description of the locus $\mathcal{M}_{S}$ in $\mathcal{M}$, calculate the base $\mathcal{A}_{S}$, and, in the case of Dixmier sheets, describe the abelianised fibration $h_{S}^{ab}:\mathcal{M}_{S}^{ab}\rightarrow\mathcal{A}_{S}$ over the component $\mathcal{A}_{S}^{0}$ in terms of Hitchin fibrations for smaller groups. For the $GL_{n}$ cases, we consider these constructions via spectral data and give a description for the abelianisation map $Ab_{S}$ in terms of abelianised spectral data on a normalisation of the spectral curve. Thus, the abelianised fibration can be viewed as an analogue to the semi-abelian data described in [49], [51], [52] and [32].

We also apply these results to the Hitchin fibration for a real form $G_{\mathbb{R}}$ of $G$, building on the work of [37] and [42]. Each real form is associated to an involution $\theta$ on $G$, and this determines the isotropy representation of the fixed point subgroup $H:=G^{\theta}$ on the vector subspace $\mathfrak{m}$ of $\theta$-anti-invariants in $\mathfrak{g}$. The $G_{\mathbb{R}}$-Hitchin fibration is the corresponding generalised Hitchin fibration; we denote the moduli stack of $G_{\mathbb{R}}$-Higgs bundles by $\mathcal{M}(G_{\mathbb{R}})$ and the $G_{\mathbb{R}}$-Hitchin map by $h_{\mathbb{R}}:\mathcal{M}(G_{\mathbb{R}})\rightarrow\mathcal{A}(G_{\mathbb{R}})$. There is also a natural map from the stack $\mathcal{M}(G_{\mathbb{R}})$ to the moduli stack $\mathcal{M}$ of $G$-Higgs bundles.

There is a notion of regularity for $G_{\mathbb{R}}$-Higgs bundles which determines a dense open substack $\mathcal{M}^{reg}(G_{\mathbb{R}})$ of $\mathcal{M}(G_{\mathbb{R}})$; but the image of $\mathcal{M}^{reg}(G_{\mathbb{R}})$ in $\mathcal{M}$ may be disjoint with $\mathcal{M}^{reg}$; this happens precisely when the real form $G_{\mathbb{R}}$ is non-quasi-split. However, for every real form $G_{\mathbb{R}}$ we show that there is a unique Dixmier sheet $S_{H}$ such that $\mathcal{M}^{reg}$ is contained in $\mathcal{M}_{S_{H}}$. We factorise $h_{\mathbb{R}}^{reg}:\mathcal{M}^{reg}(G_{\mathbb{R}})\rightarrow\mathcal{A}(G_{\mathbb{R}})$ through an abelianised fibration $h_{\mathbb{R}}^{ab}:\mathcal{M}^{ab}(G_{\mathbb{R}})\rightarrow\mathcal{A}(G_{\mathbb{R}})$, and interpret the fibres of $h_{\mathbb{R}}^{ab}$ using a $\theta$-equivariant analogue of cameral data, as in [37]. We explicitly determine the abelianised fibrations in the cases $G_{\mathbb{R}}=SU(p,q)$ for $|p-q|>1$ and $G_{\mathbb{R}}=SO^{*}(4m+2)$; the Hitchin fibres in these cases have not been previously described in the literature.

As an additional application of our local theory, we outline some results relating to the representation theory of the Lie algebra $\mathfrak{g}$. We assume that $G$ is semisimple, so that $\mathfrak{g}$ is identified with $\mathfrak{g}^{*}$ under the Killing form.

The applications relate to two variants of the orbit method [58]. The first involves the construction of Dixmier maps for sheets $\mathfrak{D}_{S}:S/G\rightarrow\mathscr{X}_{\mathfrak{g}}$ from the space of $G$-orbits of $S$ to the space $\mathscr{X}_{\mathfrak{g}}$ of primitive ideals of the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ [10] (see [24], [25] for the original setting). In order to construct $\mathfrak{D}_{S}$, one must choose a polarisation, a certain type of parabolic subalgebra associated to $S$. We prove a relationship between the polarisations for $S$ and the Katsylo group $F$, using the Grothendieck-Springer theory for sheets. Moreover, this allows us to rewrite a theorem of [9] on the asymptotic behaviour of the multiplicity function $M:\mathfrak{g}/G\times\mathbb{N}\rightarrow\mathbb{N}$ in terms of the group $F$. We note that a similar connection between the geometry of Hitchin systems and polarisations has also been observed for parabolic Higgs bundles [88], [89].

The second variant of the orbit method we consider constructs a map $\mathfrak{I}:\mathfrak{g}/G\rightarrow\mathscr{X}_{\mathfrak{g}}$ [66]. We give a formula for the multiplicity $\mu$ of the ideal $\mathfrak{I}(\mathcal{O})$ for any adjoint orbit contained in a non-singular sheet in terms of the Katsylo group $F$. If $\mathfrak{g}$ is classical, this is related to an action of $F$ on the space of one-dimensional representations for an associated finite $W$-algebra [87]. Combining this with our description for the multiplicity function gives the following relationship between these two apparently distinct notions.

I would like to thank the following people for helpful discussions and correspondence at various stages of this project: M. Bulois, M. Chaffe, A. Fernandez Herrero, S. Goodwin, J. Kimberley, I. McIntosh, B.C Ngô, J. Summerfield, L. Topley, and G. Wilkin. I would like to thank T. Pantev for his generous hospitality during a research visit to the University of Pennsylvania in February 2025. I would especially like to thank my supervisor A. Peón-Nieto for her constant support and guidance, and for providing careful comments on numerous drafts of this paper.

We will use $G$ to denote a connected reductive algebraic group over $\mathbb{C}$, and $\mathfrak{g}$ to denote its Lie algebra. Similarly, for an arbitrary algebraic group denoted by a capital letter (e.g. $A$), the lowercase gothic script (e.g. $\mathfrak{a}$) will denote its Lie algebra, unless otherwise stated. For any $g\in G$, the automorphism on $G$ defined by conjugation by $g$ will be denoted $I_{g}$.

We will always fix a Cartan subgroup $T\leq G$, and denote by $W$ the group $N_{G}(T)/T$, which for our purposes is the Weyl group of $G$ with respect to $T$. Moreover, if $L$ is a Levi subgroup of $G$, we will denote by $W_{L}$ the group $N_{G}(L)/L$, sometimes known as the relative Weyl group for $L$.

We will denote by $\Sigma$ a non-singular connected projective curve over $\mathbb{C}$ of genus $g>1$, and denote by $K$ its canonical line bundle.

We adopt the following shorthand for restriction of schemes: if $X$ is a $Y$-scheme, and $Y^{\prime}$ is a subscheme of $Y$, we will denote by $X_{Y^{\prime}}$ the fibre product $X\times_{Y}Y^{\prime}$. For a map of stacks $f:\mathcal{X}\rightarrow\mathcal{Y}$ and a $\mathbb{C}$-point $a\in\mathcal{Y}(\mathbb{C})$, we will abuse notation and write $f^{-1}(a)$ to denote the fibre product $\mathbb{C}\times_{a,f}\mathcal{X}$.

We recall the necessary constructions from the theory of sheets, as developed in [9] and [11], and two constructions of their quotients under the adjoint action, one which generalises the Chevalley restriction theorem [11] and one which generalises Kostant’s section [56]. A readable and comprehensive overview of the theory of sheets from a purely algebraic viewpoint can be found in Sections 1 and 2 of [50].

The adjoint action of $G$ on $\mathfrak{g}$ determines the centraliser group scheme $\mathcal{I}$ over $\mathfrak{g}$; the fibre $\mathcal{I}_{x}$ of $\mathcal{I}$ over a point $x$ in $\mathfrak{g}$ is the centraliser group $C_{G}(x)$. The adjoint and scaling actions on $\mathfrak{g}$ determine actions of $G$ and $\mathbb{G}_{m}$ on $\mathcal{I}$, and these actions commute.

We can use the dimension of $\mathcal{I}$ to stratify the Lie algebra $\mathfrak{g}$; we decompose $\mathfrak{g}$ as

where

In general, the strata $\mathfrak{g}_{d}$ are not irreducible (e.g. see Example 2.25); thus we have the following definition from [9].

The related notion of decomposition classes is important for the classification of sheets. We define an equivalence relation $\sim$ on $\mathfrak{g}$ where $x\sim y$ if there exists $g\in G$ such that

and $Ad_{g}(x_{n})=y_{n}$. Here $x=x_{ss}+x_{n}$ and $y=y_{ss}+y_{n}$ are the Jordan decompositions of $x$ and $y$ into semisimple and nilpotent parts.

The following class of sheets are of particular interest for our purposes.

We will now consider the quotient for the $G$-action on a sheet $S$ given in [11]. Let $(L,\mathcal{O})$ be the decomposition data for $S$, with $L$ chosen such that its maximal torus is $T$. Let $\mathfrak{z}$ be the centre of $\mathfrak{l}$, which by our choice of $L$ is contained in $\mathfrak{t}$.

Let $\overline{S}$ be the closure of $S$ in $\mathfrak{g}$. Then $\overline{S}$ is an affine variety, and the $G$-action on $S$ extends to an action on $\overline{S}$. So, in particular, we can consider the affine GIT quotient $\chi_{\overline{S}}:\overline{S}\rightarrow\overline{S}//G$, where

Since $G$ is reductive, the closed embedding $\iota:\overline{S}\hookrightarrow\mathfrak{g}$ induces a commutative diagram

where the lower horizontal arrow is a closed embedding; in (2.4), we have used the Chevalley restriction theorem to identify $\mathfrak{g}//G$ with $\mathfrak{t}/W$. Then, Borho’s generalisation of the Chevalley restriction theorem is given by the following proposition.

The situation is complicated by the fact that $\overline{S}$ may not be normal (even when $S$ is normal), e.g., see Example 2.22. Nonetheless, if the sheet itself is normal, there is a geometric quotient for $S$ compatible with Proposition 2.9.

For the regular sheet, there is a section to the Chevalley map $\chi$ defined in [61]; there is an analogous construction for an arbitrary sheet [56].

Let $e\in S$ be a nilpotent element (which always exists by [9, Korollar 3.2]), and complete it to an $\mathfrak{sl}_{2}$-triple $(e,h,f)$, i.e. $h\in\mathfrak{g}$ is semisimple, $f\in\mathfrak{g}$ is nilpotent and the following relations are satisfied:

We recall that a transverse slice at $e$ to the adjoint action on $\mathfrak{g}$ is given by the Slodowy slice [86, Section 7.4]:

The first key property of the Katsylo slice is that it is a global transverse slice for the sheet (not just a slice at $e$).

There is also a natural $\mathbb{G}_{m}$-action on $\mathfrak{K}$, which is a restriction of a $\mathbb{G}_{m}$-action on $\mathfrak{S}$. We require the following lemma, whose content is contained in [86, Sections 7.3 & 7.4]. Let

be the weight decomposition for the adjoint action of $h$ on $\mathfrak{g}$, i.e. $[h,x]=wx$ for all $x\in\mathfrak{g}^{w}$.

The second key property of the Katsylo slice is that its intersection with a given $G$-orbit in $S$ is the orbit of a group acting on the slice.

The restriction of the adjoint action of $G$ to an action of $A$ on $\mathfrak{g}$ defines an $A$-action on $\mathfrak{K}$, which commutes with the Kazhdan action.

This gives a second description for the geometric quotient for the $G$-action on a sheet $S$. Consider the finite group

the map $\Gamma\rightarrow C_{G}(e)/C_{G}^{\circ}(e)$ induced by the inclusion $A\hookrightarrow C_{G}(e)$ is an isomorphism, i.e. $\Gamma$ is the component group of $e$.

We recall the descriptions of the sheets in $\mathfrak{gl}_{n}$ and $\mathfrak{sp}_{4}$, and note how far the geometric features of the regular sheet carry over to these cases. These examples form the basis for Section 6.

We begin by considering the sheets for $G=GL_{n}$, which have much in common with the regular case. Sheets in Dynkin type A have been well-studied, e.g. see [73] and [62].

The sheets of $\mathfrak{gl}_{n}$ are particularly well-behaved and display many similarities with the regular sheet, but this is somewhat unrepresentative of the situation for general $G$. The low-rank example of $Sp_{4}$ provides a glimpse of the complications which can arise in general.

We conclude this section by stating the main theorem of [50] which guarantees smoothness for sheets in classical Lie algebras (this is not true in general, see e.g. [86, 8.11]). We make precise our terminology: by a classical Lie algebra we mean a reductive Lie algebra whose semisimple part is a sum of copies of $\mathfrak{sl}_{n}$, $\mathfrak{so}_{n}$ and $\mathfrak{sp}_{2n}$. By a classical group we mean any reductive group $G$ with classical Lie algebra $Lie(G)$. This is broader than the usual definition.

For the main body of this work, we will focus exclusively on sheets which are non-singular. This ensures that $\chi_{S}:S\rightarrow\mathfrak{c}_{S}$ is a geometric quotient (by Theorem 2.11) and for any Katsylo slice $\mathfrak{K}$ there is a canonical identification $\mathfrak{K}/\Gamma\cong\mathfrak{c}_{S}$ (by Remark 2.21).

It is well known that the centraliser is always smooth over the regular sheet (see e.g. [27]). This does not extend to the general case, but we can consider instead a finite-index subgroup scheme of the centraliser which is smooth. We will use the technology and terminology of groupoid schemes and quotients throughout this section (see Sections 1 and 2 of [57]).

Motivated by smoothness arguments in the regular case using the Kostant section (see e.g. [79, Section 3.3]), we first consider the restriction of the centraliser to a Katsylo slice. We begin by stating a simple but important lemma of [76].

In fact, if $S$ is non-singular the Katsylo slice is an affine space by Corollary 8.7 below.

Let $S$ be a non-singular sheet of $\mathfrak{g}$ and fix a choice of $\mathfrak{sl}_{2}$-triple $(e,h,f)$ with $e\in S$, using the notation of (2.6). Let $\mathfrak{K}$ be the Katsylo slice to $e$ as in Definition 2.13 and let $A$ be the reductive centraliser as in Definition 2.18. We define a group which plays a key role in the structure of the centraliser group scheme and the adjoint quotient stack.

We define the following $\mathfrak{K}$-group scheme from the $F$-action on $\mathfrak{K}$.