# Automorphisms of moduli spaces of principal bundles over a smooth curve

Roberto Fringuelli  Dipartimento di Matematica, Università di Roma “Tor
Vergata”,
Via della Ricerca Scientifica 1, I-00133 Roma, Italy,
E-mail<EMAIL_ADDRESS>
###### Abstract.

For any almost-simple group $G$ over an algebraically closed field $k$ of
characteristic zero, we describe the automorphism group of the moduli space of
semistable $G$-bundles over a connected smooth projective curve $C$ of genus
at least $4$. The result is achieved by studying the singular fibers of the
Hitchin fibration. As a byproduct, we provide a description of the irreducible
components of two natural closed subsets in the Hitchin basis: the divisor of
singular cameral curves and the divisor of singular Hitchin fibers.

## Introduction

Let $k$ be an algebraically closed field of characteristic zero. The moduli
space $\overline{\mathcal{M}}_{G}(C)$ of semistable $G$-bundles over a
connected smooth projective $k$-curve $C$ of genus $g\geq 2$, where $G$
connected reductive $k$-group, is a proper $k$-scheme. These moduli spaces
have been studied extensively, due to their connections with non-abelian Hodge
theory, conformal field theory, geometric Langlands program and, more
recently, mirror symmetry.

The automorphisms of these moduli spaces are interesting objects, especially
those ones arising from the geometry of the curve $C$ and the group $G$. For
instance, some of these automorphisms are constructed starting from
automorphisms of the group $G$ itself and their loci of fixed points may be
described as subvarieties in $\overline{\mathcal{M}}_{G}(C)$ of $G$-bundles
admitting reductions to suitable subgroups of $G$, see [AS18b], [AS18a],
[SS21], [GPR19], [GPB20]. Furthermore, the group of automorphisms of these
moduli spaces has been already described for some groups, see [KP95], [HR04],
[BGM12], [BGMn12], [AS18c].

The aim of this paper is to provide a complete description of the automorphism
group of the connected components of the moduli space
$\overline{\mathcal{M}}_{G}(C)$, where $G$ is almost-simple and $C$ has genus
at least $4$.

Before presenting the main theorem, we fix some notations. There are three
natural ways to produce automorphisms for the moduli space
$\overline{\mathcal{M}}_{G}(C)$ of semistable $G$-bundles.

  1. (a)

Pull-back along an automorphism $\sigma:C\to C$ of the curve, i.e.
$E\mapsto\sigma^{*}E.$

  2. (b)

Action of an outer automorphism $\rho\in\operatorname{Out}(G)$ of the group,
i.e. $E\mapsto\rho(E):=E\times^{\rho,G}G,$ where $E\times^{\rho,G}G$ denotes
the product $E\times G$ modulo the equivalence relation
$(eh,g)\sim(e,\rho(h)g)$, for any $e\in E$ and $g,h\in G$.

  3. (c)

Twisting by a $\mathscr{Z}(G)$-bundle $Z$ over $C$ (with
$\mathscr{Z}(G)\subset G$ center), i.e. $E\mapsto E\otimes Z,$ where $E\otimes
Z$ denotes the fibered product $E\times_{C}Z$ modulo the equivalence relation
$(et,z)\sim(e,zt)$, for any $e\in E$, $z\in Z$ and $t\in\mathscr{Z}(G)$.

For more details, see Section 5. The connected components
$\overline{\mathcal{M}}^{\delta}_{G}(C)$ of the moduli space
$\overline{\mathcal{M}}_{G}(C)$ are in bijection with the elements of the
fundamental group $\pi_{1}(G)$. In general, the automorphisms described in
(a), (b) and (c) may move these components. We are only interested in those
automorphisms, which fix a given component. When $G$ is semisimple, the
automorphisms of type (a) and (c) fix the connected components. In particular,
they descends to automorphisms of the irreducible variety
$\overline{\mathcal{M}}_{G}^{\delta}(C)$, for any $\delta\in\pi_{1}(G)$. On
the other hand, the automorphisms of $\overline{\mathcal{M}}_{G}(C)$ of type
(b), preserving the connected component
$\overline{\mathcal{M}}_{G}^{\delta}(C)$, are given by elements in the
subgroup of elements in $\operatorname{Out}(G)$ fixing $\delta\in\pi_{1}(G)$,
i.e.

$\operatorname{Out}(G,\delta):=\\{\rho\in\operatorname{Out}(G)|\pi_{1}(\rho)(\delta)=\delta\\}\subset\operatorname{Out}(G).$

The main result of this paper is the following.

######  Theorem A (= Proposition 5.7 and Theorem 7.8).

Let $C$ be a connected smooth projective $k$-curve of genus $g\geq 4$, $G$ be
an almost-simple $k$-group and $\delta\in\pi_{1}(G)$. The morphism

$\begin{array}[]{ccl}H^{1}(C,\mathscr{Z}(G))\rtimes\Big{(}\operatorname{Out}(G,\delta)\times\operatorname{Aut}(C)\Big{)}&\xrightarrow{\sim}&\operatorname{Aut}\left(\overline{\mathcal{M}}_{G}^{\delta}(C)\right)\\\
(Z,\rho,\sigma)&\mapsto&\Big{(}E\mapsto\rho(\sigma^{*}E)\otimes
Z\Big{)}\end{array}$

is an isomorphism of groups, where:

  1. (i)

$\operatorname{Aut}(C)$ acts on $H^{1}(C,\mathscr{Z}(G))$ by pull-back, i.e.
$\sigma.Z=\sigma^{*}Z$, for any $\sigma\in\operatorname{Aut}(C)$,

  2. (ii)

$\operatorname{Out}(G,\delta)$ acts on $H^{1}(C,\mathscr{Z}(G))$ as follows:
$\rho.Z=Z\times^{\rho,\mathscr{Z}(G)}\mathscr{Z}(G)$, for any
$\rho\in\operatorname{Out}(G)$.

The same holds for the open locus $\mathcal{M}^{\delta}_{G}(C)$ of stable and
simple $G$-bundles.

In particular, the automorphism group of the moduli space
$\overline{\mathcal{M}}_{G}^{\delta}(C)$ is generated by the transformations
of type (a), (b) and (c).

Using the classification of almost-simple groups, we are able to give a more
explicit description of these groups. For any $l\in\mathbb{N}$, we denote by
$\operatorname{Pic}(C)[l]$ the group of $l$-torsion line bundles over $C$.
Note that $\operatorname{Pic}(C)[l]\cong(\mathbb{Z}/l\mathbb{Z})^{2g}$ as
groups. We then have the following corollary.

######  Corollary B (= Theorems 8.1, 8.2, 8.3, 8.4 and 8.5).

Let $C$ be a connected smooth projective $k$-curve of genus $g\geq 4$. Then,
the isomorphism of Theorem A gives the following table.

$\mathfrak{g}$ | $G$ | $\delta\in\pi_{1}(G)$ | $\operatorname{Aut}(\overline{\mathcal{M}}^{\delta}_{G}(C))=\operatorname{Aut}(\mathcal{M}^{\delta}_{G}(C))$
---|---|---|---
$A_{1}$ | $\operatorname{SL}_{2}$ | $\delta\in\\{0\\}$ | $\operatorname{Pic}(C)[2]\rtimes\operatorname{Aut}(C)$
| $\operatorname{PSL}_{2}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Aut}(C)$
$A_{n-1},\;\scriptstyle{n\geq 3}$ | $\operatorname{SL}_{n}/\mu_{r},\;\scriptstyle{\text{s.t. }r|n}$ | $2\delta=0\in\mathbb{Z}/r\mathbb{Z}$ | $\operatorname{Pic}(C)[n/r]\rtimes\left(\mathbb{Z}/2\times\operatorname{Aut}(C)\right)$
| $\operatorname{SL}_{n}/\mu_{r},\;\scriptstyle{\text{s.t. }r|n}$ | $2\delta\neq 0\in\mathbb{Z}/r\mathbb{Z}$ | $\operatorname{Pic}(C)[n/r]\rtimes\operatorname{Aut}(C)$
$B_{n},\;\scriptstyle{n\geq 2}$ | $\operatorname{Spin}_{2n+1}$ | $\delta\in\\{0\\}$ | $\operatorname{Pic}(C)[2]\rtimes\operatorname{Aut}(C)$
| $\operatorname{SO}_{2n+1}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Aut}(C)$
$C_{n},\;\scriptstyle{n\geq 3}$ | $\operatorname{Sp}_{2n}$ | $\delta\in\\{0\\}$ | $\operatorname{Pic}(C)[2]\rtimes\operatorname{Aut}(C)$
| $\operatorname{PSp}_{2n}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Aut}(C)$
$D_{4}$ | $\operatorname{Spin}_{8}$ | $\delta\in\\{0\\}$ | $(\operatorname{Pic}(C)[2])^{2}\rtimes(S_{3}\times\operatorname{Aut}(C))$
| $\operatorname{SO}_{8}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Pic}(C)[2]\rtimes(\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C))$
| $\operatorname{PSO}_{8}$ | $\delta=(0,0)\in(\mathbb{Z}/2\mathbb{Z})^{2}$ | $S_{3}\times\operatorname{Aut}(C)$
| $\operatorname{PSO}_{8}$ | $\delta\neq(0,0)\in(\mathbb{Z}/2\mathbb{Z})^{2}$ | $\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C)$
$D_{2n},\;\scriptstyle{n\geq 3}$ | $\operatorname{Spin}_{4n}$ | $\delta\in\\{0\\}$ | $(\operatorname{Pic}(C)[2])^{2}\rtimes(\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C))$
| $\operatorname{SemiSpin}_{4n}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Pic}(C)[2]\rtimes\operatorname{Aut}(C)$
| $\operatorname{SO}_{4n}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Pic}(C)[2]\rtimes(\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C))$
| $\operatorname{PSO}_{4n}$ | $\delta=(0,0),(1,1)\in(\mathbb{Z}/2\mathbb{Z})^{2}$ | $\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C)$
| $\operatorname{PSO}_{4n}$ | $\delta=(1,0),(0,1)\in(\mathbb{Z}/2\mathbb{Z})^{2}$ | $\operatorname{Aut}(C)$
$D_{2n+1},\;\scriptstyle{n\geq 2}$ | $\operatorname{Spin}_{4n+2}$ | $\delta\in\\{0\\}$ | $\operatorname{Pic}(C)[4]\rtimes(\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C))$
| $\operatorname{SO}_{4n+2}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Pic}(C)[2]\rtimes(\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C))$
| $\operatorname{PSO}_{4n+2}$ | $\delta=0,2\in\mathbb{Z}/4\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C)$
| $\operatorname{PSO}_{4n+2}$ | $\delta=1,3\in\mathbb{Z}/4\mathbb{Z}$ | $\operatorname{Aut}(C)$
$E_{6}$ | $\mathbb{E}_{6}^{\operatorname{sc}}$ | $\delta\in\\{0\\}$ | $\operatorname{Pic}(C)[3]\rtimes\left(\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C)\right)$
| $\mathbb{E}_{6}^{\operatorname{ad}}$ | $\delta=0\in\mathbb{Z}/3\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}\times\operatorname{Aut}(C)$
| $\mathbb{E}_{6}^{\operatorname{ad}}$ | $\delta\neq 0\in\mathbb{Z}/3\mathbb{Z}$ | $\operatorname{Aut}(C)$
$E_{7}$ | $\mathbb{E}_{7}^{\operatorname{sc}}$ | $\delta\in\\{0\\}$ | $\operatorname{Pic}(C)[2]\rtimes\operatorname{Aut}(C)$
| $\mathbb{E}_{7}^{\operatorname{ad}}$ | $\delta\in\mathbb{Z}/2\mathbb{Z}$ | $\operatorname{Aut}(C)$
$E_{8}$ | $\mathbb{E}_{8}$ | $\delta\in\\{0\\}$ | $\operatorname{Aut}(C)$
$F_{4}$ | $\mathbb{F}_{4}$ | $\delta\in\\{0\\}$ | $\operatorname{Aut}(C)$
$G_{2}$ | $\mathbb{G}_{2}$ | $\delta\in\\{0\\}$ | $\operatorname{Aut}(C)$

We also prove a Torelly-type theorem for these moduli spaces.

######  Theorem C (= Theorem 4.3 and Corollary 4.4).

Let $C_{1}$ and $C_{2}$ be two connected smooth projective $k$-curves of genus
$g\geq 2$ and $G$ be an almost-simple $k$-group, with the only exception of
the cases where $g=2$ and $G$ is either $\operatorname{SL_{2}}$ or
$\operatorname{PGL}_{2}$, which are excluded.

If $\overline{\mathcal{M}}_{G}^{\delta_{1}}(C_{1})$ is isomorphic to
$\overline{\mathcal{M}}_{G}^{\delta_{2}}(C_{2})$, for some
$\delta_{1},\delta_{2}\in\pi_{1}(G)$, then $C_{1}$ is isomorphic to $C_{2}$.
The same holds for the locus of stable and simple bundles.

We remark that Theorem C has been already proved in greater generality by
Biswas and Hoffmann [BH12], with different methods. In particular, they showed
that the same statement holds if $G$ is a non-abelian reductive group.
Furthermore, Theorem A and Corollary B have been already proved in some cases.
When $G=\operatorname{SL}_{n}$, the computation has been carried out by
Kouvidakis and Pantev [KP95]. An alternative proof has been provided by Hwang
and Ramanan [HR04]. Later, Biswas, Gómez and Muñoz, after simplifying the
proof for $G=\operatorname{SL}_{n}$ in [BGM12], extended the result to the
symplectic group $\operatorname{Sp}_{2n}$ [BGMn12]. The cases $G$ simply-
connected of type $E_{6}$ and $F_{4}$ have been studied by Antón-Sancho
[AS18c], by adapting the Biswas-Gómez-Muñoz strategy. All the proofs rely on
the study of the singular fibers of the Hitchin fibration.

Our contribution consists in providing a unified version of Biswas-Gómez-Muñoz
proof, which holds for any almost-simple group $G$. The main difference
between our paper and the other ones consists in the study of the Hitchin
fibration. Instead of using the theory of spectral covers, we will use the
theory of cameral covers.

The main advantage of the cameral covers is that the picture is more uniform
with respect to the spectral covers, as the group varies. Just for having an
idea: it is well-known that, when
$G=\operatorname{GL}_{n},\operatorname{SL}_{n}$, a spectral curve is smooth if
and only if the corresponding Hitchin fiber is an abelian variety. For other
groups, this statement is false. For example, when $G=\operatorname{SO}_{n}$,
all the spectral curves are singular but the generic Hitchin fiber is an
abelian variety (see [Hit87, §5.15, 5.17]). Hence, the study of the relation
between spectral curves and Hitchin fibers requires a case-by-case analysis.
On the other hand, the generic cameral cover is smooth and the corresponding
Hitchin fiber is smooth, for any reductive group $G$. However, unless $G$ has
semi-simple rank $1$ (i.e. the derived subgroup $\mathscr{D}(G)\subset G$ is
equal to either $\operatorname{SL_{2}}$ or $\operatorname{PGL_{2}}$), there
are singular cameral covers, whose corresponding Hitchin fiber is an abelian
variety. Despite this complication, the theory of cameral covers allows us to
provide a uniform description of the locus in the Hitchin basis corresponding
to abelian Hitchin fibers, for any reductive group $G$, avoiding the case-by-
case analysis. For more details, see Section 3 and references therein.

We now briefly explains the proof of Theorem A. We only treat the
surjectivity, which is the most delicate part. We start with an isomorphism of
the moduli space

$\varphi:\overline{\mathcal{M}}_{G}^{\delta}(C)\xrightarrow{\sim}\overline{\mathcal{M}}_{G}^{\delta}(C).$

The codifferential $d\varphi^{*}$ gives a $\mathbb{G}_{m}$-equivariant
birational morphism of the moduli space $\mathcal{H}^{\delta}_{G}(C)$ of
semistable $G$-Higgs bundles, where $\mathbb{G}_{m}$ acts on
$\mathcal{H}^{\delta}_{G}(C)$ by scaling the Higgs field. Furthermore, the
morphism $d\varphi^{*}$ sits in a commutative diagram of
$\mathbb{G}_{m}$-equivariant morphisms of the following shape

$\textstyle{\mathcal{H}^{\delta}_{G}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{d\varphi^{*}}$$\textstyle{\mathcal{H}^{\delta}_{G}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\mathbb{A}}}$$\textstyle{\mathbb{A}}$

where $h$ is the Hitchin fibration and $\mathbb{G}_{m}$ acts on the Hitchin
basis $\mathbb{A}=\oplus_{i=1}^{\operatorname{rank}(G)}\mathbb{A}_{i}$ with
weight $d_{i}$ on the subspace $\mathbb{A}_{i}:=H^{0}(C,\omega^{d_{i}})$ (in
the paper $\omega$ is the cotangent bundle of the curve). In particular, the
isomorphism $\varphi_{\mathbb{A}}$ preserves:

  1. (i)

the eigenspace $\mathbb{A}_{\max}$ in $\mathbb{A}$ of highest weight (i.e.
$\max$ is the integer such that $d_{\max}$ is the highest number among
$d_{1},\ldots,d_{\operatorname{rank}(G)}$),

  2. (ii)

the closed subset $\mathbb{D}^{\operatorname*{ab}}$ of $\mathbb{A}$, where the
fibers of the Hitchin fibration $h$ are not abelian varieties.

The projectivization
$\mathbb{P}(\mathbb{D}^{\operatorname*{ab}}\cap\mathbb{A}_{\max})$ of the
intersection between these two loci is equal to the dual variety of $C$ with
respect to the linear system $\mathbb{A}_{\max}$. Since the curve is equal to
its double dual, the isomorphism $\varphi_{\mathbb{A}}$, restricted to the
intersection $\mathbb{D}^{\operatorname*{ab}}\cap\mathbb{A}_{\max}$, produces
an isomorphism $\sigma$ of the curve $C$. Using this fact, after some work, we
are able to give a more explicit description of $\varphi_{\mathbb{A}}$. In
particular, we get:

$\varphi_{\mathbb{A}}\left(\bigoplus_{i=1}^{r}H^{0}(C,\omega^{d_{i}}_{C}(-\sigma(p))\right)=\bigoplus_{i=1}^{r}H^{0}(C,\omega^{d_{i}}_{C}(-p)),\text{
for any }p\in C.$

From this, we are able to show that for a generic $G$-bundle $E$ with image
$\varphi(E)$, there exists a canonical isomorphism of adjoint bundles

$\operatorname*{ad}(E)\xrightarrow[\Psi]{\sim}\operatorname*{ad}(\varphi(E))\Longrightarrow\mathcal{N}_{E}\xrightarrow[\Psi]{\sim}\mathcal{N}_{\varphi(E)},$

which preserves the subsets $\mathcal{N}_{E}$ and $\mathcal{N}_{\varphi(E)}$
of nilpotent elements of $\operatorname*{ad}(E)$ and
$\operatorname*{ad}(\varphi(E))$, respectively. Using the Springer resolution
of the nilpotent cone in the Lie algebra $\mathfrak{g}$, we then show that
$\Psi$ induces an isomorphism of $G/B$-fibrations over $C$

$E/B\xrightarrow[\widehat{\Psi}]{\sim}\varphi(E)/B,\quad\text{ where }B\text{
Borel subgroup of }G.$

Thanks to a result of Demazure, there exists a canonical isomorphism between
the relative tangent bundle of the $G/B$-fibration $E/B$ (resp.
$E/\varphi(E)$) and the adjoint bundle $\operatorname*{ad}(E)$ (resp.
$\operatorname*{ad}(\varphi(E))$) and it preserves the Lie-brackets. Hence,
the (relative) differential

$\operatorname*{ad}(E)=\mathcal{T}_{(E/B)/C}\xrightarrow[d\widehat{\Psi}]{\sim}\mathcal{T}_{(\varphi(E)/B)/C}=\operatorname*{ad}(\varphi(E))$

is an isomorphism of Lie algebra bundles. If this happens the $G$-bundles $E$
and $\varphi(E)$ differ by a combination of the transformations of type (a),
(b) and (c). Since $E$ was chosen generic and it sits in the same connected
component of $\overline{\mathcal{M}}_{G}(C)$ containing $\varphi(E)$, the
isomorphism $\varphi$ must be in the image of the homomorphism described in
Theorem A.

Further generalizations. There are (at least) four ways of generalizing this
problem. (1) What happens for a general reductive group? In this generality,
the picture is more complicated. As an example, when $G=G_{1}\times G_{2}$ is
the product of two almost-simple groups, the automorphism group of
$\overline{\mathcal{M}}_{G}(C)$ should contain two copies of
$\operatorname{Aut}(C)$. Note that case $G=\operatorname{GL}_{n}$ has been
computed in [KP95]. (2) Another version of the problem would be to study the
automorphism group for the so-called twisted moduli spaces of $G$-bundles
(e.g. moduli spaces of vector bundles of fixed rank with determinant
isomorphic to a fixed, non-trivial, line bundle). For instance, the twisted
case has been treated in [KP95] and [BGM12] for $G=\operatorname{SL}_{n}$ and
in [BGMn12] for $G=\operatorname{Sp}_{2n}$. (3) Another generalization is the
case of parabolic (or, more generally, parahoric) $G$-bundles. When
$G=\operatorname{GL}_{n},\operatorname{SL}_{n}$, the problem has been studied
in [BHK10], [AFKM19], [AG21]. (4) An alternative path could be the study of
the automorphisms of the moduli stack $\mathbf{Bun}_{G}$ of all $G$-bundles.
In this case, the automorphisms form a category (more precisely a $2$-group).
We hope to come back on these problems in future works.

Outline of the paper. In Section 1, we collect some results about reductive
groups and their adjoint representations on their own Lie algebras. In Section
2, we introduce the moduli stacks of (Higgs) bundles on curves and we recall
their main properties, following the works of Faltings [Fal93] and Ngô
[Ngo10]. In section 3, we give a description of the closed subset
$\mathbb{D}^{\diamondsuit}$, resp. $\mathbb{D}^{\operatorname*{ab}}$, of the
Hitchin basis $\mathbb{A}$ parameterizing singular cameral curves, resp. non-
abelian Hitchin fibers. Then, we derive some consequences on the Hitchin
discriminant. In particular, we show that generic Hitchin fiber is either an
abelian variety or birational to a $\mathbb{P}^{1}$-fibration over an abelian
variety. In section 4, we prove the Torelli Theorem (= Theorem C) for
$G$-bundles. In Section 5, we show that the homomorphism of Theorem A is well-
defined and injective. In Section 6, we recall the necessary facts about the
nilpotent cone of a Lie algebra and the tangent bundle of the flag variety.
The surjectivity of the homomorphism of Theorem A is proved in Section 7. In
Section 8, we compute the explicit presentations of the automorphism group
$\operatorname{Aut}(\overline{\mathcal{M}}_{G}(C))$ contained in Corollary B.

######  Notations 1.

In the paper, $k$ will be always an algebraically closed field of
characteristic zero. We usually denote by $G$ a (connected) reductive group
over $k$, by $T_{G}$ a maximal torus of $G$ and by $B_{G}$ a Borel subgroup
containing $T_{G}$. We denote by $\mathscr{Z}(G)$ the center of $G$, by
$\mathscr{D}(G)$ the derived subgroup of $G$, and by $G^{\operatorname{ad}}$
the adjoint group $G/\mathscr{Z}(G)$. As usual, we denote by
$W_{G}:=N(T_{G})/T_{G}$ its Weyl group. The rank of $G$ (i.e. $\dim T_{G}$)
will be denoted by the letter $r$.

######  Notations 2.

We denote by $\mathfrak{g}$, $\mathfrak{b}_{G}$ and $\mathfrak{t}_{G}$ the Lie
algebras of $G$, $B_{G}$ and $T_{G}$, respectively. The set of roots (resp.
coroots) will be denoted by $\Phi_{G}$ (resp. $\Phi^{\vee}_{G}$). For any root
$\alpha$, we denote by $\mathfrak{g}_{\alpha}$ the 1-dimensional sublinear
space of $\mathfrak{g}$, where the torus $T_{G}$ acts with weight $\alpha$.
For any root $\alpha:T_{G}\to\mathbb{G}_{m}$, we denote by
$d\alpha:\mathfrak{t}_{G}\to k$ its Lie-algebra counterpart. We denote by
$\mathfrak{g}^{\operatorname*{reg}}$ the subset of regular elements of
$\mathfrak{g}$, i.e. those elements $x\in\mathfrak{g}$ such that the kernel of
the adjoint homomorphism $[x,-]:\mathfrak{g}\to\mathfrak{g}$ has (minimal)
dimension equal to $r$.

######  Notations 3.

As usual
$\pi_{1}(G):=\operatorname{Hom}(\mathbb{G}_{m},T_{G})/\langle\Phi^{\vee}\rangle$
denote the fundamental group of $G$, which is defined as the quotient of the
cocharacter lattice of $T_{G}$ by the sublattice $\langle\Phi^{\vee}\rangle$
generated by integral linear combinations of coroots.

When it is clear from the context, we will remove the reference to the group
$G$ from the notation (e.g. $T$ instead of $T_{G}$, $\mathfrak{t}$ instead of
$\mathfrak{t}_{G}$, $\Phi$ instead of $\Phi_{G}$, etc).

######  Notations 4.

All the schemes and stacks will be supposed over $k$. We usually denote by $C$
a (connected) smooth projective curve over $k$, by $g:=g(C)$ its geometric
genus and by $\omega_{C}$ (or $\omega$, when it is clear from the context) the
cotangent bundle of $C$. A $G$-bundle over $C$ is a $G$-torsor over $C$, where
$G$ acts on the right. Given a $G$-bundle $E$ over $C$ and a group-homorphism
$\phi:G\to H$, we denote by

$E\times^{\phi,G}H:=(E\times H)/\sim$

the associated $H$-bundle over $C$, where $\sim$ is the equivalence relation
$(eg,h)\sim(e,\phi(g)h)$. Similarly, given a $G$-bundle $E$ over $C$ and a
group-homorphism $\phi:G\to\operatorname{GL}(V)$, we denote by

$E\times^{\phi,G}V:=(E\times V)/\sim$

the associated vector bundle over $C$, where $\sim$ is the equivalence
relation $(eg,v)\sim(e,\phi(g)v)$. When it is clear from the context, we
remove the reference to the homomorphism $\phi$ from the notation (e.g.
$E\times^{G}H$ instead of $E\times^{\phi,G}H$).

##  1\. Lie algebras.

In this section, we collect some facts about the adjoint action of a reductive
group $G$ on its Lie algebra $\mathfrak{g}$.

######  Theorem 1.1.

Let $G$ be a reductive group over $k$. The following facts are true.

  1. (i)

The inclusion $\mathfrak{t}\subset\mathfrak{g}$ of Lie algebras induces an
isomorphism of rings $k[\mathfrak{g}]^{G}\cong k[\mathfrak{t}]^{W}$, where $G$
acts on $\mathfrak{g}$ via the adjoint representation.

  2. (ii)

The ring $k[\mathfrak{g}]^{G}$, resp. $k[\mathfrak{t}]^{W}$, is freely
generated by homogeneous polynomials $p_{1},\ldots,p_{r}$ in
$k[\mathfrak{g}]$, resp. in $k[\mathfrak{t}]$, of degrees
$d_{1},\ldots,d_{r}$. Moreover, the degrees do not depend on the choice of the
polynomials.

  3. (iii)

The morphism of affine varieties

$\begin{array}[]{cccl}\chi:&\mathfrak{g}&\longrightarrow&\mathfrak{c}:=\operatorname{Spec}(k[\mathfrak{t}]^{W})\\\
&x&\longmapsto&(p_{1}(x),\ldots,p_{r}(x))\end{array}$

is $\mathbb{G}_{m}$-equivariant, where $\mathbb{G}_{m}$ acts on $\mathfrak{g}$
with weight $1$ and it acts on $\mathfrak{c}$ as follows:
$t.(u_{1},\ldots,u_{r}):=(t^{d_{1}}u_{1},\ldots,t^{d_{r}}u_{r})$.

  4. (iv)

The morphism of affine varieties

$\begin{array}[]{cccl}\pi:&\mathfrak{t}&\longrightarrow&\mathfrak{c}\\\
&t&\longmapsto&(p_{1}(t),\ldots,p_{r}(t))\end{array}$

is a Galois cover with Galois group $W$. The discriminant (or branch divisor)
$\mathfrak{D}$ is given by the zeroes of the $W$-invariant function
$\prod_{\alpha\in\Phi}d\alpha.$

  5. (v)

For any simple root $\alpha$, we fix a non-zero vector
$x_{\alpha}\in\mathfrak{g}_{\alpha}$, we set $x_{+}:=\sum_{\alpha\text{ simple
}}x_{\alpha}$. Then, there exists a unique $\mathfrak{s}l_{2}$-triple
$\\{x_{-},h,x_{+}\\}$ in $\mathfrak{g}$, with $h\in\mathfrak{t}$.

  6. (vi)

With the notations of Point $(\ref{T:konstant5})$. Let $\mathfrak{g}^{x_{+}}$
be the Lie algebra of the centraliser of $x_{+}$. Then, the affine subspace
$x_{-}+\mathfrak{g}^{x_{+}}\subset\mathfrak{g}$ is contained in the open
subset $\mathfrak{g}^{\operatorname*{reg}}$. Furthermore, the restriction of
$\chi$ to this subspace

$x_{-}+\mathfrak{g}^{x_{+}}\to\mathfrak{c}$

is an isomorphism. Its inverse
$\epsilon:\mathfrak{c}\to\mathfrak{g}^{\operatorname*{reg}}$ is called
_Kostant section_.

  7. (vii)

Let $P\subset G$ be a parabolic subgroup with Levi subgroup $M$. Let
$\mathfrak{p}\subset\mathfrak{g}$ be the corresponding parabolic subalgebra
and $\mathfrak{m}$ be its Levi subalgebra. Then the inclusions
$\mathfrak{t}\subset\mathfrak{m}\subset\mathfrak{p}$ induce the isomorphisms
of rings $k[\mathfrak{p}]^{P}\cong k[\mathfrak{m}]^{M}\cong
k[\mathfrak{t}]^{W_{M}}$, where $W_{M}$ is the Weyl group of $M$.

###### Proof.

All the facts are proved in [Kos63] (see also [Ngo06, Theorem 2.1]), with the
only exception of the isomorphism $k[\mathfrak{p}]^{P}\cong
k[\mathfrak{m}]^{M}$, which has been proved in [CL12, Lemme 2.7.1]. ∎

The Weil group $W$ acts on the set of roots $\Phi$. We denote by
$\Phi_{1},\ldots,\Phi_{m}$ the sets of $W$-orbits in $\Phi$. This will give us
the irreducible decomposition of the discriminant

$\mathfrak{D}=\mathfrak{D}_{1}\cup\cdots\cup\mathfrak{D}_{m},$

where $\mathfrak{D}_{i}$ is the zero-locus of the $W$-invariant function
$\prod_{\alpha\in\Phi_{i}}d\alpha$. Furthermore, the singular locus
$\mathfrak{D}^{\operatorname*{sing}}$ is the zero loci of the polynomial
system

$\begin{cases}\prod_{\alpha\in\Phi}d\alpha=0,\\\
\sum_{\alpha\in\Phi}\left(\prod_{\beta\in\Phi\setminus\\{\alpha\\}}d\beta\right)=0.\end{cases}$

In other words, it may be described as the image of points in $\mathfrak{t}$
contained in at least two distinct hyperplane-roots. In particular, it is a
union of irreducible closed subsets in $\mathfrak{c}$ of dimension $r-2$. We
denote these components by
$\mathfrak{D}^{\operatorname*{sing}}_{1},\ldots,\mathfrak{D}^{\operatorname*{sing}}_{n}$.
It follows from the discussion that the irreducible components of
$\mathfrak{D}^{\operatorname*{sing}}$ are in bijection with orbits of the
diagonal action of the Weil group $W$ on the product $\Phi\times\Phi$.

######  Remark 1.2.

Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$. Here, we collect
some facts about the degrees $d_{1},\ldots,d_{r}$ introduced in Theorem 1.1.

  1. (i)

$G$ is semisimple if and only if all the degrees $d_{1},\ldots,d_{r}$ are
greater or equal than $2$.

  2. (ii)

If $G$ almost-simple, the highest degree among $d_{1},\ldots,d_{r}$ is the so-
called Coxeter number $h$, which is equal to the number of roots divided by
the rank of the maximal torus, i.e. $h:=\frac{|\Phi|}{r}$.

  3. (iii)

$G=T$ is abelian if and only if all the degrees are equal to $1$. Furthermore,
we have $\pi_{G}=\operatorname{Id}:\mathfrak{t}\to\mathfrak{t}$.

######  Remark 1.3.

In general, a reductive group $G$ admits a canonical isogeny
$\mathscr{Z}(G)^{o}\times\mathscr{D}(G)\to G$, where
$\mathscr{Z}(G)^{o}\subset G$ is the neutral component of the center
$\mathscr{Z}(G)$ and $\mathscr{D}(G)\subset G$ is the derived subgroup. Then,
we have the a commutative diagram of affine schemes

(1.1)
$\textstyle{\mathfrak{t}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G}}$
$\textstyle{\mathfrak{c}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{t}_{\mathscr{D}(G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{Id},\pi_{\mathscr{D}(G)})}$
$\textstyle{\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{c}_{\mathscr{D}(G)}}$

where the horizontal rows are the morphisms in Theorem
1.1$(\ref{T:konstant4})$.

######  Remark 1.4.

Assume $\mathfrak{g}$ simple. If we order the degrees such that
$d_{1}\leq\ldots\leq d_{r}$. We have the following equality

$\prod_{\alpha\in\Phi}d\alpha\operatorname{mod}\big{(}p_{1},\ldots,p_{r-1}\big{)}\equiv
cp_{r}^{r}\operatorname{mod}\big{(}p_{1},\ldots,p_{r-1}\big{)}$

where $c\in k\setminus\\{0\\}$. Indeed, observe that there exists an
isomorphism of rings

$k[\mathfrak{t}]\cong k[d\alpha_{1},\ldots,d\alpha_{r}],$

where $\alpha_{1},\ldots,\alpha_{r}$ are the simple roots. The polynomial
$\prod_{\alpha\in\Phi}d\alpha$ is an homogeneous polynomial of degree $|\Phi|$
in $k[\mathfrak{t}]$. On the other hand, $p_{r}$ is an homogeneous polynomial
in $k[\mathfrak{t}]$ of degree $|\Phi|/r$ (see Remark 1.2). Hence we must
have,

$\prod_{\alpha\in\Phi}d\alpha\operatorname{mod}\big{(}p_{1},\ldots,p_{r-1}\big{)}\equiv
cp_{r}^{r}\operatorname{mod}\big{(}p_{1},\ldots,p_{r-1}\big{)},$

for some $c\in k$. We will prove $c\neq 0$ by contradiction. Assume that
$c=0$. Then, any element $t\in\mathfrak{t}$ such that
$\pi(t)=(0,\ldots,0,a)\in\mathfrak{c}$, for some $a\in k$, is contained in
some hyperplane-root $d\alpha:\mathfrak{t}\to k$. In particular, for any $t$
of this shape, the kernel of the adjoint homomorphism
$[t,-]:\mathfrak{g}\to\mathfrak{g}$ contains the sublinear space
$\mathfrak{g}_{\alpha}\oplus\mathfrak{t}\oplus\mathfrak{g}_{-\alpha}$. In
particular, the kernel of $[t,-]$ have dimension at least $r+2$, i.e. $t$ is
not regular. However, if $a\neq 0$, the element $t$ must be cyclic (then
regular), see [Kos59, Corollary 9.2].

##  2\. Hitchin fibration.

Here, we recall some fundamental facts about (Higgs) bundles on curves and
their moduli stacks following the works of Faltings [Fal93] and Ngô [Ngo10].
Unless otherwise stated, in this section, $G$ is a (connected) reductive group
and $C$ is a smooth projective curve of genus $g\geq 2$.

###  2.1. The moduli stack $\mathbf{Higgs}_{G}$ of $G$-Higgs bundles.

For any $G$-bundle $E$ over $C$, we denote by
$\operatorname*{ad}(E):=E\times^{\operatorname{Ad},G}\mathfrak{g}$ the adjoint
bundle on $C$ of attached to $E$ via the adjoint representation
$\text{Ad}:G\to\operatorname{GL}(\mathfrak{g})$.

######  Definition 2.1.

A _$G$ -Higgs bundle_ is a pair $(E,\theta)$, where $E$ is a $G$-bundle over
$C$ and $\theta$ is a global section of the vector bundle
$\operatorname*{ad}(E)\otimes\omega$. A _regular $G$-Higgs bundle_ is a
$G$-Higgs bundle $(E,\theta)$ such that, for any point $p\in C$, the kernel of
the linear morphism of vector spaces

$[\theta_{p},-]:\operatorname*{ad}(E_{p})\to\operatorname*{ad}(E_{p})\otimes\omega_{p}$

have dimension $r$.

We denote by $\mathbf{Higgs}_{G}$ the moduli stack of $G$-Higgs bundles over
$C$ and by $\mathbf{Bun}_{G}$ the moduli stack of $G$-bundles over $C$. They
are algebraic stacks locally of finite type over $k$. They are related by the
forgetful morphism

$\mathbf{Higgs}_{G}\to\mathbf{Bun}_{G}:(E,\theta)\mapsto E.$

Consider the affine space $\mathbb{A}$ of sections of the vector bundle
$\mathfrak{c}_{\omega}:=\mathfrak{c}\times^{\mathbb{G}_{m}}\omega^{*}$, where
$\omega^{*}:=\omega\setminus\\{\operatorname{zero-section}\\}$ is the
$\mathbb{G}_{m}$-bundle attached to $\omega$. After fixing a set of generators
of $k[\mathfrak{g}]^{G}=k[\mathfrak{t}]^{W}$, we get the
$\mathbb{G}_{m}$-equivariant isomorphisms

$\mathfrak{c}_{\omega}\cong\omega^{d_{1}}\oplus\ldots\oplus\omega^{d_{r}}\quad\text{
and }\quad\mathbb{A}\cong H^{0}(C,\omega^{d_{1}})\oplus\ldots\oplus
H^{0}(C,\omega^{d_{r}}).$

For the rest of the paper, we will fix a such set of free generators, we also
assume that $d_{1}\leq\ldots\leq d_{r}$ and we set
$\mathbb{A}_{i}:=H^{0}(C,\omega^{d_{i}})$. From this decomposition, we deduce
that:

$\displaystyle\dim\mathbb{A}$
$\displaystyle=\sum_{i=1}^{r}d_{i}(2g-2)+r(1-g)+\\#\\{i|d_{i}=1\\}=$
$\displaystyle=\sum_{i=1}^{r}d_{i}(2g-2)+r(1-g)+\dim\mathscr{Z}(G)=$
$\displaystyle=\left(r+\frac{|\Phi|}{2}\right)(2g-2)+r(1-g)+\dim\mathscr{Z}(G)=$
$\displaystyle=\dim G(g-1)+\dim\mathscr{Z}(G).$

Indeed, the first two equalities follows by using Riemann-Roch and Remarks 1.2
and 1.3. We recall that the integers $d_{1},\ldots,d_{r}$ are the degrees of a
set of free generators for the ring of $W$-invariants $k[\mathfrak{t}]^{W}$.
Since the Weyl group $W$ is generated by $|\Phi|/2$ reflections, we have
$\sum_{i=1}^{r}(d_{i}-1)=|\Phi|/2$ (see [ST54]) and, so, the third equality
follows immediately. The last equality follows because $\dim
G=\dim\mathfrak{g}=r+|\Phi|$.

We call _Hitchin morphism_ the following map:

$\begin{array}[]{cccl}h:&\mathbf{Higgs}_{G}&\longrightarrow&\mathbb{A}\\\
&(E,\theta)&\longmapsto&\left(p_{1}(\theta),\ldots,p_{r}(\theta)\right).\end{array}$

Given a section $a\in\mathbb{A}$, we call _Hitchin fiber over $a$_ the inverse
image $\mathbf{Higgs}_{G}(a):=h^{-1}(a)$. The choice of a theta-characteristic
$\kappa$ (i.e. a line bundle $\kappa$ such that $\kappa^{2}=\omega$) and the
Kostant section (see Theorem 1.1(v)) gives a section of the Hitchin morphism

(2.1) $\epsilon_{\kappa}:\mathbb{A}\to\mathbf{Higgs}_{G},$

called the _Hitchin-Kostant section_ , for more details see [Ngo10, Lemma
2.2.5]. The image of $\epsilon_{\kappa}$ is contained in the locus
$\mathbf{Higgs}_{G}^{\operatorname*{reg}}$ of regular $G$-Higgs bundles, see
[Ngo10, §4.3.4]. In particular, we get a family of regular $G$-Higgs bundles
over $\mathbb{A}$. For any $a\in\mathbb{A}$, we denote by
$(E_{a},\theta_{a}):=\epsilon_{\kappa}(a)$, the corresponding regular
$G$-Higgs bundle given by the image of the Hitchin-Kostant section.

We conclude the subsection with some basic facts about these moduli stacks.
These facts are probably well-known to the experts. However, we did not find a
reference in this generality. For this reason, we added a sketch of the proof.

######  Proposition 2.2.

The following facts hold.

  1. (i)

The connected components of the moduli stacks $\mathbf{Higgs}_{G}$ and
$\mathbf{Bun}_{G}$ are in bijection with $\pi_{1}(G)$.

  2. (ii)

For any $\delta\in\pi_{1}(G)$, the corresponding connected component
$\mathbf{Bun}_{G}^{\delta}$ of the moduli stack $\mathbf{Bun}_{G}$ is a smooth
algebraic stack locally of finite type over $k$. Furthermore, it has dimension
$\dim G(g-1)$.

  3. (iii)

For any $\delta\in\pi_{1}(G)$, the corresponding connected component
$\mathbf{Higgs}_{G}^{\delta}$ of the moduli stack $\mathbf{Higgs}_{G}$ is an
irreducible algebraic stack locally of finite type over $k$. Furthermore, it
is a locally complete intersection of pure dimension $2\dim
G(g-1)+\dim\mathscr{Z}(G)$.

  4. (iv)

The Hitchin morphism $h^{\delta}:\mathbf{Higgs}_{G}^{\delta}\to\mathbb{A}$ is
faithfully flat with geometrically connected fibers of pure dimension $\dim
G(g-1)$.

###### Proof.

$(\ref{P:hit-prop2})$ The moduli stack $\mathbf{Bun}_{G}$ is locally of finite
type over $k$ by [Beh90, Proposition 4.4.5] and smooth by [Beh90, Proposition
4.5.1]. For the dimension see [Beh90, Corollary 8.1.9].
$(\ref{P:hit-prop1})$+$(\ref{P:hit-prop3})$ The statement of $(i)$ for
$\mathbf{Bun}_{G}$ can be found in [Hof10]. We now focus on the moduli stack
$\mathbf{Higgs}_{G}$. When $G$ is semi-simple, a proof can be found in [BD91].
More precisely, by [BD91, Proposition 1.1.1], the moduli stack
$\mathbf{Bun}_{G}$ is _very good_ , i.e. for any integer $n>0$, we have that
the substack

(2.2)
$\\{E\in\mathbf{Bun}_{G}|\dim\operatorname{Aut}(E)=n\\}=\\{E\in\mathbf{Bun}_{G}|,\dim
H^{1}(C,\operatorname*{ad}(E)\otimes\omega)=n\\}$

has codimension strictly greater than $n$ in $\mathbf{Bun}_{G}$. We remark
that the two sets above coincide by Serre duality and by the equality $\dim
H^{0}(C,\operatorname*{ad}(E))=\dim\operatorname{Aut}(E)$. The property of
being very good implies that the forgetful morphism
$\mathbf{Higgs}_{G}\to\mathbf{Bun}_{G}$ induces a bijection of
irreducible/connected components and also that $\mathbf{Higgs}_{G}$ is an
algebraic stack locally of finite type and a locally complete intersection of
pure dimension $2\dim G(g-1)$, see [BD91, Section 1]. Now, we assume $G$
reductive. Observe that the moduli stack $\mathbf{Higgs}_{G}$ sits in the
following cartesian diagram

(2.3)
$\textstyle{\mathbf{Higgs}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{Higgs}_{G^{\operatorname{ad}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(C,\mathfrak{t}_{\mathscr{Z}(G)}\otimes\omega)\times\mathbf{Bun}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{Bun}_{G^{\operatorname{ad}}},}$

where the horizontal map on the bottom sends the pair $(s,E)$ to the
$G^{\operatorname*{ad}}$-bundle
$E\times^{G}G^{\operatorname*{ad}}=E/\mathscr{Z}(G)$. By [Hof10, Corollary
4.2], the horizontal arrows are smooth of relative dimension

$\displaystyle\dim
H^{0}(C,\mathfrak{t}_{\mathscr{Z}(G)}\otimes\omega)+\dim\mathbf{Bun}_{\mathscr{Z}(G)}$
$\displaystyle=\dim\mathscr{Z}(G)g+\dim\mathscr{Z}(G)(g-1)=$
$\displaystyle=\dim\mathscr{Z}(G)(2g-1).$

Since $G^{\operatorname*{ad}}$ is semi-simple, we have the equalities

$\displaystyle\dim\mathbf{Higgs}_{G}$
$\displaystyle=\dim\mathbf{Higgs}_{G^{\operatorname*{ad}}}+\dim\mathscr{Z}(G)(2g-1)=$
$\displaystyle=2\dim G^{\operatorname*{ad}}(g-1)+\dim\mathscr{Z}(G)(2g-1)=$
$\displaystyle=2\dim G(g-1)+\dim\mathscr{Z}(G).$

Furthermore, the smoothness of the horizontal maps implies that
$\mathbf{Higgs}_{G}$ is a locally complete intersection of pure dimension
$2\dim G(g-1)+\dim\mathscr{Z}(G)(g-1)$. Now we compute the
irreducible/connected components of $\mathbf{Higgs}_{G}$. From the semi-simple
case (see (2.2)), we deduce that, for any $n>0$, the substack

$\mathbf{Bun}_{G,=n}:=\\{E\in\mathbf{Bun}_{G}|\,H^{1}(\operatorname*{ad}(E)\otimes\omega)=n+\dim\mathscr{Z}(G)\\}\subset\mathbf{Bun}_{G}$

has codimension strictly greater than $n$ in $\mathbf{Bun}_{G}$. From this
fact, we deduce that the locus $\mathbf{Higgs}_{G}^{o}$ of $G$-Higgs bundles
such that $\dim
H^{1}(C,\operatorname*{ad}(E)\otimes\omega)=\dim\mathscr{Z}(G)$ (i.e. of
minimal dimension) is dense in $\mathbf{Higgs}_{G}$. We proceed by
contradiction. Assume that $\mathbf{Higgs}_{G}^{o}$ is not dense in
$\mathbf{Higgs}_{G}$. Then, there exists an irreducible component of
$\mathbf{Higgs}_{G}$ with an open subset $\mathbf{Higgs}_{G}^{\\#}$ mapping
dominantly onto the substack $\mathbf{Bun}_{G,=n}$, for some $n>0$. In
particular, there exists a $G$-bundle $E$ in $\mathbf{Bun}_{=n}$ such that

$\displaystyle\dim\mathbf{Higgs}_{G}^{*}$
$\displaystyle=\dim\mathbf{Bun}_{G,=n}+\dim
H^{0}(C,\operatorname*{ad}(E)\otimes\omega)<$
$\displaystyle<(-n+\dim\mathbf{Bun}_{G})+\chi(\operatorname*{ad}(E)\otimes\omega)+\dim
H^{1}(C,\operatorname*{ad}(E)\otimes\omega)=$
$\displaystyle=(-n+\dim\mathbf{Bun}_{G})+\dim G(g-1)+(n+\dim\mathscr{Z}(G))=$
$\displaystyle=2\dim G(g-1)+\dim\mathscr{Z}(G)=\dim\mathbf{Higgs}_{G}.$

Since all the irreducible components of $\mathbf{Higgs}_{G}$ have the same
dimension, we get a contradiction. Hence, the connected components of
$\mathbf{Higgs}_{G}$ are in bijection with those of $\mathbf{Higgs}_{G}^{o}$.
The point $(i)$ follows by observing that the forgetful morphism
$\mathbf{Higgs}_{G}^{o}\to\mathbf{Bun}_{G}$ is a vector bundle. This also show
that all the connected components are irreducible, concluding the proof of the
point $(iii)$.
$(\ref{P:hit-prop4})$ When $G$ semi-simple, a proof of the assertion about the
flatness and the dimension of the fibers can be found in [BD91, Theorem
2.2.4(iii)]. When $G$ is reductive, the assertion can be deduced from the
observation that the statement holds for the adjoint (and semi-simple) group
$G^{\operatorname*{ad}}$, together with the cartesian diagram (2.3). It
remains to show that the fibers are connected. As consequence of [Fal93,
Corollary III.3] (see also Subsection 3.4), there exists an open and dense
substack
$\mathbf{Higgs}_{G}^{\delta,\operatorname{ss}}\subset\mathbf{Higgs}^{\delta}_{G}$
such that the restriction
$h^{\delta,\operatorname{ss}}:\mathbf{Higgs}_{G}^{\delta,\operatorname{ss}}\to\mathbb{A}$
of the Hitchin morphism induces an isomorphism of structure sheaves
$\mathcal{O}_{\mathbb{A}}\to
h^{\delta,\operatorname{ss}}_{*}\mathcal{O}_{\mathbf{Higgs}_{G}^{\delta,\operatorname{ss}}}$.
Note that this homomorphism factors through
$h_{*}\mathcal{O}_{\mathbf{Higgs}_{G}^{\delta}}$. In particular, the Hitchin
morphism induces an isomorphisms of structure sheaves
$\mathcal{O}_{\mathbb{A}}\to
h^{\delta}_{*}\mathcal{O}_{\mathbf{Higgs}_{G}^{\delta}}$. This implies that
all the Hitchin fibers must be connected. ∎

###  2.2. Cameral covers.

To any element $a=(a_{1},\ldots,a_{r})\in\mathbb{A}$, the corresponding
_cameral curve_ $C_{a}$ is the curve in the total space of the vector bundle
$\mathfrak{t}_{\omega}:=\mathfrak{t}\times\omega$ defined by the equations

$\begin{cases}p_{1}(t)=a_{1}(x),\\\ \vdots\\\ p_{r}(t)=a_{r}(x),\end{cases}$

where $x$ is a (local) coordinate for $C$ and $t$ are the tautological
coordinates $(t_{1}\otimes dx,\ldots,t_{r}\otimes dx)$ along the fibers of the
projection $\mathfrak{t}_{\omega}\to C$. In other words, the cameral curve
sits in the following cartesian diagram

(2.4)
$\textstyle{C_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{a}}$$\textstyle{\mathfrak{t}_{\omega}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathfrak{c}_{\omega}.}$

In the paper, we use the same symbol $\pi$ for the morphism
$\mathfrak{t}\to\mathfrak{c}$ in Theorem 1.1$(\ref{T:konstant4})$ and its
$\omega$-twisted version $\pi:\mathfrak{t}_{\omega}\to\mathfrak{c}_{\omega}$
in (2.4). We call _cameral cover_ the morphism $\pi_{a}:C_{a}\to C$ in (2.4).

The _discriminant divisor_
$\mathfrak{D}_{\omega}:=\mathfrak{D}\times^{\mathbb{G}_{m}}\omega^{*}$ is an
hypersurface in $\mathfrak{c}_{\omega}$. The image $a(C)$ of a section
$a\in\mathbb{A}$ always meets the divisor $\mathfrak{D}_{\omega}$, because
$\omega$ is ample. So, we may define some subsets of sections $a\in\mathbb{A}$
depending on the position of the curve $a(C)$ with respect to the discriminant
divisor $\mathfrak{D}_{\omega}$. For instance, let $\mathbb{A}^{\heartsuit}$
be the subset of $\mathbb{A}$ whose sections are not contained in
$\mathfrak{D}_{\omega}$, i.e.

$\mathbb{A}^{\heartsuit}:=\\{a\in\mathbb{A}\,|a(C)\nsubseteq\mathfrak{D}_{\omega}\\}.$

The next proposition resumes the main facts about this set.

######  Proposition 2.3.

The open subset $\mathbb{A}^{\heartsuit}$ is not-empty. Moreover, if
$a\in\mathbb{A}^{\heartsuit}$, then the cameral cover $C_{a}$ is reduced and
connected.

The proof of this fact is due to Ngô [Ngo10] under different hypotheses on the
line bundle $\omega$. We briefly explain how to adapt the strategy on
_loc.cit._ in our situation.

###### Proof.

By Remark 1.3, the morphism $\pi_{G}:\mathfrak{t}_{G}\to\mathfrak{c}_{G}$ can
be re-written

$\left(\operatorname{Id},\pi_{\mathscr{D}(G)}\right):\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{t}_{\mathscr{D}(G)}\to\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{c}_{\mathscr{D}(G)}.$

Hence, we may assume $G$ semi-simple. The non-emptiness will follow from
Proposition 3.3. The cameral curve is reduced by [Ngo10, Lemme 4.5.1] and
connected by [Ngo10, Prop. 4.6.1]. We remark that the canonical bundle
$\omega$ does not satisfy the hypothesis $\deg(\mathcal{L})\geq 2g$ in
_loc.cit._ However, the same result, with the exact same proof, still holds if
we assume $\mathcal{L}^{n}$ very ample for $n=d_{1},\ldots,d_{r}$. Since we
are assuming $G$ semi-simple, we have $d_{1},\ldots,d_{r}\geq 2$ (see Remark
1.2). Hence, we get the assertion, because the cotangent bundle $\omega^{n}$
of a smooth curve of genus at least two is very ample for any $n\geq 2$. ∎

###  2.3. Regular centralizer.

Here, we define a moduli stack of torsors under an abelian group scheme
$J_{a}$ on $C$, which depends on the choice of the section $a\in\mathbb{A}$.
It turns out that these moduli stacks completely describe the regular locus of
the Hitchin fibers.

######  Definition 2.4.

Let $\pi:\mathfrak{t}_{\omega}\to\mathfrak{c}_{\omega}$ be the
$\omega$-twisted quotient in (2.4). The _group of $W$-equivariant maps over
$\mathfrak{c}_{\omega}$_, denoted by $J^{1}$, is the smooth and commutative
group scheme over $\mathfrak{c}_{\omega}$

$\begin{array}[]{cccl}J_{a}^{1}:&\operatorname*{Sch}/\mathfrak{c}_{\omega}&\longrightarrow&(\operatorname*{Grps})\\\
&(S\to\mathfrak{c}_{\omega})&\longmapsto&\pi_{*}\left(\mathfrak{t}_{\omega}\times
T\right)^{W}(S):=\text{Hom}_{W}(S\times_{\mathfrak{c}_{\omega}}\mathfrak{t}_{\omega},T).\end{array}$

The smooth and commutative group scheme $J$ over $\mathfrak{c}_{\omega}$ is
the subfunctor of $J_{a}^{1}$, which associates to any
$\mathfrak{c}_{\omega}$-scheme $S$, the subset $J(S)\subset J^{1}(S)$ of the
$W$-equivariant morphisms

$f:S\times_{\mathfrak{c}_{\omega}}\mathfrak{t}_{\omega}\to T$

such that for any geometric point $x\in
S\times_{\mathfrak{c}_{\omega}}\mathfrak{t}_{\omega}$ fixed by the hyperplane-
root reflection $s_{\alpha}$ for some $\alpha$, we have $\alpha(f(x))\neq-1$.
For any section $a\in\mathbb{A}$, we denote by $J_{a}=a^{*}J$ the group scheme
over $C$ obtained as a pull-back of $J$ along the section
$a:C\to\mathfrak{c}_{\omega}$.

######  Remark 2.5.

The quotient $J^{1}/J$ is a $\mathbb{Z}_{2}$-torsion sheaf. Furthermore, the
condition of being in $J$ always holds for those roots whose corresponding
coroots are primitive (i.e. they are injective as cocharacters). In
particular, if the derived subgroup of $G$ does not contain the odd special
orthogonal group $\operatorname{SO}_{2n+1}$ (note that
$\operatorname{PGL}_{2}=\operatorname{SO}_{3}$) as a direct factor, the groups
$J$ and $J^{1}$ coincide, see [BD91, Remark at page 123].

For any $a\in\mathbb{A}$, we denote by _$\mathbf{Bun}_{J_{a}}$ the moduli
stack of $J_{a}$-torsors over $C$_. It is a group-stack and it acts on the
Hitchin fiber $\mathbf{Higgs}_{G}(a)$, by twisting the $G$-Higgs bundle.

######  Remark 2.6.

To be honest, the action is defined using a moduli stack of torsors with
respect to an another group scheme: the universal centralizer group scheme,
see [Ngo10, §2]. However, these two groups (and so the moduli stacks of
torsors under these groups) are isomorphic, see [Ngo10, Proposition 2.4.7]. We
have chosen this interpretation, because it shows more explicitly the role of
the cameral curve in the geometry of the moduli stack, see for instance
[Ngo10, §4.5.2].

The next proposition shows the relationship between these group-stacks and the
Hitchin fibers.

######  Proposition 2.7.

Fix $a\in\mathbb{A}$. The substack
$\mathbf{Higgs}_{G}^{\operatorname*{reg}}(a)\subset\mathbf{Higgs}_{G}(a)$ of
regular $G$-Higgs bundles is a smooth open substack of the Hitchin fiber over
$a$. Furthermore, $\mathbf{Higgs}_{G}^{\operatorname*{reg}}(a)$ is a trivial
$\mathbf{Bun}_{J_{a}}$-torsor.

###### Proof.

See [Ngo10, Proposition 4.3.3]. ∎

As consequence of the above proposition, the moduli stack
$\mathbf{Bun}_{J_{a}}$ gives us some information about the geometry of the
Hitchin fiber $\mathbf{Higgs}_{G}(a)$. The next two propositions collect the
main facts about this group-stack.

######  Proposition 2.8.

Fix $a\in\mathbb{A}^{\heartsuit}$. Then the following facts hold.

  1. (i)

There exists a surjective morphism of stacks
$\mathbf{Bun}_{J_{a}}\to\mathbf{Bun}_{J_{a}}^{\operatorname*{ab}}$, where
$\mathbf{Bun}_{J_{a}}^{\operatorname*{ab}}$ is a group-stack, whose neutral
component is an abelian stack (i.e. quotient stack of an abelian variety by
the trivial action of a diagonalizable group).

  2. (ii)

The fiber over the identity of the morphism
$\mathbf{Bun}_{J_{a}}\to\mathbf{Bun}_{J_{a}}^{\operatorname*{ab}}$ is an
affine group scheme, which will be denoted by
$\mathbf{Bun}_{J_{a}}^{\operatorname{aff}}$.

###### Proof.

See [Ngo10, Proposition 4.8.2]. ∎

The next formula gives the dimension of the affine part
$\mathbf{Bun}_{J_{a}}^{\operatorname{aff}}$ of the moduli stack
$\mathbf{Bun}_{J_{a}}$. The result has been obtained by Ngô (see [Ngo10,
Proposition 4.9.5]) based upon a result of Bezrukavnikov [Bez96]. Here, we
present the formula as stated in [CL10, Lemme 9.12.1], where they also provide
an alternative proof.

######  Proposition 2.9 (Ngô-Bezrukavnikov formula).

Fix $a\in\mathbb{A}^{\heartsuit}$. Let

  1. (i)

$\pi_{a}:C_{a}\to C$ be the corresponding cameral cover,

  2. (ii)

$B_{a}\subset C$ be the branch locus of the cameral cover,

  3. (iii)

$\rho_{a}:\widetilde{C}_{a}\to C_{a}$ be a normalization of $C_{a}$.

We then have

(2.5) $\delta(a):=\dim\mathbf{Bun}_{J_{a}}^{\operatorname{aff}}=\dim
H^{0}\Big{(}C,\mathfrak{t}\otimes_{\mathcal{O}_{C}}\big{(}\rho_{a*}(\mathcal{O}_{\widetilde{C}_{a}})/\pi_{a*}(\mathcal{O}_{C_{a}}\big{)}\Big{)}^{W}=\sum_{p\in
B_{a}}\delta_{p}(a),$

where

$\delta_{p}(a)=\frac{1}{2}\left\\{\deg_{p}(a^{*}\mathfrak{D})-\dim\mathfrak{t}+\dim\mathfrak{t}^{W_{x}}\right\\}.$

Here, $W_{x}$ is the stabilizer of a point $x$ in the normalized curve
$\widetilde{C}_{a}$ over the point $p\in C$.

######  Remark 2.10.

The previous proposition allows us to define a function

(2.6) $\delta:\mathbb{A}^{\heartsuit}\to\mathbb{Z}_{\geq
0}:a\mapsto\delta(a)=\dim\mathbf{Bun}_{J_{a}}^{\operatorname{aff}}.$

By the results in [Ngo10, §5.6], the function $\delta$ is upper-
semicontinuous.

###  2.4. Product Formula.

In this section, we will present the _product formula_ , due to Ngô. Roughly
speaking, it describes the Hitchin fiber in terms of the group-stack of the
previous subsection and certain affine Springer fibers.

Before presenting the formula, we need some notations. Fix an element
$a\in\mathbb{A}^{\heartsuit}$. Let $\pi_{a}:C_{a}\to C$ be the corresponding
cameral cover. We denote by $B_{a}\subset C$ the branch locus of the cameral
cover. For any point $p\in B_{a}$, we define the following objects:

  * •

the formal disk
$\Delta_{p}:=\operatorname*{Spec}(\widehat{\mathcal{O}}_{C,p})$ centered at
$p$ and the punctured formal disc
$\Delta_{p}^{*}:=\Delta_{p}\setminus\\{p\\}$,

  * •

the group scheme $J_{a,p}\to\Delta_{p}$ obtained from the group scheme
$J_{a}\to C$ (see Definition 2.4) via pull-back along the natural morphism
$\Delta_{p}\to C$.

  * •

the group-functor $\operatorname{Gr}_{J_{a,p}}$, which assigns, to any
noetherian affine $k$-scheme $S$, the isomorphism classes of pairs
$(P,\varphi)$, where $P$ is a $J_{a,p}$-torsor over $\Delta_{p}\times S$ and
$\varphi$ is an isomorphisms of $J_{a,p}$-torsors between $P$ and the trivial
one over $\Delta_{p}^{*}\times S$.

######  Definition 2.11.

Let $a\in\mathbb{A}^{\heartsuit}$ and let $(E_{a},\theta_{a})$ be the regular
$G$-Higgs bundle over $C$ given by the Hitchin-Kostant section, see (2.1). The
_affine Springer fiber over $a$_ is the functor $\operatorname{Spr}_{p}(a)$,
which assigns, to any noetherian affine $k$-scheme $S$, the isomorphism
classes of triples $(E,\theta,\varphi)$, where

  1. (i)

$E$ is a $G$-bundle over $\Delta_{p}\times S$,

  2. (ii)

$\theta$ is a section of the adjoint bundle $\operatorname*{ad}(E)$ over
$\Delta_{p}\times S$,

  3. (iii)

$\varphi$ is an isomorphism of $G$-bundles between $E$ and $E_{a}$ over
$\Delta^{*}_{p}\times S$, which commutes with the sections $\theta$ and
$\theta_{a}$.

We denote by $\operatorname{Spr}^{\operatorname*{reg}}_{a}(a)$ the subfunctor
of $\operatorname{Spr}_{a}(a)$ of regular elements, i.e. those elements where
the kernel of the morphism
$[\theta_{p},-]:\operatorname*{ad}(E)\to\operatorname*{ad}(E)$ at the fiber
over $p$ has dimension $r$.

The group-functor $\operatorname{Gr}_{J_{a,p}}$ acts on the affine Springer
fiber $\operatorname{Spr}_{p}(a)$. Both functors are representable by
(possibly non-reduced) schemes. We denote by
$\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}$ and
$\operatorname{Spr}^{\operatorname{red}}_{p}(a)$ their reduced counterparts,
respectively. The next theorem clarifies the relation between the Hitchin
fibers and the affine Springer fibers.

######  Theorem 2.12 (Product Formula).

Let $a\in\mathbb{A}^{\heartsuit}$ and $B_{a}$ be the branch locus of the
corresponding cameral cover. We have a homeomorphism of stacks

$\prod_{p\in
B_{a}}\operatorname{Spr}^{\operatorname{red}}_{p}(a)\stackrel{{\scriptstyle\prod_{p\in
B_{a}}\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}}}{{\times}}\mathbf{Bun}_{J_{a}}\longrightarrow\mathbf{Higgs}_{G}(a).$

Furthermore, the restriction to the open substacks of regular elements

$\prod_{p\in
B_{a}}\operatorname{Spr}^{\operatorname{red,reg}}_{p}(a)\stackrel{{\scriptstyle\prod_{p\in
B_{a}}\operatorname{Gr}_{J_{a,p}}^{\operatorname{red,reg}}}}{{\times}}\mathbf{Bun}_{J_{a}}\subset\prod_{p\in
B_{a}}\operatorname{Spr}^{\operatorname{red}}_{p}(a)\stackrel{{\scriptstyle\prod_{p\in
B_{a}}\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}}}{{\times}}\mathbf{Bun}_{J_{a}}$

is an isomorphism onto the open locus of regular elements
$\mathbf{Higgs}_{G}^{\operatorname*{reg}}(a)$.

###### Proof.

We sketch the proof as explained in [Ngo10, §4.15]. By gluing along the
Hitchin-Kostant section, we get a morphism

(2.7) $\prod_{p\in
B_{a}}\operatorname{Spr}^{\operatorname{red}}_{p}(a)\to\mathbf{Higgs}_{G}(a),$

which sends regular elements in regular $G$-Higgs bundles. With a similar
argument, we get a homomorphism of groups $\prod_{p\in
B_{a}}\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}\to\mathbf{Bun}_{J_{a}}$,
compatible with (2.7). Note that the morphism induces an action of the source
on the target. Furthermore, twisting the Hitchin-Kostant section by a
$J_{a}$-torsor gives a morphism

(2.8)
$\mathbf{Bun}_{J_{a}}\to\mathbf{Higgs}_{G}^{\operatorname*{reg}}(a)\hookrightarrow\mathbf{Higgs}_{G}(a).$

Combining the morphisms (2.7) and (2.8) together and then doing the quotient
by the diagonal action of the group $\prod_{p\in
B_{a}}\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}$, we get the desired
morphism (2.12). By [Ngo10, Theorem 4.6], it induces an equivalence over the
$k$-points. Hence, it is a homeomorphism. The second assertion follows by
construction. ∎

##  3\. The Hitchin discriminant.

In this section, we study the geometry of the Hitchin fibers with respect to
their image in the Hitchin basis $\mathbb{A}$ (see Propositions 3.9 and 3.17).
As a consequence, we show that the locus of non-abelian Hitchin fibers is
covered by rational curves (see Theorem 3.22).

To achieve this, we focus our attention on two subsets of $\mathbb{A}$. The
first one, denoted by $\mathbb{A}^{\diamondsuit}$, has been already introduced
and studied in [Ngo10]. It parameterizes sections, whose corresponding cameral
curves are smooth (see Proposition 3.3). By the results in _loc.cit._ , the
Hitchin fibers over $\mathbb{A}^{\diamondsuit}$ are always abelian stacks.
However, unless $G$ has semi-simple rank $1$, there are sections, whose
Hitchin fibers are abelian stacks and their cameral curves are singular. For
these reasons, we will introduce another open subset, denoted by
$\mathbb{A}^{\operatorname{ab}}$, which will parameterize sections, whose
corresponding Hitchin fiber is an abelian stack (see Proposition 3.10).

Unless otherwise stated, in this section, $G$ is a reductive group and $C$ is
a smooth projective curve of genus $g\geq 2$.

###  3.1. A characterization of the regular Hitchin fibers.

In this subsection, we present a criterion to check when a Hitchin fiber (over
$\mathbb{A}^{\heartsuit}$) contains non regular $G$-Higgs bundles.

######  Proposition 3.1.

Let $a\in\mathbb{A}^{\heartsuit}$. The following facts are equivalent

  1. (i)

$\mathbf{Higgs}_{G}^{\operatorname*{reg}}(a)=\mathbf{Higgs}_{G}(a)$.

  2. (ii)

the connected components of the Hitchin fiber $\mathbf{Higgs}_{G}(a)$ are
abelian stacks.

  3. (iii)

the dimension of the affine part of the moduli stack $\mathbf{Bun}_{J_{a}}$ of
$J_{a}$-torsors is zero, i.e.

$\delta(a)=\dim\mathbf{Bun}_{J_{a}}^{\operatorname{aff}}=0.$

  4. (iv)

for any point $p\in C$ such that $a(p)\in\mathfrak{D}_{\omega}$, the image of
the differential of the section $a:C\to\mathfrak{c}_{\omega}$ at the point
$p$, i.e.

$da_{p}:\mathcal{T}_{p}C\to\mathcal{T}_{a(p)}\mathfrak{c}_{\omega},$

is not contained in the tangent cone $\mathcal{C}_{a(p)}\mathfrak{D}_{\omega}$
of $\mathfrak{D}_{\omega}$ at the point $a(p)$.

###### Proof.

$(\ref{P:descr-reg1})\Leftrightarrow(\ref{P:descr-reg3})$ By Product Formula
(= Theorem 2.12), the point $(\ref{P:descr-reg1})$ is equivalent to the
condition

$\operatorname{Spr}_{p}(a)=\operatorname{Spr}^{\operatorname*{reg}}_{p}(a),\quad\forall
p\in B_{a}.$

Let $\delta_{p}(a)$ be the local delta-invariant defined in the Ngô-
Bezrukavnikov formula (= Proposition 2.9). It is known that
$\delta_{p}(a)=\dim\operatorname{Spr}_{p}(a)=\dim\operatorname{Gr}_{J_{a,p}}$
(see [Ngo10, Proposition 3.7.5]). Hence, the assertion is equivalent to prove
the following

$\operatorname{Spr}_{a}(a)=\operatorname{Spr}^{\operatorname*{reg}}_{a}(a)\iff\delta_{p}(a)=0.$

The implication $\Leftarrow$ is exactly the content of [Ngo10, Corollary
3.7.2]. Conversely, assume that
$\operatorname{Spr}_{p}(a)=\operatorname{Spr}^{\operatorname*{reg}}_{p}(a)$.
By [Ngo10, Proposition 3.4.1], there exists a finite abelian subgroup
$\Gamma\subset\operatorname{Gr}_{J_{a,p}}$ such that the quotient
$\operatorname{Spr}^{\operatorname*{red}}(a)/\Gamma$ is a projective variety.
On the other hand, the scheme
$\operatorname{Spr}^{\operatorname*{red},\operatorname*{reg}}(a)/\Gamma$ of
regular elements is a trivial
$\operatorname{Gr}_{J_{a,p}}^{\operatorname{red},\operatorname{reg}}/\Gamma$-torsor
(see [Ngo10, Lemma 3.3.1]). Since the group scheme
$\operatorname{Spr}^{\operatorname{red},\operatorname{reg}}(a)/\Gamma$ is
affine (see [Ngo10, Lemma 3.8.1]), we have that
$\operatorname{Spr}^{\operatorname{red}}(a)/\Gamma=\operatorname{Spr}^{\operatorname{red,reg}}(a)/\Gamma$
is an affine and projective variety. In particular, it is zero-dimensional
and, so, $\delta_{p}(a)=\dim\operatorname{Spr}_{a,p}(a)=0$.
($\ref{P:descr-reg2})\Leftrightarrow(\ref{P:descr-reg3})$ It follows from
Propositions 2.8 and 2.9.
$(\ref{P:descr-reg4})\Rightarrow(\ref{P:descr-reg1})$ See [Li20, Lemma
C.0.1(3)].
$(\ref{P:descr-reg3})\Rightarrow(\ref{P:descr-reg4})$ First, assume there
exists a point $p\in C$ such that the curve $a(C)$ meets the smooth locus of
$\mathfrak{D}_{\omega}$ at $a(p)$ and the image of the differential of the
section $a:C\to\mathfrak{c}_{\omega}$ is contained in the tangent space of
$\mathfrak{D}_{\omega}$ at $a(p)$. In other words, the curve $a(C)$ meets the
smooth locus of the discriminant with multiplicity at least two. Arguing as in
the proof of [Ngo10, Lemme 4.7.3], we may show that the cameral cover
$\pi_{a}:C_{a}\to C$, locally at a point $v$ over $p$, is given by the
following morphism

$\operatorname*{Spec}(k\llbracket
t,x\rrbracket/(t^{2}-x^{m}))\to\operatorname*{Spec}(k\llbracket
x\rrbracket),\quad\text{for some }m>1.$

The stabilizer $W_{x}$ of a point $x$ into the normalization of $C_{a}$ over
$v$ is trivial, if $m$ is even and it is cyclic of order two, generated by a
hyperplane-root reflection $s_{\alpha}\subset W$ for some root $\alpha$, if
$m$ odd. By Proposition 2.9, we get

$\delta(a)\geq\delta_{p}(a)=\frac{1}{2}\left(\deg(mp)-\dim\mathfrak{t}+\dim\mathfrak{t}^{W_{x}}\right)=\begin{cases}{m/2,}&\text{if
}m\text{ even},\\\ (m-1)/2,&\text{if }m\text{ odd}.\end{cases}$

Since $m>1$, we get $\delta(a)>0$ and so the assertion. Now, assume that the
curve $a(C)$ never meets the smooth locus of the discriminant. By hypothesis,
there exists $p\in C$ such that the curve $a(C)$ meets the singular locus of
$\mathfrak{D}_{\omega}$ at $a(p)$ and the image of
$da_{p}(\nu)\in\mathcal{T}_{a(p)}\mathfrak{c}_{\omega}$ is contained in the
tangent cone $\mathcal{C}_{a(p)}\mathfrak{D}_{\omega}$. By Lemma 3.2 below,
there exists a family of sections
$a_{R}:C_{R}:=C\times\operatorname{Spec}(R)\to\mathfrak{c}_{\omega}$ over a
complete DVR $R$, such that

  * •

the image of the generic fiber $a_{\eta}:C_{\eta}\to\mathfrak{c}_{\omega}$
meets the smooth locus of the discriminant $\mathfrak{D}_{\omega}$ at
$a_{\eta}(p)$ with multiplicity at least two,

  * •

the special fiber $a_{s}:C_{s}\to\mathfrak{c}_{\omega}$ coincides with the
section $a$ (after the pull-back along $s\to\operatorname{Spec}k$).

By upper-semicontinuity of the function
$\delta:\mathbb{A}^{\heartsuit}\to\mathbb{Z}_{\geq 0}$, we have
$\delta(a)=\delta(a_{s})\geq\delta(a_{\eta})$ (see Remark 2.10). Arguing as
above, we get $\delta(a_{\eta})>0$ and, so, $\delta(a)>0$. ∎

######  Lemma 3.2.

Let $a\in\mathbb{A}$ be a section such that $a(C)$ meets the discriminant
divisor $\mathfrak{D}_{\omega}$ at $a(p)$, for some point $p$, and the image
of the differential
$d\alpha_{p}:\mathcal{T}_{p}C\to\mathcal{T}_{a(p)}\mathfrak{c}_{\omega}$ is
contained in the tangent cone $\mathcal{C}_{a(p)}\mathfrak{D}_{\omega}$ of
$\mathfrak{D}_{\omega}$ at $a(p)$. Then, there exists a complete DVR $R$ and a
family of sections

$a_{R}:C_{R}:=C\times\operatorname{Spec}(R)\to\mathfrak{c}_{\omega},$

with the following properties:

  1. (i)

if $\eta$ is the generic point of $\operatorname{Spec}(R)$, the image of the
section $a_{\eta}:C_{\eta}\to\mathfrak{c}_{\omega}$ meets the smooth locus of
the discriminant $\mathfrak{D}_{\omega}$ at $a_{\eta}(p)$ with multiplicity at
least two;

  2. (ii)

if $s$ is the special point of $\operatorname{Spec}(R)$, the section
$a_{s}:C_{s}\to\mathfrak{c}_{\omega}$ coincides with the section $a$ (after
doing pull-back along $s\to\operatorname{Spec}k$).

###### Proof.

Let $\Delta:=\operatorname{Spec}(k\llbracket t\rrbracket)$ be the formal disc
with special point $s$ and generic point $\eta$. Set $q:=a(p)\in\mathfrak{D}$.
By [GH94, Proposition 20.2], for any vector $\nu$ in the tangent cone
$\mathcal{C}_{q}\mathfrak{D}$ of $\mathfrak{D}$ in $q$, there exists a
morphism
$\varphi:\Delta\to\mathcal{T}\mathfrak{D}\subset\mathcal{T}\mathfrak{c}$ such
that $\varphi(s)=(q,v)$ and $\varphi(\eta)$ is a tangent vector over the
smooth locus $\mathfrak{D}^{\operatorname*{sm}}$ of $\mathfrak{D}$.

Fix a such morphism
$\varphi:\Delta\to\mathcal{T}\mathfrak{D}\subset\mathcal{T}\mathfrak{c}$.
Consider the morphism of vector bundles
$C\times\mathbb{A}\to\mathfrak{c}_{\omega}$. By Remark 1.3, the morphism
$\pi_{G}:\mathfrak{t}_{G}\to\mathfrak{c}_{G}$ decomposes as follows

$(\operatorname{Id},\pi_{\mathscr{D}(G)}):\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{t}_{\mathscr{D}(G)}\to\mathfrak{c}_{\mathscr{Z}(G)}\oplus\mathfrak{c}_{\mathscr{D}(G)}.$

Furthermore, the discriminant divisor $\mathfrak{D}_{G}$ is the zero set of a
polynomial defined by the hyperplane-roots. This means that $\mathfrak{D}_{G}$
is the pull-back (along the obvious projection) of the discriminant divisor
$\mathfrak{D}_{\mathscr{D}(G)}\subset\mathfrak{c}_{\mathscr{D}(G)}$. In
particular, the tangent cone decomposes as
$\mathcal{C}_{q}\mathfrak{D}_{G}=\mathcal{T}_{q_{1}}\mathfrak{c}_{\mathscr{Z}(G)}\oplus\mathcal{C}_{q_{2}}\mathfrak{D}_{\mathscr{D}(G)}$.
Consider the composition

$\mathbb{A}_{G}\xrightarrow{D}\mathbb{A}_{\mathscr{D}(G)}\xrightarrow{\operatorname{ev_{2p}}}\mathfrak{c}_{\mathscr{D}(G)}\otimes\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}^{2}=\mathcal{T}\mathfrak{c}_{\mathscr{D}(G)}:b\mapsto(D(b)(p),D(b^{\prime})(p)).$

It is a surjective linear morphism of vector spaces. In particular, it is
smooth. Thus, using the morphism $\varphi:\Delta\to\mathcal{T}\mathfrak{c}$,
we may lift $a:C\to\mathfrak{c}_{\omega}$ to a family of sections
$C\times\Delta\to\mathfrak{c}_{\omega}$ with the properties requested by the
lemma. ∎

###  3.2. The subset $\mathbb{A}^{\diamondsuit}\subset\mathbb{A}$.

We denote by $\mathbb{A}^{\diamondsuit}$ the subset of
$\mathbb{A}^{\heartsuit}$, whose sections cut the divisor
$\mathfrak{D}_{\omega}$ transversally, i.e.

$\mathbb{A}^{\diamondsuit}:=\\{a\in\mathbb{A}\,|a(C)\text{ intersects
}\mathfrak{D}_{\omega}\text{outside of the singular locus, with multiplicity
}1\\}.$

Consider its complementary subset
$\mathbb{D}^{\diamondsuit}:=\mathbb{A}\setminus\mathbb{A}^{\diamondsuit}$. It
turns out that $\mathbb{D}^{\diamondsuit}$ is a divisor in $\mathbb{A}$.
Furthermore, its irreducible components are naturally bijection with the
irreducible components of the discriminant divisor and its singular locus.
More precisely, set

$\begin{array}[]{llll}\mathfrak{D}_{\omega}&=&\mathfrak{D}_{1,\omega}\cup\cdots\cup\mathfrak{D}_{m,\omega},&\text{
such that
}\mathfrak{D}_{i,\omega}:=\mathfrak{D}_{i}\times^{\mathbb{G}_{m}}\omega^{*},\\\
\mathfrak{D}^{\operatorname*{sing}}_{\omega}&=&\mathfrak{D}^{\operatorname*{sing}}_{1,\omega}\cup\cdots\cup\mathfrak{D}^{\operatorname*{sing}}_{n,\omega},&\text{
such that
}\mathfrak{D}^{\operatorname*{sing}}_{i,\omega}:=\mathfrak{D}^{\operatorname*{sing}}_{i}\times^{\mathbb{G}_{m}}\omega^{*},\end{array}$

where $\mathfrak{D}_{1},\ldots\mathfrak{D}_{m}$, resp.
$\mathfrak{D}^{\operatorname*{sing}}_{1},\ldots,\mathfrak{D}^{\operatorname*{sing}}_{n}$,
are the irreducible components of the discriminant divisor $\mathfrak{D}$,
resp. of the singular locus $\mathfrak{D}^{\operatorname*{sing}}$ of the
discriminant divisor (see Section 1). The main result of this subsection is
the following.

######  Proposition 3.3.

A section $a\in\mathbb{A}$ is contained in $\mathbb{D}^{\diamondsuit}$ if and
only if the corresponding cameral curve $C_{a}$ is singular. Furthermore, the
set $\mathbb{D}^{\diamondsuit}$ is the union of the irreducible hypersurfaces
in $\mathbb{A}$

$\mathbb{D}^{\operatorname*{ab}}_{1},\ldots,\mathbb{D}^{\operatorname*{ab}}_{m},\,\mathbb{D}^{\operatorname*{extra}}_{1},\ldots,\mathbb{D}^{\operatorname*{extra}}_{n},$

where:

  1. (i)

a section $a\in\mathbb{A}$ is contained in
$\mathbb{D}_{i}^{\operatorname*{ab}}$ if and only if there exists a point
$p\in C$ such that $a(p)\in\mathfrak{D}_{i,\omega}$ and the image of
differential
$d\alpha_{p}:\mathcal{T}_{p}C\to\mathcal{T}_{a(p)}\mathfrak{c}_{\omega}$ is
contained in the tangent cone $\mathcal{C}_{a(p)}\mathfrak{D}_{\omega}$ of
$\mathfrak{D}_{\omega}$ at $a(p)$.

  2. (ii)

a section $a\in\mathbb{A}$ is contained in
$\mathbb{D}_{i}^{\operatorname*{extra}}$ if and only if there exists a point
$p\in C$ such that $a(p)\in\mathfrak{D}^{\operatorname*{sing}}_{i,\omega}$.

Before proving the proposition, we present some consequences.

######  Corollary 3.4.

The generic section of $\mathbb{D}^{\diamondsuit}$ meets
$\mathfrak{D}_{\omega}$ transversally, with the exception of a unique point
$p\in C$, where either it meets the smooth locus of $\mathfrak{D}_{\omega}$
with multiplicity $2$ at $a(p)$ or it meets singular locus
$\mathfrak{D}_{\omega}^{\operatorname*{sing}}$ transversally at $a(p)$. More
precisely:

  1. (i)

the generic section in $\mathbb{D}^{\operatorname*{ab}}_{i}$ meets
$\mathfrak{D}_{\omega}$ transversally, with the exception of a unique point in
$p\in C$, where it meets the smooth locus
$\mathfrak{D}^{\operatorname*{sm}}_{\omega}$ along the irreducible component
$\mathfrak{D}_{i,\omega}$ with multiplicity 2 at the point $a(p)$;

  2. (ii)

the generic section in $\mathbb{D}^{\operatorname*{extra}}_{i}$ meets
$\mathfrak{D}_{\omega}$ transversally, with the exception of a unique point
$p\in C$, where it meets the irreducible component
$\mathfrak{D}^{\operatorname*{sing}}_{i,\omega}$ at $a(p)$ and it meets the
singular locus $\mathfrak{D}_{\omega}^{\operatorname*{sing}}$ transversally at
the point $a(p)$.

By the previous corollary, we also get the following result.

######  Corollary 3.5.

The following facts hold

  1. (i)

For any section $a\in\mathbb{A}^{\diamondsuit}$, the corresponding cameral
curve is integral.

  2. (ii)

For a generic section $a\in\mathbb{D}^{\diamondsuit}$, the corresponding
cameral curve is integral.

Note that the integrality of the smooth cameral curves has been already proved
in [Ngo10]. Indeed, as in _loc.cit._ , it is a consequence of Propositions 2.3
and 3.3. Here we present an alternative proof, which holds for both the
hypotheses of Corollary 3.5. A similar argument can be found in the proof of
[Fal93, Theorem III.2(i)].

###### Proof of Corollary 3.5.

Consider the universal cameral cover

(3.1)
$\textstyle{C_{\mathbb{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\mathbb{A}}}$$\textstyle{\mathfrak{t}_{\omega}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C\times\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{c}_{\omega}}$

The horizontal maps in (3.1) are smooth and surjective and their fibers are
affine spaces. In particular, the scheme $\mathcal{C}$ is a connected smooth
variety. Let $\mathbb{A}^{*}$ be the open subset in $\mathbb{A}$, whose
sections are either in $\mathbb{A}^{\diamondsuit}$ or generic in
$\mathbb{D}^{\diamondsuit}$ in the sense of Corollary 3.4. Let $U\subset
C\times\mathbb{A}^{*}$ be the maximal open subset such that the restriction
$\pi_{\mathbb{A}}^{-1}(U)\to U$ is étale. Since $C_{\mathbb{A}}$ is connected,
the restriction of $\pi_{\mathbb{A}}$ over $U$ is a Galois cover with Galois
group $W$. Equivalently, the induced homomorphism
$\pi_{1}(U,u)\xrightarrow{f_{u}}W$, for some $u\in U$, from the (étale)
fundamental group to the Weyl group is surjective. Note that the projection on
the first factor $U\hookrightarrow
C\times\mathbb{A}^{*}\xrightarrow{\operatorname{pr}_{1}}\mathbb{A}^{*}$ is
surjective and the complement of $U$ in $C\times\mathbb{A}^{*}$ is a relative
normal crossing divisor over $\mathbb{A}^{*}$. By [Gro63, XIII, 4.1+4.4], we
get an exact sequence of fundamental groups

(3.2) $\pi_{1}(C\setminus
B_{a},u)\xrightarrow{\iota}\pi_{1}(U,u)\to\pi_{1}(\mathbb{A}^{*},a)\to 1,$

where $\operatorname{pr}_{1}(u)=a\in\mathbb{A}^{*}$. To conclude the proof, it
is enough to show that the composition

(3.3) $\pi_{1}(C\setminus
B_{a},u)\xrightarrow{\iota}\pi_{1}(U,u)\xrightarrow{f_{u}}W$

is surjective. Indeed, the surjectivity implies that the étale cover
$V_{a}:=C_{a}\setminus\pi_{a}^{-1}(B_{a})\to C\setminus B_{a}$ is Galois. In
particular, $V_{a}$ (and so $C_{a}$) must be irreducible.

Fix a point $q\in B_{a}$ (different from the point $p$ in the statement of
Corollary 3.3). By definition, the image $a(q)\in\mathfrak{c}_{\omega}$ is a
smooth point of an irreducible component $\mathfrak{D}_{\omega,i}$ of the
discriminant divisor $\mathfrak{D}_{\omega}$, which corresponds to a $W$-orbit
of roots. Note that the homomorphism (3.3) maps the loop around the point
$q\in B_{a}$ to a reflection $s_{\alpha}\in W$ with respect to a root $\alpha$
in the $W$-orbit met by $a(q)$. The curve $a(C)$ meets all the connected
components of the smooth locus of $\mathfrak{D}_{\omega}$, namely all the
$W$-orbits of roots. In particular, the image
$W_{a}:=(f_{u}\circ\iota)(\pi_{1}(C\setminus B_{a},u))$ contains at least one
representative for each conjugacy class of hyperplane-root reflections in $W$.
Since the image of $\pi_{1}(C\setminus B_{a},u)$ in $\pi_{1}(U,u)$ is normal
(by sequence (3.2)) and $f_{u}$ is surjective, the image $W_{a}$ is a normal
subgroup of $W$. Hence, $W_{a}$ is a normal subgroup of $W$ and it contains
all the hyperplane-roots reflections. So, $W_{a}=W$. ∎

It remains to prove Proposition 3.3 and Corollary 3.4. Before of this, we need
some auxiliary results.

######  Lemma 3.6.

Let $a$ be a section such that $a(C)\subset\mathfrak{D}_{\omega}$, i.e.
$a\in\mathbb{D}^{\heartsuit}:=\mathbb{A}\setminus\mathbb{A}^{\heartsuit}$.
Then, the image of the differential
$da_{p}:\mathcal{T}_{p}C\to\mathcal{T}_{a(p)}\mathfrak{D}_{\omega}$ is
contained in the tangent cone $\mathcal{C}_{a(p)}\mathfrak{D}_{\omega}$ for
any point $p\in C$. Furthermore, the closed subset $\mathbb{D}^{\heartsuit}$
has codimension at least three in $\mathbb{A}$.

###### Proof.

Locally at a point $p\in C$, the section $a$ is given by a ring-homomorphism
$\varphi:k[p_{1},\ldots,p_{r}]\to k\llbracket t\rrbracket,$ such that if
$\mathfrak{D}$ is the zero-set of a polynomial $F$, we must have
$\varphi(F(p_{1},\ldots,p_{r}))=0$. Without loss of generality, we may assume
that $p$ corresponds to the ideal $(t)$ and $a(p)$ to the maximal ideal
$(p_{1},\ldots,p_{r})$, i.e. $(p_{1},\ldots,p_{r})=\varphi^{-1}(t)$. There
exists a number $s$ such that

$\varphi(p_{i})=a_{i}t^{s}\operatorname{mod}t^{s+1},\quad\text{ for
}i=1,\ldots,r,$

where $(a_{1},\ldots,a_{r})\in k^{r}\setminus\\{\underline{0}\\}$. If $s>1$,
the claim holds because $da_{p}(T_{p}C)$ is the zero-vector. Assume $s=1$ and
let $F=F_{n}+F_{n+1}+\ldots$ be the decomposition in homogeneous polynomials
of $F$. We then have:

$\displaystyle 0$
$\displaystyle=\varphi(F(p_{1},\ldots,p_{r}))=F_{n}\left(\scriptstyle{\frac{\varphi(p_{1})}{t},\ldots,\frac{\varphi(p_{r})}{t}}\right)t^{n}+F_{n+1}\left(\scriptstyle{\frac{\varphi(p_{1})}{t},\ldots,\frac{\varphi(p_{r})}{t}}\right)t^{n+1}+\ldots$
$\displaystyle=F_{n}(a_{1},\ldots,a_{r})t^{n}+\text{polynomials in }t\text{ of
degree bigger than }n.$

The equality implies the vanishing of $F_{n}\left(a_{1}\ldots,a_{r}\right)$.
Note that the vector $(a_{1},\ldots,a_{r})$ is a generator of the image of
$da_{p}$ in $T_{a(p)}\mathfrak{c}_{\omega}$. By definition, the tangent cone
at $(p_{1},\ldots,p_{r})$ is the zero set of the homogeneous polynomial
$F_{n}$ and, so, we have the first statement.

We now show the second statement. Consider the set $U$ of $4$-uples
$(q_{1},q_{2},q_{3},a)\in C^{3}\times\mathbb{A}$ such that $a(C)$ meets
$\mathfrak{D}_{\omega}$ and the image of the differential at $a(q_{i})$ is
contained in the tangent cone $\mathcal{C}_{a(q_{i})}\mathfrak{D}_{\omega}$
for $i=1,2,3$. By the previous statement, the set $\mathbb{D}^{\heartsuit}$ is
contained in the image $\operatorname{pr}(U)$ of $U$ along the projection
$\operatorname{pr}:C^{3}\times\mathbb{A}\to\mathbb{A}$. Hence, it is enough to
show that $\operatorname{pr}(U)$ has codimension at least three in
$\mathbb{A}$.

We denote by $U_{q_{1},q_{2},q_{3}}$ the fiber over a triple
$(q_{1},q_{2},q_{3})\in C^{3}$ along the composition $U\hookrightarrow
C^{3}\times\mathbb{A}\to\ C^{3}$. The fiber is canonically equal to a subset
of the inverse image of the product
$(\mathcal{T}\mathfrak{D})^{3}\subset(\mathcal{T}\mathfrak{c})^{3}$ along the
morphism

(3.4)
$\mathbb{A}\xrightarrow{\operatorname{ev}_{2q_{1},2q_{2},2q_{3}}}\bigoplus_{i=1}^{3}(\mathfrak{c}\otimes\mathcal{O}_{C,q_{i}}/\mathfrak{m}_{C,q_{i}}^{2})=(\mathcal{T}\mathfrak{c})^{3}.$

We will treat the cases $g\geq 3$ and $g=2$, separately.
$g\geq 3$ We first assume $G$ semi-simple. In particular, all the degrees are
greater or equal than two. The cokernel of the evaluation map (3.4) is equal
to

(3.5)
$\bigoplus_{i=1}^{r}H^{1}(C,\omega^{d_{i}}(-2q_{1}-2q_{2}-2q_{3}))\cong\bigoplus_{i=1}^{r}H^{0}(C,\omega^{1-d_{i}}(2q_{1}+2q_{2}+2q_{3}))^{*}.$

By assumptions on the genus, we have the following inequalities

$\deg(\omega^{1-d_{i}}(2q_{1}+2q_{2}+2q_{3}))=(1-d_{i})(2g-2)+6\leq-4d_{i}+10.$

In particular, the factors in the decomposition (3.5) vanish, if $d_{i}\geq
3$. If $d_{i}=2$, we have $\deg(\omega^{1-d_{i}}(2q_{1}+2q_{2}+2q_{3}))=8-2g$.
In particular, the remaining factors vanish (i.e. the evaluation map is
surjective) for either $g\geq 5$ and any triple in $C^{3}$, or $g=3,4$ and a
generic triple in $C^{3}$. Hence, if $g\geq 3$ and $(q_{1},q_{2},q_{3})\in
C^{3}$ generic, the fiber $U_{q_{1},q_{2},q_{3}}$ has codimension at least six
in $\mathbb{A}$. Then, the same holds for $U$ and, so, the projection
$\operatorname{pr}(U)$ has codimension at least three in $\mathbb{A}$. Now,
assume $G$ reductive. The morphism
$\pi_{G}:\mathfrak{t}_{G}\to\mathfrak{c}_{G}$ decomposes as follows

$(\operatorname{Id},\pi_{\mathscr{D}(G)}):\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{t}_{\mathscr{D}(G)}\to\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{c}_{\mathscr{D}(G)},$

see Remark 1.3. Furthermore, the discriminant divisor $\mathfrak{D}_{G}$ is
the zero set of a polynomial defined by the hyperplane-roots. It means that
$\mathfrak{D}_{G}$ is the pull-back (along the obvious projection) of the
discriminant divisor
$\mathfrak{D}_{\mathscr{D}(G)}\subset\mathfrak{c}_{\mathscr{D}(G)}$. In
particular, the subset $\mathbb{D}^{\heartsuit}$ is contained in the inverse
image of the product
$(\mathcal{T}\mathfrak{D}_{\mathscr{D}(G)})^{3}\subset\mathfrak{c}_{\mathscr{D}(G)}^{3}$
along the composition
$\mathbb{A}=\mathbb{A}_{G}\to\mathbb{A}_{\mathscr{D}(G)}\xrightarrow{\operatorname{ev}_{2q_{1},2q_{2},2q_{3}}}(\mathcal{T}\mathfrak{c}_{\mathscr{D}(G)})^{3}.$
Arguing as in the semi-simple case, we conclude the proof.
$g=2$ Assume $G$ semi-simple. Arguing as above, it can be shown that for
generic a generic triple $(q_{1},q_{2},q_{3})\in C^{3}$, the cokernel of (3.4)
is isomorphic to

$H^{0}(C,\omega^{-1}(2q_{1}+2q_{2}+2q_{3}))^{\oplus\\#\\{d_{i}=2\\}}.$

Moreover, this vector space is isomorphic to cokernel of the evaluation

$\bigoplus_{i\text{
s.t.}d_{i}=2}\mathbb{A}_{i}=H^{0}(C,\omega^{2})^{\\#\\{d_{i}=2\\}}\xrightarrow{\operatorname{ev}_{2q_{1},2q_{2},2q_{3}}}(\mathcal{T}\mathfrak{c}_{1})^{3},$

where $\mathfrak{c}_{1}$ is the linear subspace of $\mathfrak{c}$, where
$\mathbb{G}_{m}$ acts with weight $2$. We divide the proof in three steps.

  1. (a)

$G$ almost-simple of rank $1$, i.e.
$G=\operatorname{SL}_{2},\operatorname{PGL}_{2}$. In this case,
$\mathfrak{c}=\mathfrak{c}_{1}=\operatorname{Spec}k[t]$ and
$\mathfrak{D}=\\{t=0\\}$. In this case the evaluation (3.4) is injective.
Hence, the only section in $\mathbb{D}^{\heartsuit}$ is the zero-section.
Since $\dim\mathbb{A}=3$, we have the assertion.

  2. (b)

$G$ almost-simple of rank at least two. It is enough to show that the
projection
$\mathcal{T}\mathfrak{D}\subset\mathcal{T}\mathfrak{c}\to\mathcal{T}\mathfrak{c}_{1}=\operatorname{Spec}(k[p_{1},dp_{1}])$
is surjective. First observe that the projection
$\mathfrak{D}\to\mathfrak{c}_{1}$ is $\mathbb{G}_{m}$-equivariant. Then, it
must be surjective. Otherwise, we should have
$\mathfrak{D}\subset\\{p_{1}=0\\}$, which is false by Remark 1.4 and the
assumptions on the rank of $G$. A similar argument shows the surjectivity of
the projection $\mathcal{T}\mathfrak{D}\to\mathcal{T}\mathfrak{c}_{1}$.

  3. (c)

$G$ semi-simple. Up to isogeny, the group $G$ decomposes as a product
$G_{1}\times\cdots\times G_{s}$, with $G_{i}$ almost-simple group for any $i$.
Since the curve is irreducible, any section in $\mathbb{D}^{\heartsuit}$ has
image in some irreducible component $\mathfrak{D}_{i,\omega}$ of the
discriminant divisor. Hence, $\mathbb{D}^{\heartsuit}$ is the union of the
loci of sections with image in $\mathfrak{D}_{i,\omega}$ for any
$i\in\\{1,\ldots,n\\}$. By definition of the components $\mathfrak{D}_{i}$,
there exists a $j\in\\{1,\ldots,s\\}$ such that $\mathfrak{D}_{i}$ is an
irreducible component of the pull-back (along the obvious projection) of the
discriminant divisor $\mathfrak{D}_{G_{j}}\subset\mathfrak{c}_{G_{j}}$. In
particular, the subset $\mathbb{D}^{\heartsuit}$ is contained in the union of
the inverse images of the products
$(\mathcal{T}\mathfrak{D}_{G_{j}})^{3}\subset(\mathcal{T}\mathfrak{c}_{G_{j}})^{3}$
along the compositions
$\mathbb{A}=\mathbb{A}_{G}\to\mathbb{A}_{G_{j}}\xrightarrow{\operatorname{ev}_{2q_{1},2q_{2},2q_{3}}}(\mathcal{T}\mathfrak{c}_{G_{j}})^{3},$
for any $j=1,\ldots,s$. Hence, point (c) follows from the previous points.

The case $G$ reductive follows by arguing exactly as in the case $g\geq 3$. ∎

######  Lemma 3.7.

Let $\mathfrak{S}$ be a $\mathbb{G}_{m}$-invariant irreducible subvariety of
$\mathfrak{D}$. The following facts hold.

  1. (i)

The locus of pairs $(p,a)\in C\times\mathbb{A}$, such that
$a(p)\in\mathfrak{S}_{\omega}:=\mathfrak{S}\times^{\mathbb{G}_{m}}\omega^{*}$,
is an irreducible variety of codimension $r-\dim\mathfrak{S}$ in
$C\times\mathbb{A}$. If $\dim\mathfrak{S}=\dim\mathfrak{D}-1$, the projection
in $\mathbb{A}$ is an irreducible variety of codimension $1$.

  2. (ii)

The locus of triples $(p,q,a)\in C\times C\times\mathbb{A}$, such that
$a(p),a(q)\in\mathfrak{S}_{\omega}$, is an irreducible variety of codimension
$2(r-\dim\mathfrak{S})$ in $C\times C\times\mathbb{A}$.

###### Proof.

$(\ref{L:int-sect1})$ We denote the locus in the assertion by $V$. Consider
the morphism of vector bundles
$\operatorname{ev}:C\times\mathbb{A}\to\mathfrak{c}_{\omega}$. Since the line
bundle $\omega^{j}$ is base-point-free for any $j$, the evaluation at the
fibers
$\operatorname{ev_{p}}:\mathbb{A}\to\mathfrak{c}\otimes\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}=\mathfrak{c}$
is a smooth and surjective morphism with integral fibers of dimension
$\dim\mathbb{A}-r$. In particular, the inverse image
$\operatorname{ev}_{p}^{-1}(\mathfrak{S})$ is an irreducible variety of
dimension $\dim\mathfrak{S}+\dim\mathbb{A}-r$. Note that
$\operatorname{ev}_{p}^{-1}(\mathfrak{S})$ is nothing but the fiber over the
point $p$ of the composition $V\hookrightarrow
C\times\mathbb{A}\xrightarrow{\pi_{1}}C$. This implies that $V$ is an
irreducible variety in $C\times\mathbb{A}$ of dimension
$\dim\mathfrak{S}+\dim\mathbb{A}+1-r$. If $\dim S=\dim\mathfrak{D}-1$, the
composition $V\hookrightarrow C\times\mathbb{A}\to\mathbb{A}$ is generically
quasi-finite (by Lemma 3.6). Hence, the last assertion follows.
$(\ref{L:int-sect2})$ Since the composition
$\mathbb{A}=\mathbb{A}_{G}\to\mathbb{A}_{\mathscr{D}(G)}\xrightarrow{\operatorname{ev}_{p,q}}\mathfrak{c}_{\mathscr{D}(G)}^{2}$
is surjective for any pair of points in $C$, the assertion is obtained by
arguing as in the previous point. ∎

######  Lemma 3.8.

Consider the open and connected subset
$\mathfrak{U}_{i,\omega}:=\mathfrak{D}_{\omega}^{\operatorname*{sm}}\cap\mathfrak{D}_{i,\omega}$
of $\mathfrak{D}_{i,\omega}$. Then, the locus of pairs $(p,a)\in
C\times\mathbb{A}$, such that $a(C)$ meets $\mathfrak{U}_{i,\omega}$ at $a(p)$
with multiplicity $\geq 2$, is an irreducible locally-closed subset of
codimension $2$ in $C\times\mathbb{A}$. The projection in $\mathbb{A}$ is an
irreducible locally-closed subset of codimension $1$.

###### Proof.

We first assume $G$ semi-simple. Let $F$ be the (irreducible) polynomial in
$k[\mathfrak{t}]^{W}$ such that
$\operatorname{Spec}(k[\mathfrak{t}]^{W}/F)=\mathfrak{D}_{i}$. We denote the
locus in the assertion by $U$. Consider the morphism

$C\times\mathbb{A}\xrightarrow{\operatorname{ev}}\mathfrak{c}_{\omega}\xrightarrow{F}\omega^{\deg
F}.$

Since $G$ is semi-simple, all the degrees $d_{1},\ldots,d_{r}$ are strictly
bigger than $1$. In particular, the line bundle $\omega^{d_{i}}$ is very ample
for any $i=1,\ldots,r$. Hence, the evaluation
$\operatorname{ev}_{2p}:\mathbb{A}\to\mathfrak{c}\otimes\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}^{2}$
is a smooth and surjective homomorphism with integral fibers of dimension
$\dim\mathbb{A}-2r$. Let $\mathfrak{c}^{o}$ be the open subset of
$\mathfrak{c}$, where the morphism $F:\mathfrak{c}\to\mathbb{A}^{1}$ is
smooth. A section $a$ meets the smooth locus of $\mathfrak{D}_{i,\omega}$ at
$a(p)$ with multiplicity $\geq 2$ if and only if $\operatorname{ev}_{2p}$
factors through $\mathfrak{c}^{0}$ and it is contained in the fiber over
$[m_{C,p}^{2}]$ of the smooth morphism

$\mathfrak{c}^{o}\otimes\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}^{2}\xrightarrow{F_{p}}\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}^{2}.$

Observe that the locus $F^{-1}_{p}([m_{C,p}^{2}])$ is isomorphic to the
tangent bundle $\mathcal{T}\mathfrak{D}_{i,\omega}^{\operatorname*{sm}}$ over
the smooth locus of $\mathfrak{D}_{i,\omega}$. In particular, it is isomorphic
to an irreducible open subset of the (possibly reducible) affine variety of
dimension $2(r-1)$

$\operatorname*{Spec}k[p_{1},\ldots,p_{r},dp_{1},\ldots,dp_{r}]/(F,dF)\subset\mathfrak{c}\times\mathfrak{c}\cong\mathfrak{c}\otimes\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}^{2}.$

By definition, we have the inclusion
$\mathfrak{U}_{i,\omega}\subset\mathfrak{D}_{i,\omega}^{\operatorname*{sm}}$.
Then, the set
$\operatorname{ev}_{2p}^{-1}(\mathcal{T}\mathfrak{U}_{i,\omega})$ is an
irreducible locally-closed subset of $\mathbb{A}$ of dimension $\dim
T\mathfrak{D}_{i,\omega}^{\operatorname*{sm}}+\dim\mathbb{A}-2r=\dim\mathbb{A}-2$.
By definition, the set above is the fiber over the point $p$ of the
composition $U\hookrightarrow C\times\mathbb{A}\xrightarrow{\pi_{1}}C$. Hence,
$U$ is an irreducible locally-closed subset of $C\times\mathbb{A}$ of
dimension $\dim\mathbb{A}-1$. Since the composition $U\hookrightarrow
C\times\mathbb{A}\xrightarrow{\pi_{2}}\mathbb{A}$ is generically quasi-finite
(by Lemma 3.6), we have the last assertion. Now, assume $G$ reductive. The
morphism $\pi_{G}:\mathfrak{t}_{G}\to\mathfrak{c}_{G}$ decomposes as follows

$(\operatorname{Id},\pi_{\mathscr{D}(G)}):\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{t}_{\mathscr{D}(G)}\to\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathfrak{c}_{\mathscr{D}(G)},$

see Remark 1.3. Furthermore, the discriminant divisor $\mathfrak{D}_{G}$ is
the zero set of a polynomial defined by the hyperplane-roots. It means that
$\mathfrak{D}_{G}$ is the pull-back (along the obvious projection) of the
discriminant divisor
$\mathfrak{D}_{\mathscr{D}(G)}\subset\mathfrak{c}_{\mathscr{D}(G)}$. In
particular,
$\mathcal{T}\mathfrak{D}_{G}=\mathcal{T}\mathfrak{t}_{\mathscr{Z}(G)}\oplus\mathcal{T}\mathfrak{D}_{\mathscr{D}(G)}$.
Consider the composition

$\mathbb{A}_{G}\xrightarrow{D}\mathbb{A}_{\mathscr{D}(G)}\xrightarrow{\operatorname{ev_{2p}}}\mathcal{\mathfrak{c}}_{\mathscr{D}(G)}.$

By previous case, we know that the image $D(U)$ has codimension $1$ in
$C\times\mathbb{A}_{\mathscr{D}(G)}$. Since $D$ is linear and surjective and
$U=D^{-1}(D(U))$, we have that $U$ has codimension $1$ in
$C\times\mathbb{A}_{G}$. The last assertion follows because the composition
$U\hookrightarrow C\times\mathbb{A}_{G}\xrightarrow{\pi_{2}}\mathbb{A}_{G}$ is
generically quasi-finite (by Lemma 3.6).
∎

We are finally ready for proving the proposition and its corollary.

###### Proof of Proposition 3.3.

The first part is exactly the content of [Ngo10, Lemma 4.7.3]. We now focus on
the description of the irreducible components of $\mathbb{D}^{\diamondsuit}$.
Consider the loci:

  1. (a)

$U\subset C\times\mathbb{A}$ of pairs $(p,a)$ such that $a(C)$ meets
$\mathfrak{D}_{\omega}^{\operatorname*{sm}}$ at $a(p)$ with multiplicity at
least $2$,

  2. (b)

$V\subset C\times\mathbb{A}$ of pairs $(p,a)$ such that $a(C)$ meets
$\mathfrak{D}_{\omega}^{\operatorname*{sing}}$.

By definition $\mathbb{D}^{\diamondsuit}=\pi_{2}(U)\cup\pi_{2}(V)$. By Lemma
3.7, the locus $\widetilde{V}:=\pi_{2}(V)$ is the union of (distinct)
irreducible closed subsets $\widetilde{V}_{1},\ldots,\widetilde{V}_{n}$ of
codimension $1$ in $\mathbb{A}$. Where $\widetilde{V}_{i}$ is the locus of
sections meeting the irreducible component
$\mathfrak{D}^{\operatorname*{sing}}_{\omega,i}$ of the singular locus
$\mathfrak{D}_{\omega}^{\operatorname*{sing}}$. In other words,
$\widetilde{V}_{i}=\mathbb{D}^{\operatorname*{extra}}_{i}$ for $i=1,\ldots n$.
Similarly, by Lemma 3.8, the closure of the projection of $U$ into
$\mathbb{A}$, i.e.
$\widetilde{U}:=\overline{\pi_{2}(U)}=\pi_{2}(\overline{U})$, is the union of
(distinct) irreducible closed subsets
$\widetilde{U}_{1},\ldots,\widetilde{U}_{m}$ of codimension $1$ in
$\mathbb{A}$. Where $\widetilde{U}_{i}$ is the closure of the locus of
sections meeting the smooth locus $\mathfrak{D}^{\operatorname*{sm}}_{\omega}$
through the irreducible component $\mathfrak{D}_{\omega,i}$ with multiplicity
$\geq 2$ at least in one point. It remains to show that
$\widetilde{U}_{i}=\mathbb{D}^{\operatorname*{ab}}_{i}$ for $i=1,\ldots,n$. By
Lemma 3.2, any section in $\mathbb{D}^{\operatorname*{ab}}_{i}$ is the limit
of a section in $\pi_{2}(U_{i})$, i.e.
$\mathbb{D}^{\operatorname*{ab}}_{i}\subset\widetilde{U}_{i}$. Conversely, let
$a$ be a section in $\widetilde{U}_{i}$. We distinguish two cases. First case:
$a\in\mathbb{A}^{\heartsuit}\cap\widetilde{U}_{i}$. Then the condition
$a\in\mathbb{D}_{i}^{ab}$ follows by Proposition 3.1 and the upper-
semicontinuity of the function
$\delta:\mathbb{A}^{\heartsuit}\to\mathbb{Z}_{\geq 0}$ (see Remark 2.10).
Second case: $a\in\widetilde{U}_{i}$ and $a(C)\subset\mathfrak{D}_{\omega}$.
Then, the inclusion follows by Lemma 3.6. ∎

Corollary 3.4 has been already proved in [Li20, Lemma C.0.2], when $G$ is
semi-simple and $\omega=\mathcal{L}$ is a line bundle of degree greater than
$2g$. Since our assumptions are slightly different, we include a self-
contained proof.

###### Proof of Corollary 3.4.

$(\ref{C:gen-discr1})$ Fix $i=1,\ldots,m$. By Proposition 3.3, a generic point
in $\mathbb{D}_{i}^{\operatorname*{ab}}$ corresponds to those sections meeting
the irreducible divisor $\mathfrak{D}_{i,\omega}$ outside of the singular
locus of the entire discriminant divisor $\mathfrak{D}_{\omega}$. Then, it is
enough to show that

  1. (a)

the set of pairs $(p,a)\in C\times\mathbb{A}$, such that $a(C)$ meets the
smooth locus $\mathfrak{D}^{\operatorname*{sm}}_{\omega}$ of the discriminant
divisor with multiplicity greater or equal than $3$ at $a(p)$, has codimension
at least three in $C\times\mathbb{A}$;

  2. (b)

the set of sections $a\in\mathbb{A}$, such that $a(C)$ meets the smooth locus
$\mathfrak{D}^{\operatorname*{sm}}_{\omega}$ of the discriminant divisor with
multiplicity greater than $2$ in at least two points, has codimension at least
2 in $\mathbb{A}$.

Point (a): we denote the set in the assertion by $U$. The fiber $U_{p}$ over a
point $p\in C$ along the projection $U\hookrightarrow C\times\mathbb{A}\to C$
is canonically equal to an open subset of the inverse image of
$[\mathfrak{m}_{C,p}^{3}]$ along the composition

$\mathbb{A}=\mathbb{A}_{G}\to\mathbb{A}_{\mathscr{D}(G)}\xrightarrow{\operatorname{ev}_{3p}}\mathfrak{c}\otimes\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}^{3}\xrightarrow{\prod_{\alpha\in\Phi}d\alpha}\mathcal{O}_{C,p}/\mathfrak{m}_{C,p}^{3},$

Using Riemann-Roch, we see that the evaluation $\operatorname{ev}_{3p}$ is
surjective (and smooth) for any $p\in C$, if $g\geq 3$ and for $p\in C$
generic, if $g=2$. In particular, for a generic point $p$, the fiber $U_{p}$
has codimension at least $3$ in $\mathbb{A}$. Hence, the set $U$ has
codimension $3$ in $C\times\mathbb{A}$. Point (b): it follows by arguing as in
the proof of the second statement of Lemma 3.6.
$(\ref{C:gen-discr2})$ Fix $i=1,\ldots,n$. By Proposition 3.3, there exists an
open subset in $\mathbb{D}_{i}^{\operatorname{extra}}$, whose sections always
meet the discriminant divisor $\mathfrak{D}_{\omega}$ along the irreducible
subvariety $\mathfrak{D}^{\operatorname*{sing}}_{i,\omega}$. Then, it is
enough to show that

  1. (a)

the set of pairs $(p,a)\in C\times\mathbb{A}$, such that $a(C)$ meets the
singular locus
$(\mathfrak{D}^{\operatorname*{sing}}_{\omega})^{\operatorname*{sing}}$ of
$\mathfrak{D}^{\operatorname*{sing}}_{\omega}$ at $a(p)$, has codimension at
least three in $C\times\mathbb{A}$;

  2. (b)

the set of triples $(p,q,a)\in C\times C\times\mathbb{A}$, such that
$a(p),a(q)\in\mathfrak{D}^{\operatorname*{sing}}_{\omega}$, has codimension at
least four in $C\times C\times\mathbb{A}$.

Point (a) follows from Lemma 3.7$(\ref{L:int-sect1})$. On the other hand,
Point (b) follows from Lemma 3.7$(\ref{L:int-sect2})$. ∎

###  3.3. The subset $\mathbb{A}^{\operatorname*{ab}}\subset\mathbb{A}$.

In the previous section, we have defined the open subset
$\mathbb{A}^{\diamondsuit}$ of $\mathbb{A}$ and we have seen that it
parameterizes smooth cameral curves. In this subsection, we will define
another open subset in $\mathbb{A}$, containing $\mathbb{A}^{\diamondsuit}$,
and we will show that it parameterizes Hitchin fibers which are isomorphic to
abelian stacks.

We denote by $\mathbb{A}^{\operatorname*{ab}}$ the subset in $\mathbb{A}$,
whose sections satisfy condition $(\ref{P:descr-reg4})$ of Proposition 3.1,
i.e.

$\mathbb{A}^{\operatorname*{ab}}:=\\{a\in\mathbb{A}\,|\,\operatorname{Im}(da_{p})\not\subset\mathcal{C}_{a(p)}\mathfrak{D}_{\omega}\text{
for any }p\in a^{-1}(\mathfrak{D}_{\omega})\subset C\\}.$

We denote
$\mathbb{D}^{\operatorname*{ab}}:=\mathbb{A}\setminus\mathbb{A}^{\operatorname*{ab}}$
its complementary subset. The goal of the section is proving the following
proposition.

######  Proposition 3.9.

Fix $\delta\in\pi_{1}(G)$. The following facts hold.

  1. (i)

For any section $a\in\mathbb{A}^{\operatorname*{ab}}$, the Hitchin fiber
$\mathbf{Higgs}_{G}^{\delta}(a)$ is an abelian stack.

  2. (ii)

For a generic section $a\in\mathbb{D}^{\operatorname*{ab}}$, there exists a
finite birational morphism of stacks

$\mathbb{P}^{1}\stackrel{{\scriptstyle\mathbb{G}_{m}}}{{\times}}\mathbf{Bun}_{J_{a}}^{\delta}\to\mathbf{Higgs}_{G}^{\delta}(a),$

where the left-hand side is a locally trivial $\mathbb{P}^{1}$-fibration over
the abelian stack $\mathbf{Bun}_{J_{a}}^{\delta,\operatorname*{ab}}$.

Before proving this, we need the following fact.

######  Proposition 3.10.

The closed subset $\mathbb{D}^{\operatorname*{ab}}$ is the union of the
subsets

$\mathbb{D}^{\operatorname*{ab}}_{1},\ldots,\mathbb{D}^{\operatorname*{ab}}_{m},$

where $\mathbb{D}^{\operatorname*{ab}}_{i}$ is the irreducible hypersurface in
$\mathbb{A}$ described in Proposition 3.3. Furthermore, we have the chain of
inclusions
$\mathbb{A}^{\diamondsuit}\subset\mathbb{A}^{\operatorname*{ab}}\subset\mathbb{A}^{\heartsuit}$.

###### Proof.

The first statement follows from the description of the irreducible components
of $\mathbb{D}^{\diamondsuit}$ (see Proposition 3.3). In particular, we get
$\mathbb{D}^{ab}\subset\mathbb{D}^{\diamondsuit}$ and, so, the first inclusion
in the second statement holds. The second inclusion follows by Lemma 3.6. ∎

######  Remark 3.11.

By Proposition 3.10, we know that
$\mathbb{A}^{\operatorname*{ab}}\subset\mathbb{A}^{\heartsuit}$. In
particular, $a\in\mathbb{A}^{\operatorname*{ab}}$ if and only
$a\in\mathbb{A}^{\heartsuit}$ and it satisfies one of the equivalent
conditions of Proposition 3.1.

######  Remark 3.12.

Proposition 3.10 gives a criterion for counting the number of irreducible
components of the divisor $\mathbb{D}^{\operatorname*{ab}}$. In particular, we
may deduce the following facts.

  1. (i)

The divisor $\mathbb{D}^{\operatorname*{ab}}$ is empty if and only if $G$ is
abelian.

  2. (ii)

The divisor $\mathbb{D}^{\operatorname*{ab}}$ is irreducible and non-empty if
and only if the derived subgroup $\mathscr{D}(G)$ is an almost-simple group of
type ADE.

  3. (iii)

Assume $\mathscr{D}(G)$ being a non-trivial almost-simple group. The divisor
$\mathbb{D}^{\operatorname*{ab}}$ is irreducible, if $\mathscr{D}(G)$ is of
type ADE and reducible with two components, otherwise.

######  Remark 3.13.

Propositions 3.3 and 3.10 together explain how far is the inclusion
$\mathbb{A}^{\diamondsuit}\subset\mathbb{A}^{\operatorname*{ab}}$ from being
an equality. In particular, the following facts are equivalent.

  1. (i)

The inclusion
$\mathbb{A}^{\diamondsuit}\subsetneq\mathbb{A}^{\operatorname*{ab}}$ is
strict, i.e. there exists a section $a\in\mathbb{A}$, such that the cameral
curve $C_{a}$ is singular and the Hitchin fiber $\mathbf{Higgs}_{G}(a)$ is an
abelian stack.

  2. (ii)

The discriminant divisor $\mathfrak{D}\subset\mathfrak{c}$ is singular.

  3. (iii)

the group $G$ has semi-simple rank greater or equal than $2$, i.e.
$\mathscr{D}(G)\neq\operatorname{SL}_{2},\operatorname{PGL}_{2}$.

Thanks to Proposition 3.10, we can now show the main result of the subsection.

###### Proof of Proposition 3.9.

Point $(\ref{P:uni-stack1})$ follows by Remark 3.11. We now show the point
$(\ref{P:uni-stack2})$. Let $a$ be a generic section in
$\mathbb{D}^{\operatorname*{ab}}$. By Proposition 3.10 and Corollary
3.4$(\ref{C:gen-discr1})$, we may assume that $a(C)$ meets
$\mathfrak{D}_{\omega}$ transversally, with the exception of a unique point in
$p\in C$, where it meets the smooth locus
$\mathfrak{D}^{\operatorname*{sm}}_{\omega}$ along the irreducible component
$\mathfrak{D}_{\omega,i}$ with multiplicity 2 at the point $\alpha(p)$. In
particular, $a\in\mathbb{A}^{\heartsuit}$. By Product formula (= Theorem
2.12), we have a homeomorphism of stacks

(3.6) $\prod_{q\in
B_{a}}\operatorname{Spr}^{\operatorname{red}}_{q}(a)\stackrel{{\scriptstyle\prod_{p\in
B_{a}}\operatorname{Gr}_{J_{a,q}}^{\operatorname{red}}}}{{\times}}\mathbf{Bun}_{J_{a}}\to\mathbf{Higgs}_{G}(a),$

where $B_{a}$ is the branch locus of the cameral cover $C_{a}$ corresponding
to $a$. Arguing as in the proof of Proposition 3.1, we get that all the
elements in the affine Springer fiber over the point $q\neq p$ are regular,
i.e.
$\operatorname{Spr}^{\operatorname{red}}_{q}(a)=\operatorname{Spr}^{\operatorname{red},\operatorname*{reg}}_{q}(a)$
for any $q\in B_{a}\setminus\\{p\\}$. The latter is a torsor with respect to
the group $\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}$. In particular,
the homeomorphism (3.6) becomes

(3.7)
$\operatorname{Spr}^{\operatorname{red}}_{p}(a)\stackrel{{\scriptstyle\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}}}{{\times}}\mathbf{Bun}_{J_{a}}\to\mathbf{Higgs}_{G}(a).$

We now describe the affine Springer fiber
$\operatorname{Spr}^{\operatorname{red}}_{p}(a)$ following [Ngo10, Section
8.3]. There exists a co-root $\alpha^{\vee}$ such that

$\operatorname{Spr}^{\operatorname{red}}_{p}(a)=\bigsqcup_{j\in\Lambda/\langle\alpha^{\vee}\rangle_{\mathbb{Z}}}X_{j},$

where $\langle\alpha^{\vee}\rangle_{\mathbb{Z}}$ is the sublattice of the
cocharacter lattice $\Lambda:=\operatorname{Hom}(\mathbb{G}_{m},T)$ of the
maximal torus $T\subset G$ and
$X_{j}=\bigcup_{i\in<\alpha^{\vee}>_{\mathbb{Z}}}\mathbb{P}^{1}$ is an
infinite chain of rational lines. Furthermore, there exists an exact sequence
of groups

$1\to\mathbb{G}_{m}\to\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}\to\Lambda\to
1,$

where $\mathbb{G}_{m}$ acts on the affine Springer fibers by scalar
multiplication. Fix a projective line $\mathbb{P}^{1}$ in $X_{0}$, we get a
finite birational morphism of stacks

(3.8)
$\mathbb{P}^{1}\times^{\mathbb{G}_{m}}\mathbf{Bun}_{J_{a}}\to\operatorname{Spr}^{\operatorname{red}}_{p}(a)\stackrel{{\scriptstyle\operatorname{Gr}_{J_{a,p}}^{\operatorname{red}}}}{{\times}}\mathbf{Bun}_{J_{a}}.$

Then, the composition of the morphisms (3.7) and (3.8) gives the required
morphism. ∎

###  3.4. The moduli space $\mathcal{H}_{G}$ of semistable $G$-Higgs bundles.

Here, we present the analogous results of the previous sections for the (good)
moduli space of semistable bundles.

######  Definition 3.14.

A $G$-bundle $E$ is _simple_ if $\operatorname{Aut}(E)=\mathscr{Z}(G)$. A
$G$-bundle $E$ is _stable_ (resp. _semistable_) if for any reduction $F$ of
$E$ to any parabolic subgroup $P\subset G$, we have

$\deg(F\times^{\operatorname{Ad},P}\mathfrak{p})<0,\quad(\text{resp.
}\deg(F\times^{\operatorname{Ad},P}\mathfrak{p})\leq 0),$

where $\operatorname{Ad}:P\to\operatorname{GL}(\mathfrak{p})$ is the adjoint
action of $P$ on its Lie algebra $\mathfrak{p}$.
A $G$-Higgs bundle $(E,\theta)$ is _stable_ (resp. _semistable_) if for any
reduction $F$ of $E$ to any parabolic subgroup $P\subset G$ such that
$\theta\in
H^{0}(C,(F\times^{\operatorname{Ad},P}\mathfrak{p})\otimes\omega)\subset
H^{0}(C,\operatorname*{ad}(E)\otimes\omega)$, we have

$\deg(F\times^{\operatorname{Ad},P}\mathfrak{p})<0,\quad(\text{resp.
}\deg(F\times^{\operatorname{Ad},P}\mathfrak{p})\leq 0).$

The (semi)stable locus $\mathbf{Bun}_{G}^{(s)s}$ is an open substack of the
moduli stack $\mathbf{Bun}_{G}$. It is well-known that the moduli stack
$\mathbf{Bun}_{G}^{ss}$ admits a good moduli space
$\overline{\mathcal{M}}_{G}$ (or $\overline{\mathcal{M}}_{G}(C)$, when the
reference to the curve is needed). The next theorem resumes the main
properties of this moduli space.

######  Theorem 3.15.

The following facts hold.

  1. (i)

The moduli stack $\mathbf{Bun}_{G}^{\operatorname{(s)s}}$ of (semi)stable
$G$-bundles is an open and dense substack of finite type over $k$ in
$\mathbf{Bun}_{G}$.

  2. (ii)

The good moduli morphism
$\mathbf{Bun}_{G}^{\operatorname{ss}}\to\overline{\mathcal{M}}_{G}$ gives a
bijection of connected components
$\pi_{0}(\mathbf{Bun}_{G})=\pi_{0}(\overline{\mathcal{M}}_{G})=\pi_{1}(G)$.

  3. (iii)

For any $\delta\in\pi_{1}(G)$, the connected component
$\overline{\mathcal{M}}^{\delta}_{G}$ of semistable $G$-bundle of degree
$\delta$ is an irreducible normal projective variety over $k$.

  4. (iv)

For any $\delta\in\pi_{1}(G)$, the subset $\mathcal{M}^{\delta}_{G}$ of stable
and simple $G$-bundles of degree $\delta$ is contained in the smooth locus of
the connected component $\overline{\mathcal{M}}^{\delta}_{G}$.

On the other hand, the (semi)stable locus
$\mathbf{Higgs}_{G}^{\operatorname{(s)s}}$ is an open substack of the entire
moduli stack $\mathbf{Higgs}_{G}$. It is well-known that the moduli stack
$\mathbf{Higgs}_{G}^{\operatorname{ss}}$ admits a good moduli space
$\mathcal{H}_{G}$. The Hitchin morphism factors through the good moduli space

(3.9) $h:\mathbf{Higgs}_{G}\to\mathcal{H}_{G}\to\mathbb{A}.$

With abuse of notation, we denote with the same symbol $h$ the second morphism
in the composition (3.9) and we call it _Hitchin morphism_. Analogously, we
call _Hitchin fiber over $a$_ the inverse image
$\mathcal{H}_{G}(a):=h^{-1}(a)\subset\mathcal{H}_{G}$ with respect to the
section $a\in\mathbb{A}$.

######  Theorem 3.16.

The following facts hold.

  1. (i)

The moduli stack $\mathbf{Higgs}_{G}^{\operatorname{(s)s}}$ of (semi)stable
$G$-Higgs bundles is an open and dense substack of finite type over $k$ in
$\mathbf{Higgs}_{G}$.

  2. (ii)

The good moduli morphism
$\mathbf{Higgs}_{G}^{\operatorname{(s)s}}\to\mathcal{H}_{G}$ gives a bijection
of connected components
$\pi_{0}(\mathbf{Higgs}_{G})=\pi_{0}(\mathcal{H}_{G})=\pi_{1}(G)$.

  3. (iii)

Fix $\delta\in\pi_{1}(G)$. The Hitchin morphism
$h^{\delta}:\mathcal{H}_{G}^{\delta}\to\mathbb{A}$ is projective with
geometrically connected fibers of dimension $\dim\mathbb{A}=\dim
G(g-1)+\dim\mathscr{Z}(G)$. Furthermore, it is faithfully flat if we restrict
to the locus of stable $G$-Higgs bundles.

###### Proof.

The first two points are consequences of the third one, whose proof (in this
generality) can be found in [Fal93]. ∎

######  Proposition 3.17.

Fix $\delta\in\pi_{1}(G)$. The following facts hold.

  1. (i)

For any $a\in\mathbb{A}^{\operatorname*{ab}}$, the Hitchin fiber
$\mathcal{H}_{G}^{\delta}(a)$ is an abelian variety.

  2. (ii)

For a generic $a\in\mathbb{D}^{\operatorname*{ab}}$, the Hitchin fiber
$\mathcal{H}_{G}^{\delta}(a)$ is a uniruled irreducible variety.

###### Proof.

$(\ref{T:i1:geom-fiber})$ Fix a section $a\in\mathbb{A}^{\operatorname*{ab}}$.
By Proposition 3.9$(\ref{P:uni-stack1})$, we know that the stacky Hitchin
fiber $\mathbf{Higgs}_{G}^{\delta}(a)$ is an abelian stack, i.e.
$\mathbf{Higgs}_{G}^{\delta}(a)=[A/H]$, where $A$ is an abelian variety and
$H$ is a multiplicative group acting trivially on $A$. In particular, the
canonical morphism $[A/H]\to A$ makes $A$ a good moduli space of the stacky
Hitchin fiber. More precisely, it is a tame moduli space (in the sense of
Alper, see [Alp13, Definition 7.1]). Consider the open substack
$\mathbf{Higgs}_{G}^{\delta,ss}(a)\subset\mathbf{Higgs}_{G}^{\delta}(a)$ of
semistable $G$-Higgs bundles. By the above discussion, we must have
$\mathbf{Higgs}_{G}^{\delta,\operatorname{ss}}(a)=[A^{\operatorname{ss}}/H]$,
where $A^{\operatorname{ss}}$ is an open subset of $A$. By the universal
property of the good moduli space, we have
$\mathcal{H}_{G}^{\delta}(a)=A^{\operatorname{ss}}.$ Since the Hitchin
morphism is projective (see Theorem 3.16), we must have
$A^{\operatorname{ss}}=A$. Equivalently, any $G$-Higgs bundle in
$\mathbf{Higgs}_{G}^{\delta}(a)$ is semistable. This proves the point
$(\ref{T:i1:geom-fiber})$.
$(\ref{T:i2:geom-fiber})$ Let $a$ be a generic section in $\mathbb{D}^{ab}$.
By Corollary 3.5$(\ref{C:irr-cam2})$, we may assume that the corresponding
cameral curve is integral. By Lemma 3.18 below, all the $G$-Higgs bundles in
the stacky Hitchin fiber $\mathbf{Higgs}_{G}^{\delta}(a)$ are stable. We then
have the following commutative diagram

$\textstyle{\mathbb{P}^{1}\times^{\mathbb{G}_{m}}\mathbf{Bun}_{J_{a}}^{\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\mathbf{Higgs}_{G}^{\delta}(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathcal{H}_{G}}^{\delta}(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\nu}}$$\textstyle{\mathcal{H}_{G}^{\delta}(a)}$

where

  * •

$\nu$ is the finite birational morphism of stacks (see Proposition 3.9),

  * •

$\widetilde{\mathcal{H}_{G}}^{\delta}(a)$ is the tame moduli space of
$\mathbb{P}^{1}\times^{\mathbb{G}_{m}}\mathbf{Bun}_{J_{a}}^{\delta}$, which
exists because the latter is a locally trivial $\mathbb{P}^{1}$-fibration over
the abelian stack $\mathbf{Bun}_{J_{a}}^{\delta,\operatorname*{ab}}$ (see
_loc.cit._),

  * •

the vertical maps are the canonical maps onto the tame moduli spaces,

  * •

the existence of the morphism $\overline{\nu}$ follows by the universal
property of good moduli spaces.

The morphism $\overline{\nu}$ is birational, because its counterpart $\nu$ is
birational. Furthermore, the moduli space
$\widetilde{\mathcal{H}_{G}}^{\delta}(a)$ is a locally trivial
$\mathbb{P}^{1}$-fibration over the good moduli space of the abelian stack
$\mathbf{Bun}_{J_{a}}^{\delta,\operatorname*{ab}}$, which is an abelian
variety. In particular, the good moduli space $\mathcal{H}_{G}^{\delta}(a)$ is
a uniruled variety. ∎

######  Lemma 3.18.

Let $a$ be a section in $\mathbb{A}$ such that the corresponding cameral curve
is integral, then any $G$-Higgs bundle in the Hitchin fiber
$\mathbf{Higgs}_{G}(a)$ is stable.

###### Proof.

We will show a stronger result: if the cameral curve $C_{a}$ is integral, then
any $G$-Higgs bundle $(E,\theta)$ in the Hitchin fiber $\mathbf{Higgs}_{G}(a)$
does not have a reduction to a parabolic subgroup $P\subset G$ compatible with
$\theta$.

We will show this fact by contradiction. Let $(E,\theta)$ be a $G$-Higgs
bundle in $\mathbf{Higgs}_{G}(a)$ with a reduction $(F,\theta)$ to a parabolic
subgroup $P\subset G$ compatible with the section $\theta$. Let $M$ be the
Levi-subgroup of $P$. Note that having a $G$-Higgs bundle is the same as
having a morphism of stacks $C\to[\mathfrak{g}_{\omega}/G]$. Then, having the
reduction to a parabolic subgroup is equivalent to have a commutative diagram
of stacks

$\textstyle{[\mathfrak{p}_{\omega}/P]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[\mathfrak{g}_{\omega}/G]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathfrak{c}_{M,\omega}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{c}_{G,\omega}}$

The first vertical arrow follows from Theorem 1.1$(\ref{T:konstant6})$. In
particular, the section $a$ factors through $\mathfrak{c}_{M,\omega}$. By
definition of cameral cover, we have a commutative diagram of schemes

$\textstyle{\widehat{C}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{t}_{\omega}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{c}_{M,\omega}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathfrak{c}_{G,\omega}}$

where the squares are cartesian. Then, we have a contradiction, because the
cameral curve $C_{a}$ is not irreducible (it contains the proper subcurve
$\widehat{C}_{a}$). ∎

######  Remark 3.19.

From the proof of Proposition 3.17, one can also deduce that the $G$-Higgs
bundles in the stacky Hitchin fibers over $\mathbb{A}^{\operatorname*{ab}}$
are always stable. Furthermore, one can also show that the morphism
$\overline{\nu}:\widetilde{\mathcal{H}_{G}}(a)\to\mathcal{H}_{G}(a)$ in (3.4)
is the normalization of $\mathcal{H}_{G}(a)$.

######  Corollary 3.20.

Fix $\delta\in\pi_{1}(G)$. The following facts hold.

  1. (i)

The cotangent bundle $\mathcal{T}^{*}\mathcal{M}_{G}^{\delta}$ of the moduli
space $\mathcal{M}_{G}^{\delta}$ of simple and stable $G$-bundles of degree
$\delta$ is an open and dense subset of the moduli space
$\mathcal{H}_{G}^{\delta}$.

  2. (ii)

For any $a\in\mathbb{A}^{\operatorname*{ab}}$, the intersection
$\mathcal{T}^{*}\mathcal{M}_{G}^{\delta}\cap\mathcal{H}_{G}(a)$ is a non-empty
open subset of the abelian variety $\mathcal{H}_{G}^{\delta}(a)$.

  3. (iii)

For a generic $a\in\mathbb{D}^{\operatorname*{ab}}$, the intersection
$\mathcal{T}^{*}\mathcal{M}_{G}^{\delta}\cap\mathcal{H}_{G}(a)$ is a non-empty
open subset of the uniruled variety $\mathcal{H}_{G}^{\delta}(a)$.

###### Proof.

We remove the reference to the group $G$ from the notation. $(\ref{C:i1:geom-
fiber})$ By standard deformation theory, we know that the fiber of tangent
bundle $T\mathcal{M}\to\mathcal{M}$ over a simple and stable $G$-bundle $E$ is
canonically identified with $H^{1}(C,\operatorname*{ad}(E))$. By Serre
duality, the fiber of the cotangent bundle
$\mathcal{T}^{*}\mathcal{M}\to\mathcal{M}$ over the same point is the vector
space $H^{0}(C,\operatorname*{ad}(E)\otimes\omega)$. It is easy to check that
if a $G$-bundle $E$ is stable, then $(E,\theta)$ is a stable $G$-Higgs bundle
for any section $\theta\in H^{0}(C,\operatorname*{ad}(E)\otimes\omega)$. In
particular, $\mathcal{T}^{*}\mathcal{M}$ is an open subset in $\mathcal{H}$.
The density follows because the inclusion
$\mathcal{T}^{*}\mathcal{M}\subset\mathcal{H}$ gives a bijection between the
connected components, i.e.
$\pi_{0}(\mathcal{M})=\pi_{0}(\mathcal{T}^{*}\mathcal{M})=\pi_{0}(\mathcal{H})=\pi_{1}(G)$.
$(\ref{C:i2:geom-fiber})+(\ref{C:i3:geom-fiber})$ By previous point and
Theorem 3.17, we have the statement, except for the non-emptiness of the
intersection. This last property follows by [Fal93, Theorem II.5(ii)]. ∎

######  Proposition 3.21.

Let $\mathbb{D}^{\operatorname*{ab}}_{i}$ be an irreducible component of
$\mathbb{D}^{\operatorname*{ab}}$, for some $i=1,\ldots,m$. Then the inverse
image $(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab}}_{i})$ is an
irreducible divisor of $\mathcal{H}_{G}^{\delta}$.

###### Proof.

To make the notation easier to follow, we write $\mathbb{D}$ instead of
$\mathbb{D}^{\operatorname*{ab}}_{i}$. Since the fibers of the Hitchin
morphism $h^{\delta}$ are equi-dimensional, the inverse image
$(h^{\delta})^{-1}(\mathbb{D})$ is a divisor in $\mathcal{H}_{G}^{\delta}$. It
remains to prove the irreducibility. Let $\mathbb{D}^{*}\subset\mathbb{D}$ be
the open subset of generic sections in the sense of Proposition
3.17$(\ref{T:i2:geom-fiber})$. The restriction
$(h^{\delta})^{-1}(\mathbb{D}^{*})\to\mathbb{D}^{*}$ is a projective morphism
with geometrically integral fibers. In particular,
$(h^{\delta})^{-1}(\mathbb{D}^{*})$ is an irreducible open subset of the
divisor $(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab}})$. To conclude the
proof, it is enough to show that $(h^{\delta})^{-1}(\mathbb{D}^{*})$ is also
dense in the divisor $(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab}})$.
This statement is equivalent to show that the complement

$(h^{\delta})^{-1}(\mathbb{D})\setminus(h^{\delta})^{-1}(\mathbb{D}^{*})=(h^{\delta})^{-1}(\mathbb{D}\setminus\mathbb{D}^{*})$

has codimension at least two in $\mathcal{H}_{G}^{\delta}$. And this follows
because $h^{\delta}$ have equi-dimensional fibers and
$\mathbb{D}\setminus\mathbb{D}^{*}$ have codimension at least two in
$\mathbb{A}$. ∎

######  Theorem 3.22.

The inverse image $(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab}})$ is the
closure of the union of projective rational curves in
$\mathcal{H}_{G}^{\delta}$.

###### Proof.

Let $R$ be the union of projective rational curves in $\mathcal{H}_{G}$. We
want to show that

$(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab}})=\overline{R}.$

$\supset$ It is enough to show
$h^{\delta}(R)\subset\mathbb{D}^{\operatorname*{ab}}$. Let $X=\mathbb{P}^{1}$
be a projective line and $f:X\to\mathcal{H}_{G}^{\delta}$ be a (finite)
morphism. Since $X$ is integral and projective and $\mathbb{A}$ is affine, the
image $a:=h^{\delta}(f(X))\subset\mathbb{A}$ must be a point. In particular,
the curve $f(X)$ is contained in the Hitchin fiber
$\mathcal{H}_{G}^{\delta}(a)$. Since an abelian variety does not contain
rational curves, we must have $a\in\mathbb{D}^{\operatorname*{ab}}$ (see
Proposition 3.17).
$\subset$ By Proposition 3.21, it is enough to show
$(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab},*})\subset R$, where
$\mathbb{D}^{\operatorname*{ab},*}\subset\mathbb{D}^{\operatorname*{ab}}$ is
the open subset of generic sections in the sense of Proposition
3.17$(\ref{T:i2:geom-fiber})$. By _loc.cit._
$(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab},*})\to\mathbb{D}^{\operatorname*{ab},*}$
is a family of irreducible uniruled varieties. In particular,
$(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab},*})$ is a union of
projective rational curves, i.e.
$(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab},*})\subset R$. ∎

The same proof also gives the following corollary, which was already known for
$G=\operatorname{SL}_{n}$ [BGM12, Theorem 3.3], $G=\operatorname{Sp}_{2n}$
[BGMn12, Theorem 3.3] and $G$ simply-connected of type either $E_{6}$ or
$F_{4}$ [AS18c, Proposition 5]. Following _loc.cit._ , we call _Hitchin
discriminant_ the intersection
$(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab}})\cap
T^{*}\mathcal{M}_{G}^{\delta}$.

######  Corollary 3.23.

The Hitchin discriminant
$(h^{\delta})^{-1}(\mathbb{D}^{\operatorname*{ab}})\cap
T^{*}\mathcal{M}_{G}^{\delta}$ is the closure of the union of projective
rational curves in $T^{*}\mathcal{M}_{G}^{\delta}$.

###### Proof.

The proof is a copy of the proof of Theorem 3.22. The only difference is that,
instead of using Proposition 3.17, we need to use Corollary 3.20 as reference.
∎

##  4\. Torelli Teorem.

In this section, we show a Torelli-type theorem for the moduli space of
$G$-bundles. Our proof is a replica of the proof of Biswas, Gómez and Muñoz in
[BGM12], for $G=\operatorname{SL}_{n}$, in [BGMn12], for
$G=\operatorname{Sp}_{2n}$, and later adapted by Antón Sancho [AS18c] for the
cases $G=\mathbb{E}_{6}^{\operatorname{sc}},\mathbb{F}_{4}$. We also remark
that this result has been already established with different methods and in
greater generality in [BH12]. However, later in the paper, we will need some
of the results of this section (especially Corollary 4.5) for the computation
of the automorphisms of the moduli spaces $\overline{\mathcal{M}}_{G}$.

Here, we will always assume $G$ almost-simple and $C$ connected, smooth
projective curve of genus $g\geq 2$, with the only exception of the cases
where $g=2$ and $G$ is either $\operatorname{SL_{2}}$ or
$\operatorname{PGL}_{2}$, which are excluded.

######  Lemma 4.1.

The global regular functions
$H^{0}(\mathcal{T}^{*}\mathcal{M}_{G},\mathcal{O}_{\mathcal{T}^{*}\mathcal{M}_{G}})$
give a map

$h^{\delta}:\mathcal{T}^{*}\mathcal{M}_{G}^{\delta}\to\operatorname{Spec}(H^{0}(\mathcal{T}^{*}\mathcal{M}_{G}^{\delta},\mathcal{O}_{\mathcal{T}^{*}\mathcal{M}_{G}^{\delta}}))\cong\mathbb{A},$

which is the Hitchin morphism up to an automorphism of $\mathbb{A}$.
Furthermore, the standard action of $\mathbb{G}_{m}$ on the fibers of
$\mathcal{T}^{*}\mathcal{M}_{G}$ induces a unique $\mathbb{G}_{m}$-action on
$\mathbb{A}$ making the morphism $h^{\delta}$ $\mathbb{G}_{m}$-equivariant.

###### Proof.

We remove the reference to $G$, from the notation. The statement holds for the
Hitchin morphism $h:\mathcal{H}^{\delta}\to\mathbb{A}$ over the entire moduli
space of semistable Higgs-bundles. Indeed, it is a consequence of Theorem
3.4$(iii)$ and Stein factorization. The statement of
$\mathcal{T}^{*}\mathcal{M}^{\delta}$ holds because the generic fiber is
smooth and the codimension of $\mathcal{T}^{*}\mathcal{M}\subset\mathcal{H}$
on the fibers of the Hitchin morphism is at least two, see [Fal93, Theorem
II.6]. ∎

######  Proposition 4.2.

The intersection $C^{*}:=\mathbb{D}^{\operatorname*{ab}}\cap\mathbb{A}_{r}$ is
irreducible and $\mathbb{P}(C^{*})\subset\mathbb{P}(\mathbb{A}_{r})$ is the
dual variety of the curve $C\subset\mathbb{P}(\mathbb{A}_{r}^{*})$ for the
embedding given by the line bundle $\omega^{d_{r}}$.

###### Proof.

Fix a section $a=(0,0,\ldots,0,a_{r})\in\mathbb{A}_{r}\subset\mathbb{A}$. Note
that the restriction of the discriminant $\mathfrak{D}_{\omega}$ to
$\operatorname{Spec}(k[p_{1},\ldots,p_{r}]/(p_{1},\ldots,p_{r-1}))\subset\mathfrak{c}$
is given by the solution of the polynomial $p_{r}^{r}$, see Remark 1.4. Then,
the section $a$ is contained in $\mathbb{D}^{\operatorname*{ab}}$ if and only
if $a(p)=(0,\ldots,0)$ and $\operatorname{Im}(da_{p})=(0,\ldots,0)$ for some
$p\in C$. The last statement is equivalent to the condition that the section
$a_{r}\in\mathbb{A}_{r}$ vanishes at $p$ with order at least two for some
$p\in C$, i.e. $a_{r}\in H^{0}(C,\omega^{d_{r}}(-2p))\subset
H^{0}(C,\omega^{d_{r}})=\mathbb{A}_{r}$. By varying the point $p$, we have the
equality

$C^{*}=\bigcup_{p\in C}H^{0}(C,\omega^{d_{r}}(-2p))\subset\mathbb{A}_{r}.$

In particular, the projectivization
$\mathbb{P}(C^{*})\subset\mathbb{P}(\mathbb{A}_{r})$ is exactly the dual
variety of the curve $C$ with respect to the linear system $\mathbb{A}_{r}$. ∎

######  Theorem 4.3 (Torelli Theorem).

Let $C_{1}$ and $C_{1}$ be two smooth projective curves. If
$\mathcal{M}^{\delta_{1}}_{G}(C_{1})\cong\mathcal{M}^{\delta_{2}}_{G}(C_{2})$
for some $\delta_{1},\delta_{2}\in\pi_{1}(G)$, then $C_{1}\cong C_{2}$.

###### Proof.

By hypothesis, there exists an isomorphism
$\varphi:\mathcal{M}_{G}^{\delta_{1}}(C_{1})\to\mathcal{M}_{G}^{\delta_{2}}(C_{2})$.
Then, the codifferential
$d\varphi^{*}:T^{*}\mathcal{M}^{\delta_{2}}_{G}(C_{2})\cong
T^{*}\mathcal{M}^{\delta_{1}}_{G}(C_{1})$ is an isomorphism of cotangent
bundles. By Lemma 4.1, we have a cartesian diagram

$\textstyle{T^{*}\mathcal{M}^{\delta_{2}}_{G}(C_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\varphi^{*}}$$\textstyle{T^{*}\mathcal{M}^{\delta_{1}}_{G}(C_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\mathbb{A}}}$$\textstyle{\mathbb{A}}$

where the vertical maps are the Hitchin morphisms. Note that the linearity of
$d\varphi^{*}$ makes the isomorphism $\varphi_{\mathbb{A}}$
$\mathbb{G}_{m}$-equivariant, where the action on $\mathbb{A}$ is defined as
follows

$\lambda.(a_{1},\ldots,a_{r})=(\lambda^{d_{1}}a_{1},\ldots,\lambda^{d_{r}}a_{r}),\text{
for }(a_{1},\ldots,a_{r})\in\mathbb{A}\text{ and }\lambda\in\mathbb{G}_{m}.$

In particular, it preserves the linear subspace in $\mathbb{A}$ of highest
weight, i.e. $\mathbb{A}_{r}$. Thus, we get a linear isomorphism

$\varphi_{\mathbb{A}_{r}}:=(\varphi_{\mathbb{A}})_{|_{\mathbb{A}}}:\mathbb{A}_{r}\to\mathbb{A}_{r}.$

By Corollary 3.20, the divisor $\mathbb{D}^{\operatorname*{ab}}$ is preserved
by $\varphi_{\mathbb{A}_{r}}$, otherwise you will get an abelian variety
birational to a uniruled variety. Hence, the morphism
$\varphi_{\mathbb{A}_{r}}$ preserves the intersection
$\mathbb{D}^{\operatorname*{ab}}\cap\mathbb{A}_{r}$. By proposition 4.2, we
have an isomorphism of the dual varieties of the curves $C_{2}$ and $C_{1}$
with respect to the embedding into $\mathbb{P}(\mathbb{A}_{r})$. Since the
double dual of a curve is the curve itself, we get the desired isomorphism
$\sigma:C_{1}\to C_{2}$ of curves. ∎

######  Corollary 4.4.

Let $C_{1}$ and $C_{2}$ be two smooth projective curves. If
$\overline{\mathcal{M}}^{\delta_{1}}_{G}(C_{1})\cong\overline{\mathcal{M}}^{\delta_{2}}_{G}(C_{2})$,
then $C_{1}\cong C_{2}$.

###### Proof.

Given an isomorphism
$\varphi:\overline{\mathcal{M}}^{\delta_{1}}_{G}(C_{1})\cong\overline{\mathcal{M}}^{\delta_{2}}_{G}(C_{2})$,
we obtain an isomorphism
$\mathcal{M}^{\delta_{1}}_{G}(C_{1})\cong\varphi(\mathcal{M}^{\delta_{1}}_{G}(C_{1}))$.
Both sides are smooth varieties, by Theorem 3.15. Furthermore, the tangent
space $\mathcal{T}^{*}(\varphi(\mathcal{M}^{\delta_{1}}_{G}(C_{1})))$
satisfies the analogous statement of Lemma 4.1. Then, by repeating the same
argument of Theorem 4.3, we get the corollary. ∎

In the proof of Torelli Theorem (= Theorem 4.3), we constructed a morphism of
sets

(4.1)
$\operatorname{Aut}(\mathcal{M}_{G}^{\delta})\to\operatorname{Aut}(\mathbb{A}):\varphi\mapsto\varphi_{\mathbb{A}}.$

A priori, we know that $\varphi_{\mathbb{A}}$ is $\mathbb{G}_{m}$-equivariant
with respect to the canonical $\mathbb{G}_{m}$-action on $\mathbb{A}$, induced
by the one of $\mathfrak{c}_{\omega}$. We will show later that this
isomorphism comes from the composition of an isomorphism of the vector bundle
$\mathfrak{c}_{\omega}$ together with the pull-back along an automorphism of
$C$. At the moment, we are only able to show the following fact.

######  Corollary 4.5.

Let $\varphi$ be an automorphism of $\mathcal{M}^{\delta}_{G}(C)$ and
$\varphi_{\mathbb{A}}$ be the induced automorphism on the Hitchin basis, given
by (4.1). Then there exists $\lambda\in k\setminus\\{0\\}$ and
$\sigma\in\operatorname*{Aut}(C)$ such that the restriction

$\varphi_{\mathbb{A}_{r}}:=(\varphi_{\mathbb{A}})_{|_{\mathbb{A}_{r}}}:\mathbb{A}_{r}\to\mathbb{A}_{r}$

is equal to $(\lambda\cdot Id_{\mathbb{A}_{r}})\circ\sigma^{*}$, where
$\sigma^{*}$ is the isomorphism on $\mathbb{A}_{r}$ given by the pull-back of
the line bundle $\omega^{d_{r}}$ along $\sigma$.

###### Proof.

In the proof of Theorem 4.3, we deduced that the the projective isomorphism
$\overline{\varphi}_{\mathbb{A}}:\mathbb{P}(\mathbb{A}_{r})\to\mathbb{P}(\mathbb{A}_{r})$
preserves the dual variety of $C$. Furthermore, the dual map
$\overline{\varphi}^{*}_{\mathbb{A}}$ preserves the embedding
$C\hookrightarrow\mathbb{P}(\mathbb{A}^{*}_{r})$. Since $C$ is not contained
in any hyperplane of $\mathbb{P}(\mathbb{A}^{*}_{r})$ and
$\overline{\varphi}^{*}_{\mathbb{A}}$ is linear, we get
$\overline{\varphi}^{*}_{\mathbb{A}}\in\operatorname{Aut}(C)$. ∎

##  5\. The tautological automorphisms of $\overline{\mathcal{M}}_{G}$.

The aim of this section is to define the homomorphism of Theorem A and proving
its injectivity. Unless otherwise stated, in this section, $G$ is a reductive
group and $C$ is a smooth projective curve of genus $g\geq 2$.

Given a $G$-bundle $E$, we may construct some new (possibly isomorphic)
$G$-bundles, by using the automorphisms of the curves, the automorphisms of
the group and the $\mathscr{Z}(G)$-bundles on $C$. More precisely, given a
$G$-bundle $E$:

  1. (a)

if $\sigma:C\cong C$ is an isomorphism, the pull-back $\sigma^{*}E$ is a
$G$-bundle,

  2. (b)

if $\rho:G\cong G$ is a group-isomorphism, then we define the $G$-bundle
$\rho(E):=E\times^{\rho,G}G$,

  3. (c)

if $Z$ is a $\mathscr{Z}(G)$-bundle, the tensorization by $Z$, denoted by
$E\otimes Z$, is a $G$-bundle, where $E\otimes Z$ is the fibered product
$E\times_{C}Z$ modulo the equivalence relation $(et,z)\sim(e,zt)$, for any
$e\in E$, $z\in Z$ and $t\in\mathscr{Z}(G)$.

In particular, they define automorphisms for the moduli stack
$\mathbf{Bun}_{G}$ of $G$-bundles. We remark that some of these automorphisms
may be trivial, as the next lemma shows.

######  Lemma 5.1.

If $\rho\in\operatorname{Aut}(G)$ is an inner automorphism (i.e.
$\rho\in\operatorname{Inn}(G)$), then $E\cong\rho(E)$ for any $G$-bundle.

###### Proof.

By hypothesis, there exists $h\in G$ such that $hgh^{-1}=\rho(g)$ for any
$g\in G$. Then, the map

$\displaystyle E$ $\displaystyle\xrightarrow{f}\rho(E):=E\times^{\rho,G}G$
$\displaystyle e$ $\displaystyle\mapsto[eh,1]$

is a well-defined morphism of $G$-bundles and it gives the desidered
isomorphism. Indeed, for any $e\in E$ and $g\in G$, we have
$f(eg)=[egh,1]=[ehh^{-1}gh,1]=[eh\rho^{-1}(g),1]=[eh,g]\stackrel{{\scriptstyle\operatorname{def}}}{{=}}f(e)g.$
∎

In particular, the isomorphism
$\mathbf{Bun}_{G}\to\mathbf{Bun}_{G}:E\mapsto\rho(E)$, induced by an
automorphism $\rho\in\operatorname{Aut}(G)$, depends only on the class
$[\rho]$ in the quotient
$\operatorname{Out}(G):=\operatorname{Aut}(G)/\operatorname{Inn}(G)$. With
abuse of notation, we denote by the same symbol $\rho$ an automorphism $\rho$
of $G$ and its equivalence class $[\rho]$ in $\operatorname{Out}(G)$. Next, we
want to show that all these automorphisms of $\mathbf{Bun}_{G}$ actually
descend to automorphisms of the moduli space $\overline{\mathcal{M}}_{G}$ of
semistable bundles.

######  Lemma 5.2.

Let $E$ be a simple, resp. stable, resp. semistable, $G$-bundle. Then the
following $G$-bundles are simple, resp. stable, resp. semistable.

  1. (i)

$\sigma^{*}(E)$, where $\sigma\in\operatorname{Aut}(C)$,

  2. (ii)

$\rho(E)$, where $\rho\in\operatorname{Out}(G)$,

  3. (iii)

$E\mapsto E\otimes Z$, where $Z$ is a $\mathscr{Z}(G)$-bundle over $C$.

###### Proof.

We recall that the operations above define automorphisms of the moduli stack
$\mathbf{Bun}_{G}$. In particular, all the bundles in the list have
automorphism groups equal to $\operatorname{Aut}(E)$. Hence, if $E$ simple,
the same holds for the other bundles in the list. We now focus on the
(semi)stable case. Fix a parabolic subgroup $P\subset G$.
$(\ref{L:sss1})$ The $P$-reductions of $E$ are in bijection with the
$P$-reductions of $\sigma^{*}E$. Indeed, $F$ is a $P$-reduction of $E$ if and
only if $\sigma^{*}F$ is a $P$-reduction of $\sigma^{*}E$. Furthermore, the
pull-back along $\sigma$ commutes with the formation of the adjoint bundles
and preserves the degree, i.e.

$\deg\left(F\times^{P}\mathfrak{p}\right)=\deg\left(\sigma^{*}F\times^{P}\mathfrak{p}\right).$

Putting all together, we get that $E$ is (semi)stable if and only if
$\sigma^{*}E$ is (semi)stable.
$(\ref{L:sss2})$ Since $\rho$ is an isomorphism, it induces a bijection
between the parabolic subgroups of $G$, i.e. $P$ is parabolic if and only if
$\rho(P)$ is parabolic. Moreover, similarly to the previous point, the
$P$-reductions of $E$ are in bijection with the $\rho(P)$-reductions of
$\rho(E)$. Indeed, $F$ is a $P$-reduction of $E$ if and only if