# Infinitesimal deformations of opers
in positive characteristic and
the de Rham cohomology of symmetric products

Yasuhiro Wakabayashi

###### Abstract.

The Eichler-Shimura isomorphism describes a certain cohomology group with
coefficients in a space of polynomials by using holomorphic modular/cusp
forms. It determines a canonical decomposition of the corresponding de Rham
cohomology group associated to a specific oper on a Riemann surface. One
purpose of the present paper is to establish its analogue for opers in
positive characteristic. We first discuss some basic properties on the
(parabolic) de Rham cohomology groups and deformation spaces of $G$-opers
(where $G$ is a semisimple algebraic group of adjoint type) in a general
formulation. In particular, it is shown that the deformation space of a
$G$-oper induced from an $\mathrm{SL}_{2}$-oper decomposes into a direct sum
of the (parabolic) de Rham cohomology groups of its symmetric products. As a
consequence, we obtain an Eichler-Shimura-type decomposition for dormant opers
on general pointed stable curves by considering a transversal intersection of
related spaces in the de Rham moduli space.

††footnotetext: Y. Wakabayashi: Graduate School of Information Science and
Technology, Osaka University, Suita, Osaka 565-0871, Japan;††footnotetext:
e-mail: wakabayashi@ist.osaka-u.ac.jp;††footnotetext: 2020 Mathematical
Subject Classification: Primary 14H60, Secondary 14G17;††footnotetext: Key
words: Eichler-Shimura isomorphism, infinitesimal deformation, oper, positive
characteristic, de Rham cohomlogy

###### Contents

  1. 1 Introduction
    1. 1.1 The Eichler-Shimura isomorphism for a modular curve
    2. 1.2 Analogous decomposition in positive characteristic
    3. 1.3 Outline of our argument
    4. 1.4 Notation and Conventions
  2. 2 Flat $G$-bundles and opers on a pointed stable curve
    1. 2.1 Flat $G$-bundles on a pointed stable curve
    2. 2.2 The Cartan decomposition of a Lie algebra
    3. 2.3 Residues and radii
    4. 2.4 $G$-opers on a pointed stable curve
  3. 3 The moduli space of opers and change of structure group
    1. 3.1 The moduli space of $G$-opers
    2. 3.2 Change of structure group
    3. 3.3 The affine structure on the moduli space
    4. 3.4 The $\mathrm{SL}_{n}$-oper associated to an $\mathrm{SL}_{2}$-oper
    5. 3.5 The $\mathrm{SL}_{2l+1}$-oper associated to a $\mathrm{PGL}_{2}$-oper
  4. 4 Infinitesimal deformations of opers
    1. 4.1 De Rham cohomology/Parabolic de Rham cohomology
    2. 4.2 Duality for de Rham cohomology
    3. 4.3 Duality for parabolic de Rham cohomology
    4. 4.4 The deformation space of a $G$-oper I
    5. 4.5 The deformation space of a $G$-oper II
  5. 5 Cohomology of the symmetric products of opers
    1. 5.1 The de Rham cohomology of an $\mathrm{SL}_{n}$-oper
    2. 5.2 Duality for symmetric products of opers
    3. 5.3 The irreducible decomposition of the adjoint bundle
  6. 6 Dormant opers and their deformations
    1. 6.1 Dormant opers on pointed stable curves
    2. 6.2 Example: a dormant $\mathrm{PGL}_{2}$-oper on a Shimura curve
    3. 6.3 The deformation space of a dormant $G$-oper
    4. 6.4 Ordinariness for dormant $G$-opers
  7. 7 Eichler-Shimura-type isomorphisms for dormant opers
    1. 7.1 The irreducible decomposition of the deformation space
    2. 7.2 The canonical decomposition derived from ordinariness

##  1\. Introduction

###  1.1. The Eichler-Shimura isomorphism for a modular curve

The Eichler-Shimura isomorphism describes a certain cohomology group with
coefficients in a space of polynomials by using holomorphic modular/cusp
forms. To be precise, let $N$ be an integer with $N\geq 5$, and $\Gamma$ the
level $N$ congruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$. Given each
$l\geq 2$, we have the $\mathbb{C}$-vector space

(1) $\displaystyle V_{l-2}:=\mathrm{Sym}^{l-2}(\mathbb{C}X+\mathbb{C}Y)$

of homogenous degree $l-2$ polynomials in $\mathbb{C}[X,Y]$. It has a
$\Gamma$-action given by $\gamma(P)(X,Y)=P(aX+cY,bX+dY)$ for any $P\in
V_{l-2}$ and $\gamma:=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma$.

The Eichler-Shimura isomorphism (cf. [Eic], [Shi]) asserts a canonical
isomorphism for the cohomology group $H^{1}(\Gamma,V_{l-2})$ of the resulting
$\mathbb{C}[\Gamma]$-module $V_{l-2}$:

(2) $\displaystyle
M_{l}(\Gamma,\mathbb{C})\oplus\overline{S_{l}(\Gamma,\mathbb{C})}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(\Gamma,V_{l-2}),$

where $\overline{(-)}$ denotes the complex conjugation and
$M_{l}(\Gamma,\mathbb{C})$ (resp., $S_{l}(\Gamma,\mathbb{C})$) denotes the
$\mathbb{C}$-vector space of modular forms (resp., cusp forms) of weight $l$
and level $N$.

This isomorphism can be interpreted in geometric terms, as follows. Let
$Y_{\Gamma}$ denote the Riemann surface defined as the quotient
$\Gamma\backslash\mathbb{H}$ of the upper half plane
$\mathbb{H}:=\left\\{z\in\mathbb{C}\,|\,\mathrm{Im}(z)>0\right\\}$ by the
natural $\Gamma$-action. Also, let
$X_{\Gamma}:=\Gamma\backslash\mathbb{H}^{*}$, where
$\mathbb{H}^{*}:=\mathbb{H}\cup\mathbb{P}^{1}(\mathbb{Q})$ (as a subset of
$\mathbb{P}^{1}(\mathbb{C})$, equipped with a natural
$\mathrm{SL}_{2}(\mathbb{R})$-action) has a topology inducing a structure of
compact Riemann surface on the quotient $X_{\Gamma}$. The moduli
interpretation of $Y_{\Gamma}$ brings us a universal family of complex tori
$\pi:E\rightarrow Y_{\Gamma}$ of level $N$.

The relative de Rham cohomology of this family forms a holomorphic vector
bundle $\mathcal{F}_{Y}:=\mathbb{R}^{1}\pi_{*}\Omega_{E/Y_{\Gamma}}^{\bullet}$
of rank $2$; it carries a (flat) connection $\nabla_{Y}$ (called the Gauss-
Manin connection), as well as a $2$-step decreasing filtration
$\\{\mathcal{F}_{Y}^{j}\\}_{j=0}^{2}$ (called the Hodge filtration) given by
$\mathcal{F}^{0}_{Y}:=\mathcal{F}_{Y}$, $\mathcal{F}^{2}_{Y}:=0$, and
$\mathcal{F}^{1}_{Y}:=\pi_{*}(\Omega_{E/Y_{\Gamma}})$, considered as a line
subbundle of $\mathcal{F}_{Y}$ via the injection
$\mathcal{F}^{1}_{Y}\hookrightarrow\mathcal{F}_{Y}$ arising from the Hodge to
de Rham spectral sequence
${}^{\prime}E_{1}^{a,b}:=\mathbb{R}^{b}\pi_{*}\Omega^{a}_{E/Y_{\Gamma}}\Rightarrow\mathbb{R}^{a+b}\pi_{*}\Omega_{E/Y_{\Gamma}}^{\bullet}$.

Each object in the data
$(\mathcal{F}_{Y},\nabla_{Y},\\{\mathcal{F}^{j}_{Y}\\}_{j=0}^{2})$ has a
natural extension over $X_{\Gamma}$ by admitting simple poles along the cusps
$D:=X_{\Gamma}\setminus Y_{\Gamma}$; the resulting collection

(3)
$\displaystyle\mathscr{F}_{X}^{\heartsuit}:=(\mathcal{F}_{X},\nabla_{X},\\{\mathcal{F}_{X}^{j}\\}_{j=0}^{2})$

specifies a canonical example of an $\mathrm{SL}_{2}$-oper (cf. Remark 2.1),
and the line bundle $\varTheta:=\mathcal{F}_{X}^{1}$ satisfies
$\varTheta^{\otimes 2}\cong\Omega_{X}(D)$.

We shall write $\mathrm{Sym}^{l-2}\mathscr{F}_{X}$ for the $(l-2)$-nd
symmetric power of the flat bundle
$\mathscr{F}_{X}:=(\mathcal{F}_{X},\nabla_{X})$. The $1$-st de Rham cohomology
group $H_{\mathrm{dR}}^{1}(X,\mathrm{Sym}^{l-2}\mathscr{F}_{X})$ of
$\mathrm{Sym}^{l-2}\mathscr{F}_{X}$ fits into the short exact sequence

(4) $\displaystyle 0\longrightarrow H^{0}(X,\varTheta^{\otimes
l})\longrightarrow
H_{\mathrm{dR}}^{1}(X,\mathrm{Sym}^{l-2}\mathscr{F}_{X})\longrightarrow
H^{0}(X,\varTheta^{\otimes l}(-D))^{\vee}\longrightarrow 0$

arising from the filtration on $\mathrm{Sym}^{l-2}\mathscr{F}_{X}$ induced by
$\\{\mathcal{F}_{X}^{j}\\}_{j}$. Since $\mathrm{Sym}^{l-2}\mathscr{F}_{X}$
corresponds to the local system determined by the $\mathbb{C}[\Gamma]$-module
$V_{l-2}$ via the Riemann-Hilbert correspondence, there exists a canonical
identification

(5) $\displaystyle
H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{l-2}\mathscr{F}_{X})=H^{1}(\Gamma,V_{l-2})$

(cf. [Del, Chap. II, Corollaire 6.10]). On the other hand, the elements of
$H^{0}(X,\varTheta^{\otimes l})$ can be interpreted, via descent along the
covering map $\mathbb{H}\twoheadrightarrow Y_{\Gamma}$, as those of
$M_{l}(\Gamma,\mathbb{C})$, and moreover, the Petersson inner product yields
$H^{0}(X,\varTheta^{\otimes
l}(-D))^{\vee}=\overline{S_{l}(\Gamma,\mathbb{C})}$.

Under these identifications, the Eichler-Shimura isomorphism (2) may be
regarded as a canonical choice of a splitting of the short exact sequence (4)
given by taking periods. See, e.g., [BrHa], [KaSc], [Sch] for algebraic
treatments of the Eichler-Shimura isomorphism in terms of de Rham cohomology.

###  1.2. Analogous decomposition in positive characteristic

The present paper aims to investigate the short exact sequence (4) for opers
in positive characteristic (including the case of parabolic de Rham
cohomology) in relation to their deformation spaces, as well as to establish
an analogue of the Eichler-Shimura type decomposition for such opers. The
present paper contains many arguments based on those in [Wak4] and can be
positioned as a continuation of that work.

Let $p$ be an odd prime, $l$ an integer with $3\leq l\leq p-1$, $k$ an
algebraically closed field of characteristic $p$, $(g,r)$ a pair of
nonnegative integers with $2g-2+r>0$, and
$\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i=1}^{r})$ an $r$-pointed stable curve over
$k$ of genus $g$, where $X$ denotes a nodal curve over $k$ and
$\\{\sigma_{i}\\}_{i}$ is a set of marked points on $X$. (The present paper
deals not only with smooth curves but also with pointed and stable curves,
since the same arguments can be applied. However, it is not so essential to
consider such a fairly general situation.)

Suppose that we are given an $\mathrm{SL}_{2}$-oper
$\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\\{\mathcal{F}^{j}\\}_{j})$ on
$\mathscr{X}$. Then, the $2$-fold tensor product of the line bundle
$\varTheta:=\mathcal{F}^{1}\left(\subseteq\mathcal{F}\right)$ is isomorphic to
the sheaf $\Omega_{X^{\mathrm{log}}}$ of logarithmic $1$-forms on the log
curve $X^{\mathrm{log}}$ determined by $\mathscr{X}$. Just as in the complex
case, $\mathscr{F}^{\heartsuit}$ associates the $1$-st de Rham cohomology
$H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{l-2}\mathscr{F})$ of the $(l-2)$-nd
symmetric power $\mathrm{Sym}^{l-2}\mathscr{F}$ of
$\mathscr{F}:=(\mathcal{F},\nabla)$, fitting into the short exact sequence

(6) $\displaystyle 0\longrightarrow H^{0}(X,\varTheta^{\otimes
l})\longrightarrow
H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{l-2}\mathscr{F})\longrightarrow
H^{0}(X,\varTheta^{\otimes l}(-D))^{\vee}\longrightarrow 0$

(cf. Theorem 5.2, (i)), where $D$ denotes the divisor on $X$ given by the
union of the $\sigma_{i}$’s. (Similarly, a version for the parabolic de Rham
cohomology of $\mathrm{Sym}^{l-1}\mathscr{F}$ holds, as asserted in Theorem
5.2, (ii).)

Our question regarding this sequence is described as follows:

> For what kind of $\mathrm{SL}_{2}$-opers $\mathscr{F}^{\heartsuit}$ can we
> construct a canonical splitting of (6) arising from geometry such as the
> classical Eichler-Shimura isomorphism?

Even when $\mathscr{F}^{\heartsuit}$ is taken to be the mod $p$ reduction of
the “$\mathscr{F}_{X}^{\heartsuit}$” defined on a modular curve as above, the
classical Eichler-Shimura isomorphism would not be appropriate in constructing
a canonical splitting because of its non-algebraic construction.

To get an answer to the above question, we discuss opers with vanishing
$p$-curvature, called dormant opers. Here, recall that $p$-curvature is an
invariant for (flat) connections measuring the obstruction to the
compatibility of $p$-power structures appearing in certain associated spaces
of infinitesimal symmetries; this is a key ingredient of the $p$-adic
Teichmuller theory developed by S. Mochizuki (cf. [Moc1], [Moc2]), in which
various moduli spaces of $\mathrm{SL}_{2}$-opers (or $\mathrm{PGL}_{2}$-opers)
characterized by the degree of vanishing $p$-curvature were investigated. That
work suggests that $\mathrm{SL}_{2}$-opers whose $p$-curvature are close to
zero may be thought of as candidates for suitable analogues of the canonical
$\mathrm{SL}_{2}$-opers on modular curves. In fact, we prove the following
assertion, which is one of the main results in the present paper (cf. Theorem
7.7 for a similar assertion with slightly relaxed assumptions).

######  Theorem A (cf. Corollary 7.8).

Let us keep the above notation. Moreover, suppose that $l$ is even,
$\mathscr{F}^{\heartsuit}$ is dormant (i.e. has vanishing $p$-curvature), and
$\mathscr{X}$ is general in the moduli space $\overline{\mathcal{M}}_{g,r}$ of
$r$-pointed stable curves of genus $g$. Then, there exists a canonical
splitting of (6). In particular, we obtain a canonical decomposition

(7) $\displaystyle
H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{l-2}\mathscr{F})=H^{0}(X,\varTheta^{\otimes
l})\oplus H^{0}(X,\varTheta^{\otimes l}(-D))^{\vee}$

of the $1$-st de Rham cohomology
$H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{l-2}\mathscr{F})$ of
$\mathrm{Sym}^{l-2}\mathscr{F}$.

Other relatively important results are Theorems 3.1, 4.9, 4.12, 5.2, 5.3, 7.2,
7.7, and Corollaries 7.5, 7.6.

###  1.3. Outline of our argument

The present paper develops a general theory of opers on pointed stable curves
and their infinitesimal deformations, that may be positioned as a continuation
of the related work in [Wak4]. (The first half of our discussion proceeds
under the situation where the characteristic of the base field is not
necessarily positive.) Here, we briefly describe the arguments and results
leading to the above theorem. (There are many assertions proved in this text
that are not directly involved in proving Theorem A. For example, the study of
parabolic de Rham cohomology groups and their self-duality assertion are
included; see Corollary 4.6. Regarding the self-duality, we can find some
previous results involved, e.g., [Can, Theorem 5.2, (a)], [Ogu2, Remark 3.2],
and [Sch, Theorem 2.7, (i)].)

Let $G$ be a semisimple algebraic group of adjoint type with certain
properties (cf. the assumption imposed at the beginning of § 1.4). The GIT
quotient of its Lie algebra $\mathfrak{g}:=\mathrm{Lie}(G)$ by the adjoint
$G$-action defines a $k$-scheme $\mathfrak{c}$. We shall fix an element $\rho$
of $\mathfrak{c}(k)^{\times r}$, which lifts to an element
$\widetilde{\rho}\in\mathfrak{g}^{\times r}$ via the Kostant section (cf.
(39)).

Denote by

(8) $\displaystyle\mathcal{C}onn_{G}$

(cf. (20)) the moduli stack classifying flat $G$-bundles on $\mathscr{X}$. As
proved in Theorem 3.1, the moduli stack

(9) $\displaystyle\mathcal{O}p_{G}\ \left(\text{resp.,}\
\mathcal{O}p_{G,\rho}\right)$

classifying $G$-opers (resp., $G$-opers of radii $\rho$) forms a locally
closed substack of $\mathcal{C}onn_{G}$.

If we are given a $G$-oper $\mathscr{E}^{\spadesuit}$ classified by a closed
point $q\in\mathcal{O}p_{G,\rho}$, then the inclusion
$\mathcal{O}p_{G}\hookrightarrow\mathcal{C}onn_{G}$ induces, via
differentiation, a $k$-linear inclusion $T_{q}\mathcal{O}p_{G}\hookrightarrow
T_{q}\mathcal{C}onn_{G}$ between the tangent spaces at $q$. We investigate
these spaces, as well as the relationship between them, by describing relevant
infinitesimal deformations in terms of various (de Rham) cohomology groups
associated to $\mathscr{E}^{\spadesuit}$. The resulting description allows us
to obtain a short exact sequence

(10) $\displaystyle 0\longrightarrow T_{q}\mathcal{O}p_{G}$
$\displaystyle\xrightarrow{\mathrm{inclusion}}T_{q}\mathcal{C}onn_{G}\longrightarrow
T_{q}^{\vee}\mathcal{O}p_{G,\rho}\longrightarrow 0$

(cf. Theorem 4.9), where $T^{\vee}_{q}(-)$ denotes the cotangent space at $q$.

Next, since the base field $k$ is of characteristic $p$, one can define the
closed substack

(11) $\displaystyle\mathcal{C}onn_{G}^{\psi=0}$

(cf. (242)) of $\mathcal{C}onn_{G}$ classifying flat $G$-bundles with
vanishing $p$-curvature. The intersection

(12)
$\displaystyle\mathcal{O}p_{G}^{{}^{\mathrm{Zzz...}}}:=\mathcal{O}p_{G}\cap\mathcal{C}onn_{G}^{\psi=0}\
$

(cf. (243), (244)) is nothing but the moduli space of dormant $G$-opers and
was substantially investigated in [Wak4] from a viewpoint of enumerative
geometry. One important fact proved in loc. cit. is the étaleness of
$\mathcal{O}p_{G}^{{}^{\mathrm{Zzz...}}}\\!$ for a sufficiently general
$\mathscr{X}$ (cf. [Wak4, Theorem 8.19]). (This fact also played an essential
role in the proof of Joshi’s conjecture; see [Wak4, Theorem 9.10].) Since the
equality of dimensions

(13)
$\displaystyle\mathrm{dim}(T_{q}\mathcal{O}p_{G})+\mathrm{dim}(T_{q}\mathcal{C}onn_{G}^{\psi=0})=\mathrm{dim}(T_{q}\mathcal{C}onn_{G})$

holds (cf. Proposition 6.1), the ètaleness means that $\mathcal{O}p_{G}$ and
$\mathcal{C}onn_{G}^{\psi=0}$ intersect transversally. Hence, the composite

(14) $\displaystyle T_{q}\mathcal{C}onn_{G}^{\psi=0}\hookrightarrow
T_{q}\mathcal{C}onn_{G}\twoheadrightarrow T^{\vee}_{q}\mathcal{O}p_{G,\rho}$

is an isomorphism, and the composite of its inverse and the inclusion
$T_{q}\mathcal{C}onn_{G}^{\psi=0}\hookrightarrow T_{q}\mathcal{C}onn_{G}$
determines a split injection of (10). It follows that there exists a canonical
decomposition

(15) $\displaystyle T_{q}\mathcal{O}p_{G}\oplus
T_{q}^{\vee}\mathcal{O}p_{G}=T_{q}\mathcal{C}onn_{G}\ $

of $T_{q}\mathcal{C}onn_{G}$.

Finally, let us consider the case of $G=\mathrm{PGL}_{n}$ with
$n=\frac{p-1}{2}$. (This is only the setting considered here for simplicity,
and the main text will deal with a general $G$.) Suppose further that
$\mathscr{E}^{\spadesuit}$ comes from the “$\mathscr{F}^{\heartsuit}$” as in
the statement of Theorem A via change of structure group
$\mathrm{SL}_{2}\rightarrow\mathrm{PGL}_{n}$ obtained by integrating a
suitable principal $\mathfrak{s}\mathfrak{l}_{2}$-subalgebra of
$\mathfrak{p}\mathfrak{g}\mathfrak{l}_{n}$. Then, a natural affine structure
on $\mathcal{O}p_{G}$ yields direct sum decompositions

(16) $\displaystyle
T_{q}\mathcal{O}p_{G}=\bigoplus_{j=1}^{n-1}H^{0}(X,\varTheta^{\otimes
2j}),\hskip
28.45274ptT_{q}^{\vee}\mathcal{O}p_{G,\rho}=\bigoplus_{j=1}^{n-1}H^{0}(X,\varTheta^{\otimes
2j}(-D))^{\vee}$

(cf. [Wak4, Theorem 2.24]). On the other hand, we show (cf. (92), (230)) that
the $k$-vector space $T_{q}\mathcal{C}onn_{G}$ decomposes into a direct sum

(17) $\displaystyle
T_{q}\mathcal{C}onn_{G}=\bigoplus_{j=1}^{n-1}H_{\mathrm{dR}}^{1}(X,\mathrm{Sym}^{2j}\mathscr{F})$

according to the irreducible decomposition of the
$\mathfrak{s}\mathfrak{l}_{2}$-algebra
$\mathfrak{p}\mathfrak{g}\mathfrak{l}_{n}$. By various definitions involved,
(15) turns out to be compatible with the respective structures of direct sum
decomposition. Thus, the required decomposition (7) can be obtained by
considering its $(l/2)$-th factor.

######  Remark 1.1.

As observed in [Wak8], it can be seen that the construction of the canonical
decomposition (7) discussed above is entirely similar to that of the classical
Eichler-Shimura isomorphism, by regarding $\mathcal{C}onn_{G}^{\psi=0}$ as a
positive characteristic analogue of the moduli space of $G$-local systems with
real monodromy.

###  1.4. Notation and Conventions

The basic terminology and notation in our discussion follow [Wak4]. Throughout
the present paper, we fix an algebraically closed field $k$, a semisimple
algebraic group $G$ over $k$, a maximal torus $T$ of $G$, and a Borel subgroup
$B$ of $G$ containing $T$. Write $\mathfrak{g}$, $\mathfrak{b}$, and
$\mathfrak{t}$ for the Lie algebras of $G$, $B$, and $T$, respectively.
Whenever we discuss the case of $G=\mathrm{SL}_{n}$ or $\mathrm{PGL}_{n}$, the
group $B$ (resp., $T$) is assumed to be the subgroup of $G$ consisting of
upper triangular matrices (resp., diagonal matrices).

Regarding the algebraic group $G$, we assume that either
“$\mathrm{char}(k)=0$” or “$\mathrm{char}(k)=p>2h_{G}$” or
“$G=\mathrm{PGL}_{n}$ with $1<n<\mathrm{char}(k)$” is fulfilled, where $h_{G}$
denotes $1$ plus the height of a root with maximum height, as defined in
[Wak4, § 1.4.1]. (If $G$ is simple, then $h_{G}$ coincides with its Coxeter
number.) In particular, when $\mathrm{char}(k)=p>0$, the prime $p$ is very
good for $G$, in the sense of [KiWe, Chap. VI, Definition 1.6] (cf. the proof
of Lemma 3.2).

Next, given an fs log scheme $T^{\mathrm{log}}$ and an fs log scheme
$Y^{\mathrm{log}}$ over $T^{\mathrm{log}}$, we shall write
$\Omega_{Y^{\mathrm{log}}/T^{\mathrm{log}}}$ for the sheaf of logarithmic
$1$-forms on $Y^{\mathrm{log}}$ over $T^{\mathrm{log}}$, and write
$\mathcal{T}_{Y^{\mathrm{log}}/T^{\mathrm{log}}}$ for its dual, i.e., the
sheaf of locally defined logarithmic derivations on $\mathcal{O}_{Y}$ relative
to $T^{\mathrm{log}}$.

We fix a pair of nonnegative integers $(g,r)$ with $2g-2+r>0$ and an
$r$-pointed stable curve $\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i=1}^{r})$ of
genus $g$ over $k$, where $X$ denotes the underlying curve and
$\\{\sigma_{i}\\}_{i=1}^{r}$ denotes the set of its marked points. Recall from
[Kato, Theorem 2.6] that $\mathrm{Spec}(k)$ and $X$ admit canonical log
structures pulled-back, via the classifying map of $\mathscr{X}$, from the
moduli stack of $r$-pointed stable curves of genus $g$ and the universal
family of curves over that stack, respectively; the resulting log schemes are
denoted by $\mathrm{Spec}(k)^{\mathrm{log}}$ and $X^{\mathrm{log}}$,
respectively. For simplicity, we write
$\Omega:=\Omega_{X^{\mathrm{log}}/\mathrm{Spec}(k)^{\mathrm{log}}}$ and
$\mathcal{T}:=\mathcal{T}_{X^{\mathrm{log}}/\mathrm{Spec}(k)^{\mathrm{log}}}$.
Also, write $d$ for the logarithmic universal derivation
$\mathcal{O}_{X}\rightarrow\Omega$.

Let $D$ denote the divisor $D$ defined as the union of the $\sigma_{i}$’s. For
each $\mathcal{O}_{X}$-module $\mathcal{F}$, we set
${{}^{c}}\mathcal{F}:=\mathcal{F}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}(-D)$.
In particular, ${{}^{c}}\Omega$ is isomorphic to the dualizing sheaf
$\omega_{X}$ of $X/k$.

##  2\. Flat $G$-bundles and opers on a pointed stable curve

In this section, we recall the definition of a flat $G$-bundle, as well as a
$G$-oper, on a pointed stable curve. We refer the reader to, e.g., [Wak4] for
detailed discussions on $G$-opers in such a general formulation. See also
[BeDr], [JoPa] and [Moc2].

###  2.1. Flat $G$-bundles on a pointed stable curve

We shall fix a (right) $G$-bundle $\pi:\mathcal{E}\rightarrow X$ on $X$. Given
a $k$-vector space $\mathfrak{h}$ equipped with a $G$-action, we write
$\mathfrak{h}_{\mathcal{E}}$ for the vector bundle on $X$ associated to the
affine space
$\mathcal{E}\times^{G}\mathfrak{h}\left(:=(\mathcal{E}\times_{k}\mathfrak{h})/G\right)$.
In the case where $\mathfrak{h}$ is taken to be the Lie algebra $\mathfrak{g}$
equipped with the $G$-action via the adjoint representation
$\mathrm{Ad}:G\rightarrow\mathrm{GL}(\mathfrak{g})$, the resulting vector
bundle $\mathfrak{g}_{\mathcal{E}}$ is called the adjoint bundle of
$\mathcal{E}$.

By pulling-back the log structure of $X^{\mathrm{log}}$ via $\pi$, one obtains
a log structure on $\mathcal{E}$; the resulting log scheme is denoted by
$\mathcal{E}^{\mathrm{log}}$. The $G$-action on $\mathcal{E}$ induces a
$G$-action on the direct image
$\pi_{*}(\mathcal{T}_{\mathcal{E}^{\mathrm{log}}/\mathrm{Spec}(k)^{\mathrm{log}}})$
of $\mathcal{T}_{\mathcal{E}^{\mathrm{log}}/\mathrm{Spec}(k)^{\mathrm{log}}}$.
We shall set
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}:=\pi_{*}(\mathcal{T}_{\mathcal{E}^{\mathrm{log}}/\mathrm{Spec}(k)^{\mathrm{log}}})^{G}$,
i.e., the subsheaf of $G$-invariant sections of
$\pi_{*}(\mathcal{T}_{\mathcal{E}^{\mathrm{log}}/\mathrm{Spec}(k)^{\mathrm{log}}})$.
The differential of $\pi$ gives rise to a canonical short exact sequence of
$\mathcal{O}_{X}$-modules

(18) $\displaystyle
0\longrightarrow\mathfrak{g}_{\mathcal{E}}\longrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}\xrightarrow{d_{\mathcal{E}}}\mathcal{T}\longrightarrow
0$

(cf. [Wak4, § 1.2.5]).

Recall that a log $k$-connection (or, a log connection, for short) on
$\mathcal{E}$ is, by definition, an $\mathcal{O}_{X}$-linear morphism
$\nabla:\mathcal{T}\rightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}$
with
$d_{\mathcal{E}}\circ\nabla=\mathrm{id}_{\mathcal{T}_{X^{\mathrm{log}}}}$.
Since $\Omega$ is a line bundle, any log connection is flat, i.e., has
vanishing curvature (cf. [Wak4, Definition 1.23] for the definition of
curvature). By a flat $G$-bundle on $\mathscr{X}$, we shall mean a pair
$(\mathcal{E},\nabla)$ consisting of a (right) $G$-bundle $\mathcal{E}$ on $X$
and a (flat) log connection $\nabla$ on $\mathcal{E}$. For example, any flat
$\mathrm{SL}_{n}$-bundle (where $n\geq 1$) on $\mathscr{X}$ can be identified
with a collection

(19) $\displaystyle(\mathcal{F},\nabla,\delta)$

consisting of a rank $n$ vector bundle $\mathcal{F}$ on $X$, a log connection
$\nabla$ on $\mathcal{F}$ in the usual sense (i.e., a $k$-linear morphism
$\nabla:\mathcal{F}\rightarrow\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F}$
satisfying the Leibnitz rule), and an isomorphism of flat bundles
$\delta:(\mathrm{det}(\mathcal{F}),\mathrm{det}(\nabla))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathcal{O}_{X},d)$,
where $\mathrm{det}(\nabla)$ denotes the log connection on the determinant
bundle $\mathrm{det}(\mathcal{F})$ of $\mathcal{F}$ induced naturally from
$\nabla$.

We denote by

(20) $\displaystyle\mathcal{C}onn_{G}$

the moduli stack classifying flat $G$-bundles on $\mathscr{X}$. It is verified
(from, e.g., [Wan, Theorem 1.0.1]) that $\mathcal{C}onn_{G}$ may be
represented by an algebraic stack over $k$ in the sense of Artin, and locally
of finite type over $k$.

###  2.2. The Cartan decomposition of a Lie algebra

Denote by $\Gamma$ the set of simple roots in $B$ with respect to $T$. For
each character $\beta$ of $T$, we set

(21)
$\displaystyle\mathfrak{g}^{\beta}:=\left\\{x\in\mathfrak{g}\,|\,\mathrm{Ad}(t)(x)=\beta(t)\cdot
x\ \text{for all}\ t\in T\right\\}.$

Recall that the Cartan decomposition of $\mathfrak{g}$ is a Lie algebra
grading $\mathfrak{g}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j}$ such that
$\mathfrak{g}_{j}=\mathfrak{g}_{-j}=0$ for all $j>\mathrm{rk}(\mathfrak{g})$,
where $\mathrm{rk}(\mathfrak{g})$ denotes the rank of $\mathfrak{g}$, and
$\mathfrak{g}_{0}=\mathfrak{t}$,
$\mathfrak{g}_{1}=\bigoplus_{\alpha\in\Gamma}\mathfrak{g}^{\alpha}$,
$\mathfrak{g}_{-1}=\bigoplus_{\alpha\in\Gamma}\mathfrak{g}^{-\alpha}$. It
associates the decreasing filtration $\\{\mathfrak{g}^{j}\\}_{j\in\mathbb{Z}}$
on $\mathfrak{g}$ given by $\mathfrak{g}^{j}:=\bigoplus_{l\geq
j}\mathfrak{g}_{l}$.

Let us fix a collection of data
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt:=\\{x_{\alpha}\\}_{\alpha\in\Gamma}$, where each $x_{\alpha}$
denotes a generator of $\mathfrak{g}^{\alpha}$, and write
$q_{1}:=\sum_{\alpha\in\Gamma}x_{\alpha}$. (If we take two such collections,
then they are conjugate by an element of $T$. For that reason, the results
obtained in the subsequent discussions are essentially independent of the
choice of $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$.) Also, by regarding the fundamental coweight
$\check{\omega}_{\alpha}$ of $\alpha$ as an element of $\mathfrak{t}$ via
differentiation, we obtain
$\check{\rho}:=\sum_{\alpha\in\Gamma}\check{\omega}_{\alpha}\in\mathfrak{t}$.
Then, there is a unique collection $\\{y_{\alpha}\\}_{\alpha\in\Gamma}$ of
generators $y_{\alpha}\in\mathfrak{g}^{-\alpha}$, such that the set
$\\{q_{-1},2\check{\rho},q_{1}\\}$, where
$q_{-1}:=\sum_{\alpha\in\Gamma}y_{\alpha}\in\mathfrak{g}_{-1}$, forms an
$\mathfrak{s}\mathfrak{l}_{2}$-triple.

If $G=\mathrm{PGL}_{n}$, then we identify its Lie algebra
$\mathfrak{p}\mathfrak{g}\mathfrak{l}_{n}$ with $\mathfrak{s}\mathfrak{l}_{n}$
via the composite isomorphism
$\mathfrak{s}\mathfrak{l}_{n}\hookrightarrow\mathfrak{g}\mathfrak{l}_{n}\twoheadrightarrow\mathfrak{p}\mathfrak{g}\mathfrak{l}_{n}$,
and assume that the triple $\\{q_{-1},2\check{\rho},q_{1}\\}$ in
$\mathfrak{s}\mathfrak{l}_{n}$ is always taken as

(22) $\displaystyle q_{-1}=\begin{pmatrix}0&0&\cdots&0&0\\\ 1&0&\cdots&0&0\\\
0&1&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&1&0\end{pmatrix},\ 2\check{\rho}=\begin{pmatrix}n-1&0&0&\cdots&0\\\
0&n-3&0&\cdots&0\\\ 0&0&n-5&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\cdots&-(n-1)\end{pmatrix},$ $\displaystyle
q_{1}=\begin{pmatrix}0&n-1&0&0&\cdots&0\\\ 0&0&2(n-2)&0&\cdots&0\\\
0&0&0&3(n-2)&\cdots&0\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&0&\cdots&n-1\\\ 0&0&0&0&\cdots&0\end{pmatrix}.$

Denote by $\mathfrak{c}_{G}$ (or $\mathfrak{c}$, for simplicity) the GIT
quotient of $\mathfrak{g}$ by the adjoint $G$-action. In particular, there
exists a natural projection $\chi:\mathfrak{g}\twoheadrightarrow\mathfrak{c}$.
This morphism factors through the projection
$\mathfrak{g}\rightarrow[\mathfrak{g}/G]$ to the quotient stack
$[\mathfrak{g}/G]$ of $\mathfrak{g}$ by the adjoint $G$-action. The resulting
morphism of $k$-stacks $[\mathfrak{g}/G]\rightarrow\mathfrak{c}$ will be
denoted by $[\chi]$.

Next, let us consider the space $\mathfrak{g}^{\mathrm{ad}(q_{1})}$ of
$\mathrm{ad}(q_{1})$-invariants, i.e.,
$\mathfrak{g}^{\mathrm{ad}(q_{1})}:=\left\\{x\in\mathfrak{g}\,|\,\mathrm{ad}(q_{1})(x)=0\right\\}$.
It has dimension equal to $\mathrm{rk}(\mathfrak{g})$, and the Cartan
decomposition $\mathfrak{g}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j}$
restricts to a decomposition
$\mathfrak{g}^{\mathrm{ad}(q_{1})}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})}$
on $\mathfrak{g}^{\mathrm{ad}(q_{1})}$. By the assumption on $G$ imposed in §
1.4, the composite

(23)
$\displaystyle\mathrm{Kos}:\mathfrak{g}^{\mathrm{ad}(q_{1})}\xrightarrow{v\mapsto
v+q_{-1}}\mathfrak{g}\xrightarrow{\chi}\mathfrak{c}.$

becomes an isomorphism of $k$-schemes. Denote by $[0]_{G}$ (or simply, $[0]$,
if there is no fear of confusion) the element of $\mathfrak{c}(k)$ determined
by the image of the zero section $0$ of $\mathfrak{g}^{\mathrm{ad}(q_{1})}$
via $\mathrm{Kos}$.

###  2.3. Residues and radii

Let $(\mathcal{E},\nabla)$ be a flat $G$-bundle on $\mathscr{X}$. By a marking
on $(\mathcal{E},\nabla)$, we mean a collection of Lie algebra isomorphisms
$\sigma_{i}^{*}(\mathfrak{g}_{\mathcal{E}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}$
($i=1,\cdots,r$). The notion of an isomorphism between flat $G$-bundles with
marking can be defined in an evident manner.

Under the assumption that $r>0$, let us choose $i\in\\{1,\cdots,r\\}$, and
moreover choose a local function $t\in\mathcal{O}_{X}$ defining the closed
subscheme $\mathrm{Im}(\sigma_{i})$ of $X$. Then, the element

(24)
$\displaystyle\mu_{i}^{\nabla}:=\overline{\nabla\left(t\frac{d}{dt}\right)}\in\sigma_{i}^{*}(\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}})$

lies in $\sigma_{i}^{*}(\mathfrak{g}_{\mathcal{E}})$ and does not depend on
the choice of $t$ (cf. the discussion in [Wak4, § 1.6.3]). We refer to
$\mu_{i}^{\nabla}$ as the residue (or, the monodromy operator, in the
terminology of [Moc2] and [Wak4]) of $\nabla$ at $\sigma_{i}$. If
$(\mathcal{E},\nabla)$ is equipped with a marking, then $\mu_{i}^{\nabla}$ can
be regarded as an element of $\mathfrak{g}$ by passing to the isomorphism
$\sigma^{*}_{i}(\mathfrak{g}_{\mathcal{E}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}$
given by this marking. In the case of $r=0$, any log connection is regarded as
being equipped with the trivial marking and having residue $\emptyset$.

Thus, for each $r$-tuple $\mu:=(\mu_{i})_{i=1}^{r}$ of elements of
$\mathfrak{g}$ (where $\mu:=\emptyset$ if $r=0$), we obtain the moduli stack

(25) $\displaystyle\mathcal{C}onn_{G,\mu}$

classifying flat $G$-bundles on $\mathscr{X}$ with marking whose residue at
$\sigma_{i}$ coincides with $\mu_{i}$ for every $i=1,\cdots,r$.

Next, note that, for each $i=1,\cdots,r$, the pair
$(\sigma_{i}^{*}(\mathcal{E}),\mu_{i}^{\nabla})$ specifies a $k$-rational
point of $[\mathfrak{g}/G]$; it defines a $k$-rational point

(26)
$\displaystyle\rho_{i}^{\nabla}:=[\chi]\left((\sigma_{i}^{*}(\mathcal{E}),\mu_{i}^{\nabla})\right)\in\mathfrak{c}(k)$

via the morphism $[\chi]:[\mathfrak{g}/G]\rightarrow\mathfrak{c}$. This
element is well-defined without choosing any marking, and we refer to
$\rho_{i}^{\nabla}$ as the radius of $\nabla$ at $\sigma_{i}$. Given an
$r$-tuple $\rho:=(\rho_{i})_{i=1}^{r}\in\mathfrak{c}^{\times r}$, we say that
$\nabla$ is of radii $\rho$ if its radius at $\sigma_{i}$ coincides with
$\rho_{i}$ for every $i=1,\cdots,r$. In the case of $r=0$, any log connection
is regarded as having radius $\emptyset$.

The stack $\mathcal{C}onn_{G}$ admits a closed substack

(27) $\displaystyle\mathcal{C}onn_{G,\rho}$

classifying flat $G$-bundles on $\mathscr{X}$ of radii $\rho$. By assigning
the residues of each flat bundle to its radii via
$\chi:\mathfrak{g}\rightarrow\mathfrak{c}$, we have a morphism of $k$-stacks

(28)
$\displaystyle\mathcal{C}onn_{G,\mu}\rightarrow\mathcal{C}onn_{G,\chi(\mu)},$

where $\chi(\mu):=(\chi(\mu_{i}))_{i=1}^{r}$.

###  2.4. $G$-opers on a pointed stable curve

Let $\mathcal{E}_{B}$ be a $B$-bundle on $X$. Denote by $\mathcal{E}_{G}$ the
$G$-bundle obtained from $\mathcal{E}_{B}$ via change of structure group by
the inclusion $B\hookrightarrow G$. (Hence, $\mathcal{E}_{B}$ specifies a
$B$-reduction of $\mathcal{E}_{G}$.) By using the injection
$\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}$
induced by the natural inclusion
$\mathcal{E}_{B}\hookrightarrow\mathcal{E}_{G}$, we regard
$\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}}$ as an
$\mathcal{O}_{X}$-submodule of
$\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}$. Since the
$k$-vector subspace $\mathfrak{g}^{j}$ of $\mathfrak{g}$ (for each
$j\in\mathbb{Z}$) is closed under the adjoint $B$-action, it induces a
subbundle $\mathfrak{g}_{\mathcal{E}_{B}}^{j}$ of
$\mathfrak{g}_{\mathcal{E}_{B}}\left(=\mathfrak{g}_{\mathcal{E}_{G}}\right)$.
For each $j\in\mathbb{Z}$, we shall set

(29)
$\displaystyle\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{j}:=\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}}+\mathfrak{g}_{\mathcal{E}_{B}}^{j}\left(\subseteq\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}\right).$

The inclusion
$\mathfrak{g}_{\mathcal{E}_{B}}\left(=\mathfrak{g}_{\mathcal{E}_{G}}\right)\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}$
yields an isomorphism

(30)
$\displaystyle\mathfrak{g}_{\mathcal{E}_{B}}^{j}/\mathfrak{g}_{\mathcal{E}_{B}}^{j+1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{j}/\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{j+1}.$

On the other hand, since each $\mathfrak{g}^{-\alpha}$ ($\alpha\in\Gamma$) is
closed under the adjoint $B$-action, the decomposition
$\mathfrak{g}^{-1}/\mathfrak{g}^{0}=\bigoplus_{\alpha\in\Gamma}\mathfrak{g}^{-\alpha}$
yields a canonical isomorphism
$\mathfrak{g}^{-1}_{\mathcal{E}_{B}}/\mathfrak{g}^{0}_{\mathcal{E}_{B}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigoplus_{\alpha\in\Gamma}\mathfrak{g}_{\mathcal{E}_{B}}^{-\alpha}$.
By composing it with the inverse of (30) for $j=-1$, we obtain a decomposition

(31)
$\displaystyle\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{-1}/\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigoplus_{\alpha\in\Gamma}\mathfrak{g}_{\mathcal{E}_{B}}^{-\alpha}.$

Now, let us consider a pair
$\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ consisting of a
$B$-bundle $\mathcal{E}_{B}$ on $X$ and a log connection $\nabla$ on the
$G$-bundle $\mathcal{E}_{G}:=\mathcal{E}_{B}\times_{B}G$ induced by
$\mathcal{E}_{B}$. Recall that $\mathscr{E}^{\spadesuit}$ is called a $G$-oper
on $\mathscr{X}$(cf. [Wak4, Definition 2.1]) if it satisfies the following two
conditions:

  * •

$\nabla(\mathcal{T})\subseteq\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{-1}$;

  * •

For any $\alpha\in\Gamma$, the composite

(32)
$\displaystyle\mathcal{T}\xrightarrow{\nabla}\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{-1}\twoheadrightarrow\widetilde{\mathcal{T}}^{-1}_{\mathcal{E}_{G}^{\mathrm{log}}}/\widetilde{\mathcal{T}}^{0}_{\mathcal{E}_{G}^{\mathrm{log}}}\twoheadrightarrow\mathfrak{g}^{-\alpha}_{\mathcal{E}_{B}}$

is an isomorphism, where the third arrow denotes the natural projection with
respect to the decomposition (31).

If $r>0$ and $\rho$ is an element of $\mathfrak{c}(k)^{\times r}$, then we say
that a $G$-oper $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ is of
radii $\rho$ if the flat $G$-bundle $(\mathcal{E}_{G},\nabla)$ is of radii
$\rho$ (cf. [Wak4, Definition 2.32]). In the case of $r=0$, any $G$-oper is
regarded as being of radius $\emptyset$.

######  Remark 2.1 (Case of $G=\mathrm{SL}_{n}$).

Each $\mathrm{SL}_{n}$-oper can be described in terms of vector bundles
because a $B$-reduction of an $\mathrm{SL}_{n}$-bundle corresponds to a
complete flag on the associated vector bundle. To be precise, a pair
$(\mathcal{E}_{B},\nabla)$ of a $B$-bundle $\mathcal{E}_{B}$ and a connection
$\nabla$ on
$\mathcal{E}_{\mathrm{SL}_{n}}:=\mathcal{E}_{B}\times_{B}\mathrm{SL}_{n}$ can
be translated into a collection of data

(33)
$\displaystyle\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\\{\mathcal{F}^{j}\\}_{j=0}^{n},\delta)$

such that $(\mathcal{F},\nabla,\delta)$ is a triple as in (19) and
$\\{\mathcal{F}^{j}\\}_{j=0}^{n}$ denotes a decreasing filtration
$0=\mathcal{F}^{n}\subseteq\mathcal{F}^{n-1}\subseteq\cdots\subseteq\mathcal{F}^{1}\subseteq\mathcal{F}^{0}=\mathcal{F}$
of $\mathcal{F}$ whose graded pieces $\mathcal{F}^{j}/\mathcal{F}^{j+1}$
($i=0,\cdots,n-1$) are line bundles. Moreover, such a collection corresponds
to an $\mathrm{SL}_{n}$-oper if it satisfies the following two conditions:

  * •

For every $j=1,\cdots,n-1$, the inclusion relation
$\nabla(\mathcal{F}^{j})\subseteq\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F}^{j-1}$
holds;

  * •

For every $j=1,\cdots,n-1$, the $\mathcal{O}_{X}$-linear morphism

(34)
$\displaystyle\mathrm{KS}_{\mathscr{F}^{\heartsuit}}^{j}:\mathcal{F}^{j}/\mathcal{F}^{j+1}\rightarrow\Omega\otimes_{\mathcal{O}_{X}}(\mathcal{F}^{j-1}/\mathcal{F}^{j})$

induced from $\nabla$ (by taking account of the former condition) is an
isomorphism. (The isomorphism $\mathrm{KS}_{\mathscr{F}^{\heartsuit}}^{j}$ is
called the $j$-th Kodaira-Spencer map for $\mathscr{F}^{\heartsuit}$.)

Also, such a collection without a choice of an isomorphism $\delta$ is called
a $\mathrm{GL}_{n}$-oper (cf. [Wak4, Definition 4.17]). Even when dealing with
an $\mathrm{SL}_{n}$-oper, we will omit the datum “$\delta$” in its notation
for simplicity.

##  3\. The moduli space of opers and change of structure group

This section deals with the moduli stack of $G$-opers and recall a
construction of $G$-opers using $\mathrm{PGL}_{2}$-opers (or
$\mathrm{SL}_{2}$-opers).

###  3.1. The moduli space of $G$-opers

Denote by

(35) $\displaystyle\mathcal{O}p_{G}$

the moduli stack classifying $G$-opers on $\mathscr{X}$. In the rest of the
present paper, we assume that $G$ is of adjoint type unless otherwise stated.
Then, it follows from [Wak4, Theorem A] that $\mathcal{O}p_{G}$ may be
represented by an affine scheme over $k$ (cf. § 3.3 for more details). The
assignment $(\mathcal{E}_{B},\nabla)\mapsto(\mathcal{E}_{G},\nabla)$ (for each
$G$-oper $(\mathcal{E}_{B},\nabla)$) determines a morphism of $k$-stacks

(36)
$\displaystyle\mathrm{Imm}_{G}:\mathcal{O}p_{G}\rightarrow\mathcal{C}onn_{G}.$

Next, suppose that we are given an element $\rho:=(\rho_{i})_{i=1}^{r}$ of
$\mathfrak{c}(k)^{\times r}$ (where $\rho:=\emptyset$ if $r=0$). It determines
a closed subscheme

(37) $\displaystyle\mathcal{O}p_{G,\rho}$

of $\mathcal{O}p_{G}$ classifying $G$-opers of radii $\rho$. The morphism
$\mathrm{Imm}_{G}$ restricts to a morphism of $k$-stacks

(38)
$\displaystyle\mathrm{Imm}_{G,\rho}:\mathcal{O}p_{G,\rho}\rightarrow\mathcal{C}onn_{G,\rho}.$

Let us take a $G$-oper $\mathscr{E}^{\spadesuit}$ on $\mathscr{X}$ of radii
$\rho$. According to [Wak4, Proposition 2.19], there exists a unique (up to
isomorphism) pair
$({{}^{\dagger}}\mathscr{E}^{\spadesuit},\mathrm{nor}_{\mathscr{E}^{\spadesuit}})$
consisting of a $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normal $G$-oper
${{}^{\dagger}}\mathscr{E}^{\spadesuit}:=({{}^{\dagger}}\mathcal{E}_{B},\nabla)$
on $\mathscr{X}$ and an isomorphism of $G$-opers
$\mathrm{nor}_{\mathscr{E}^{\spadesuit}}:{{}^{\dagger}}\mathscr{E}^{\spadesuit}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}^{\spadesuit}$
(cf. [Wak4, Definition 2.14] for the definition of
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normality). Here, recall from [Wak4, § 2.4.3] that, for each
$i=1,\cdots,r$, there exists a canonical $G$-equivariant isomorphism
$\sigma^{*}_{i}({{}^{\dagger}}\mathcal{E}_{G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G$,
where
${{}^{\dagger}}\mathcal{E}_{G}:={{}^{\dagger}}\mathcal{E}_{B}\times^{B}G$,
that induces a Lie algebra isomorphism
$\left(\sigma^{*}_{i}(\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{G}})=\right)\sigma^{*}_{i}(\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{B}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}$.
In this way, the underlying flat $G$-bundle
$({{}^{\dagger}}\mathcal{E}_{G},\nabla)$ of
${{}^{\dagger}}\mathscr{E}^{\spadesuit}$ admits a canonical choice of a
marking.

Also, by the definition of
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normality, the residue of $\nabla$ at $\sigma_{i}$ coincides with
$q_{-1}+\mathrm{Kos}^{-1}(\rho_{i})$. If we write

(39)
$\displaystyle\widetilde{\rho}\left(=(\widetilde{\rho}_{i})_{i=1}^{r}\right):=(q_{-1}+\mathrm{Kos}^{-1}(\rho_{i}))_{i=1}^{r},$

then the resulting assignment
$\mathscr{E}^{\spadesuit}\mapsto({{}^{\dagger}}\mathcal{E}_{G},\nabla)$
determines a morphism of $k$-stacks

(40)
$\displaystyle\mathrm{Imm}_{G,\widetilde{\rho}}:\mathcal{O}p_{G,\rho}\rightarrow\mathcal{C}onn_{G,\widetilde{\rho}}.$

The following assertion generalizes the fact in [BeDr, § 3.1.11, Remark,
(iii)], dealing with the case where the underlying curve is an unpointed
smooth projective curve over the field of complex numbers.

######  Theorem 3.1.

(Recall that $G$ is assumed to be of adjoint type.) The morphism of $k$-stacks
$\mathrm{Imm}_{G}$ (resp., $\mathrm{Imm}_{G,\rho}$; resp.,
$\mathrm{Imm}_{G,\widetilde{\rho}}$) is schematic and an immersion.

###### Proof.

It suffices to consider the non-resp’d assertion because the proofs of the
remaining ones are similar. Denote by $\mathcal{B}un_{G}$ (resp.,
$\mathcal{B}un_{B}$) the moduli stack of $G$-bundles (resp., $B$-bundles) on
$X$. By forgetting the data of log connections, we obtain a morphism of
$k$-stacks $\mathcal{C}onn_{G}\rightarrow\mathcal{B}un_{G}$. Note that the
morphism $\mathcal{B}un_{B}\rightarrow\mathcal{B}un_{G}$ induced by the change
of structure group using $B\hookrightarrow G$ is schematic (cf. [Wan,
Corollary 3.2.4]), and $\mathcal{O}p_{G}$ can be represented by a locally
closed substack of the fiber product
$\mathcal{C}onn_{G}\times_{\mathcal{B}un_{G}}\mathcal{B}un_{B}$. Hence, the
morphism $\mathrm{Imm}_{G}$ is verified to be schematic. The result of Lemma
3.2 described below implies that $\mathrm{Imm}_{G}$ is a monomorphism. By
Lemma 3.3, this morphism also satisfies the valuative criterion $(*)$ in the
sense of [Moc2, Chap I, § 2.4]. In particular, it is a radimmersion (cf.
[Moc2, Chap I, Theorem 2.12]). Consequently, it follows from [Moc2, Chap I,
Corollary 2.13] that $\mathrm{Imm}_{G}$ defines an immersion. ∎

The following two lemmas were applied in the proof of the above proposition.

######  Lemma 3.2.

Let $\mathscr{E}^{\spadesuit}_{i}:=(\mathcal{E}_{B,i},\nabla_{i})$ ($i=1,2$)
be $G$-opers on $\mathscr{X}$ and
$\alpha_{G}:(\mathcal{E}_{G,1},\nabla_{1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathcal{E}_{G,2},\nabla_{2})$
(where $\mathcal{E}_{G,i}:=\mathcal{E}_{B,i}\times^{B}G$) an isomorphism of
flat $G$-bundles. Then, there exists a unique isomorphism of $G$-opers
$\alpha_{B}:\mathscr{E}^{\spadesuit}_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}^{\spadesuit}_{2}$
inducing $\alpha_{G}$ via change of structure group by $B\hookrightarrow G$.
In particular, if $(\mathcal{E}_{G},\nabla)$ is a flat $G$-bundle on
$\mathscr{X}$, then a $B$-reduction $\mathcal{E}_{B}$ of $\mathcal{E}_{G}$ for
which the pair $(\mathcal{E}_{B},\nabla)$ defines a $G$-oper is (if it exists)
uniquely determined up to isomorphism.

###### Proof.

We may assume, without loss of generality, that
$\mathcal{E}:=\mathcal{E}_{G,1}=\mathcal{E}_{G,2}$,
$\nabla:=\nabla_{1}=\nabla_{2}$, and that $\alpha_{G}$ coincides with the
identity morphism of $\mathcal{E}$. The two Borel reductions
$\mathcal{E}_{B,1}$, $\mathcal{E}_{B,2}$ of $\mathcal{E}$ define the
$\mathcal{O}_{X}$-submodules of $\mathfrak{g}_{\mathcal{E}}$, i.e.,
$\mathfrak{g}^{0}_{\mathcal{E}_{B,1}}\left(=\mathfrak{b}_{\mathcal{E}_{B,1}}\right)$
and
$\mathfrak{g}^{0}_{\mathcal{E}_{B,2}}\left(=\mathfrak{b}_{\mathcal{E}_{B,2}}\right)$.
Notice here that, for each $i\in\\{1,2\\}$ and $j\in\mathbb{Z}$, the
subquotient
$\mathfrak{g}^{j}_{\mathcal{E}_{B,i}}/\mathfrak{g}^{j+1}_{\mathcal{E}_{B,i}}$
is isomorphic to $\Omega^{\otimes j}\otimes_{k}\mathfrak{g}_{j}$. Hence, by
using $\mathrm{deg}(\Omega)>0$, we can verify the equality
$\mathfrak{g}^{1}_{\mathcal{E}_{B,1}}=\mathfrak{g}^{1}_{\mathcal{E}_{B,2}}$.

Next, let us take a closed point $x$ of $X$, and fix an identification
$\mathcal{E}_{B,1}|_{x}=B$, which extends to $\mathcal{E}|_{x}=G$ and moreover
induces $\mathfrak{g}^{1}_{\mathcal{E}_{B,1}}|_{x}=\mathfrak{g}^{1}$ via
differentiation. Then, we can write $\mathcal{E}_{B,2}|_{x}=hB\left(\subseteq
G\right)$ for some $h\in G$. It follows that
$\mathfrak{g}^{1}_{\mathcal{E}_{B,2}}|_{x}$ coincides with the image
$\mathrm{ad}(h)(\mathfrak{g}^{1})$ of $\mathfrak{g}^{1}$ via the adjoint
operator $\mathrm{ad}(h)$ given by $h$. Since the characteristic of the base
field $k$ is very good for $G$ if it is of positive characteristic, the
equality
$\mathfrak{g}^{1}_{\mathcal{E}_{B,1}}=\mathfrak{g}^{1}_{\mathcal{E}_{B,2}}$
proved above together with the Springer isomorphism (cf. e.g., [KiWe, Theorem
3.3]) implies $N=hNh^{-1}$, where $N$ denotes the unipotent radical of $B$. It
follows that $B=hBh^{-1}$ (cf. [Hum, § 23.1, Corollary D]), and we have $h\in
B$ (cf. [Hum, § 23.1, Theorem]). Therefore, the equality
$\mathcal{E}_{B,1}|_{x}=\mathcal{E}_{B,2}|_{x}$ holds for all $x$. This means
$\mathcal{E}_{B,1}=\mathcal{E}_{B,2}$, completing the proof of this assertion.
∎

######  Lemma 3.3.

Let $T$ be a trait, i.e., the spectrum of a discrete valuation ring. Denote by
$\eta$ its generic point and by $t$ its closed point. Now, let us consider a
collection

(41) $\displaystyle(\mathscr{E}_{T},\mathcal{E}_{B,\eta},\mathcal{E}_{B,t}),$

where

  * •

$\mathscr{E}_{T}:=(\mathcal{E}_{T},\nabla_{T})$ denotes a flat $G$-bundle
$\mathscr{E}_{T}:=(\mathcal{E}_{T},\nabla_{T})$ on the base-change
$\mathscr{X}_{T}:=(X_{T},\\{\sigma_{i,T}\\}_{i=1}^{r})$ of $\mathscr{X}$ over
$T$ (cf. [Wak4, § 1] for the formulation of a flat $G$-bundle on a family of
pointed stable curves);

  * •

$\mathcal{E}_{B,\eta}$ and $\mathcal{E}_{B,t}$ are $B$-reductions of
$\mathcal{E}_{\eta}:=\mathcal{E}_{T}\times_{T}\eta$ and
$\mathcal{E}_{t}:=\mathcal{E}_{T}\times_{T}t$, respectively, such that both
$(\mathcal{E}_{B,\eta},\nabla_{T}|_{\mathcal{E}_{\eta}})$ and
$(\mathcal{E}_{B,t},\nabla_{T}|_{\mathcal{E}_{t}})$ form $G$-opers.

Then, there exists a $B$-reduction $\mathcal{E}_{B,T}$ on $\mathcal{E}_{T}$
such that $\mathcal{E}_{B,T}\times_{T}\eta=\mathcal{E}_{B,\eta}$,
$\mathcal{E}_{B,T}\times_{T}t=\mathcal{E}_{T,t}$, and the pair
$(\mathcal{E}_{B,T},\nabla_{T})$ forms a $G$-oper.

###### Proof.

Denote by $X_{\eta}$ the generic fiber of $X_{T}$. Note that
$\\{\mathfrak{g}_{\mathcal{E}_{\eta}}^{j}\\}_{j}$ and
$\\{\mathfrak{g}_{\mathcal{E}_{t}}^{j}\\}_{j}$ coincide with the Harder-
Narasimhan filtrations on $\mathfrak{g}_{\mathcal{E}_{\eta}}$ and
$\mathfrak{g}_{\mathcal{E}_{t}}$, respectively. According to [HuLe, Theorem
2.3.2], $\\{\mathfrak{g}_{\mathcal{E}_{\eta}}^{j}\\}_{j}$ extends to a
decreasing filtration $\\{\mathfrak{g}_{\mathcal{E}_{T}}^{j}\\}_{j}$ (i.e.,
$\mathfrak{g}_{\mathcal{E}_{T}}^{j}|_{X_{\eta}}=\mathfrak{g}_{\mathcal{E}_{\eta}}^{j}$
for every $j$) on $\mathfrak{g}_{\mathcal{E}_{T}}$ such that the subquotients
$\mathfrak{g}_{\mathcal{E}_{T}}^{j}/\mathfrak{g}_{\mathcal{E}_{T}}^{j+1}$ are
all $T$-flat. By the upper semicontinuity of the Harder-Narasimhan type, the
special fiber of $\\{\mathfrak{g}_{\mathcal{E}_{T}}^{j}\\}_{j}$ must coincide
with $\\{\mathfrak{g}_{\mathcal{E}_{t}}^{j}\\}_{j}$. Here, recall that the
isomorphism
$G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Aut}(\mathfrak{g})^{0}$
(where $\mathrm{Aut}(\mathfrak{g})^{0}$ denotes the identity component of the
group of Lie algebra automorphisms of $\mathfrak{g}$) induced from the adjoint
representation of $G$ restricts to an isomorphism of algebraic groups
$B\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\left\\{h\in\mathrm{Aut}(\mathfrak{g})^{0}\,|\,h(\mathfrak{b})\subseteq\mathfrak{b}\right\\}$
(cf. [Wak4, Remark 1.29]). It follows that the Lie subalgebra
$\mathfrak{g}_{\mathcal{E}_{T}}^{0}$ of $\mathfrak{g}_{\mathcal{E}_{T}}$
determines a $B$-reduction $\mathcal{E}_{B,T}$ of $\mathcal{E}_{T}$, which
satisfies $\mathcal{E}_{B,T}\times_{T}\eta=\mathcal{E}_{B,\eta}$ and
$\mathcal{E}_{B,t}\times_{T}t=\mathcal{E}_{B,t}$. Since the generic and
special fibers of $(\mathcal{E}_{B,T},\nabla_{T})$ coincide with the $G$-opers
$(\mathcal{E}_{B,\eta},\nabla_{T}|_{\mathcal{E}_{\eta}})$ and
$(\mathcal{E}_{B,t},\nabla_{T}|_{\mathcal{E}_{t}})$, respectively, the pair
$(\mathcal{E}_{B,T},\nabla_{T})$ turns out to form a $G$-oper. This completes
the proof of this lemma. ∎

###  3.2. Change of structure group

In the case of $G=\mathrm{PGL}_{2}$, we use the notations $G^{\odot}$,
$B^{\odot}$, $\mathfrak{g}^{\odot}$, and $\mathfrak{b}^{\odot}$ instead of
$G$, $B$, $\mathfrak{g}$, and $\mathfrak{b}$, respectively, for simplicity.
The $\mathfrak{s}\mathfrak{l}_{2}$-triple $\\{q_{-1},2\check{\rho},q_{1}\\}$
in $\mathfrak{g}$ associated to the fixed collection $\
\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$ determines
an injection

(42)
$\displaystyle\iota_{\mathfrak{g}}:\mathfrak{g}^{\odot}\hookrightarrow\mathfrak{g}.$

To be precise, $\iota_{\mathfrak{g}}$ is given by
$\iota_{\mathfrak{g}}\left(\begin{bmatrix}0&1\\\
0&0\end{bmatrix}\right)=q_{1}$,
$\iota_{\mathfrak{g}}\left(\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}\right)=2\check{\rho}$, and
$\iota_{\mathfrak{g}}\left(\begin{bmatrix}0&0\\\
1&0\end{bmatrix}\right)=q_{-1}$. In particular, we have
$\iota_{\mathfrak{g}}(\mathfrak{b}^{\odot})\subseteq\mathfrak{b}$. According
to [Wak4, Proposition 2.10], there exists a unique injective morphism of
algebraic $k$-groups $\iota_{B}:B^{\odot}\hookrightarrow B$ whose differential
induces the restriction $\mathfrak{b}^{\odot}\hookrightarrow\mathfrak{b}$ of
$\iota_{\mathfrak{g}}$.

Let
$\mathscr{E}^{\spadesuit}_{\odot}:=(\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
be a $G^{\odot}$-oper on $\mathscr{X}$. We shall write
$\mathcal{E}_{G^{\odot}}:=\mathcal{E}_{B^{\odot}}\times^{B^{\odot}}G^{\odot}$,
$\mathcal{E}_{B}:=\mathcal{E}_{B^{\odot}}\times^{B^{\odot},\iota_{B}}B$, and
$\mathcal{E}_{G}:=\mathcal{E}_{B}\times^{B}G$. The injection
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{B^{\odot}}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{B}}$
induced by $\iota_{B}$ extends to an $\mathcal{O}_{X}$-linear injection
$d\iota_{G}:\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{G^{\odot}}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{G}}$.
The composite
$\iota_{G*}(\nabla_{\odot}):\mathcal{T}_{X^{\mathrm{log}}}\xrightarrow{\nabla_{\odot}}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{G^{\odot}}}\xrightarrow{d\iota_{G}}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{G}}$
specifies a log connection on $\mathcal{E}_{G}$, and the resulting pair

(43)
$\displaystyle\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot}):=(\mathcal{E}_{B},\iota_{G*}(\nabla_{\odot}))$

forms a $G$-oper on $\mathscr{X}$. The assignment
$\mathscr{E}_{\odot}^{\spadesuit}\mapsto\iota_{G*}(\mathscr{E}_{\odot}^{\spadesuit})$
determines a closed immersion between $k$-schemes

(44) $\displaystyle\mathcal{O}p_{G^{\odot}}\hookrightarrow\mathcal{O}p_{G}$

(cf. [Wak4, Theorem 2.24, (ii)]).

If $\iota_{\mathfrak{c}}$ denotes the morphism
$\mathfrak{c}_{G^{\odot}}\rightarrow\mathfrak{c}_{G}$ induced by
$\iota_{\mathfrak{g}}$ via the adjoint quotients, then, for each
$\rho_{\odot}:=(\rho_{\odot,i})_{i=1}^{r}\in\mathfrak{c}_{G^{\odot}}(k)^{\times
r}$, the morphism (44) restricts to a closed immersion

(45)
$\displaystyle\mathcal{O}p_{G^{\odot},\rho_{\odot}}\hookrightarrow\mathcal{O}p_{G,\iota_{\mathfrak{c}}(\rho_{\odot})}$

by putting
$\iota_{\mathfrak{c}}(\rho_{\odot}):=(\iota_{\mathfrak{c}}(\rho_{\odot,i}))_{i=1}^{r}$,
where $\rho_{\odot}:=\emptyset$ and
$\iota_{\mathfrak{c}}(\rho_{\odot}):=\emptyset$ if $r=0$ (cf. [Wak4, Remark
2.37]). In particular, we obtain a closed immersion
$\mathcal{O}p_{G^{\odot},[0]^{\times
r}}\hookrightarrow\mathcal{O}p_{G,[0]^{\times r}}$, where $[0]^{\times
r}:=([0],\cdots,[0])$ in $\mathfrak{c}_{G^{\odot}}(k)^{\times r}$ or
$\mathfrak{c}_{G}(k)^{\times r}$.

###  3.3. The affine structure on the moduli space

Note that $\mathfrak{g}^{\mathrm{ad}(q_{1})}$ is closed under the
$B^{\odot}$-action via $\iota_{B}$. Therefore, if
${{}^{\dagger}}\mathcal{E}_{B^{\odot}}$ denotes the $B^{\odot}$-bundle on $X$
constructed in [Wak4, (190)], then it induces a rank
$\mathrm{rk}(\mathfrak{g})$ vector bundle

(46)
$\displaystyle\mathcal{V}_{G}:=\Omega\otimes_{\mathcal{O}_{X}}(\mathfrak{g}^{\mathrm{ad}(q_{1})})_{{{}^{\dagger}}\mathcal{E}_{B^{\odot}}}$

on $X$ (cf. [Wak4, § 2.4.5]). For example, $\mathcal{V}_{G^{\odot}}$ is
canonically isomorphic to $\Omega^{\otimes 2}$. The decomposition
$\mathfrak{g}^{\mathrm{ad}(q_{1})}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})}$
determines a decomposition
$\mathcal{V}_{G}=\bigoplus_{j\in\mathbb{Z}}\mathcal{V}_{G,j}$; it gives a
decreasing filtration $\\{\mathcal{V}^{j}_{G}\\}_{j\in\mathbb{Z}}$ on
$\mathcal{V}_{G}$ by putting $\mathcal{V}^{j}_{G}:=\bigoplus_{j^{\prime}\geq
j}\mathcal{V}_{G,j^{\prime}}$. Also, for each $j\in\mathbb{Z}$, there exists
an isomorphism

(47)
$\displaystyle\mathcal{V}_{G,j}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega^{\otimes(j+1)}\otimes_{k}\mathfrak{g}^{\mathrm{ad}(q_{1})}_{j}.$

Recall that $\mathcal{O}p_{G^{\odot}}$ admits a structure of affine space
modeled on the space of quadratic log differentials $H^{0}(X,\Omega^{\otimes
2})\left(=H^{0}(X,\mathcal{V}_{G^{\odot}})\right)$ (cf. [Wak4, § 2.3.4]). In
what follows, let us review a canonical affine structure on $\mathcal{O}p_{G}$
generalizing this fact (cf. [Wak4, Theorem 2.24, (i)]). The
$B^{\odot}$-equivariant inclusion
$\mathfrak{g}^{\mathrm{ad}(q_{1})}\hookrightarrow\mathfrak{g}$ yields an
$\mathcal{O}_{X}$-linear injection

(48)
$\displaystyle\varsigma:\mathcal{V}_{G}\hookrightarrow\left(\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{B^{\odot}}}=\right)\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{B}},$

where
${{}^{\dagger}}\mathcal{E}_{B}:={{}^{\dagger}}\mathcal{E}_{B^{\odot}}\times^{B^{\odot},\iota_{B}}B$.
Also, $\iota_{\mathfrak{g}}$ induces an $\mathcal{O}_{X}$-linear injection
$\Omega^{\otimes
2}\left(=\mathcal{V}_{G^{\odot}}\right)\hookrightarrow\mathcal{V}_{G}$. By
using these injections, we shall regard $\mathcal{V}_{G}$ and $\Omega^{\otimes
2}$ as $\mathcal{O}_{X}$-submodules of
$\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{B}}$
and $\mathcal{V}_{G}$, respectively.

Next, let us take a $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normal $G^{\odot}$-oper
${{}^{\dagger}}\mathscr{E}_{\odot}^{\spadesuit}:=({{}^{\dagger}}\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
on $\mathscr{X}$. Given an element $R$ of $H^{0}(X,\mathcal{V}_{G})$, regarded
as an $\mathcal{O}_{X}$-linear morphism
$\mathcal{T}\rightarrow(\mathfrak{g}^{\mathrm{ad}(q_{1})})_{{{}^{\dagger}}\mathcal{E}_{B^{\odot}}}\left(\subseteq\widetilde{\mathcal{T}}_{{{}^{\dagger}}\mathcal{E}^{\mathrm{log}}_{G}}\right)$,
one may verify that the collection

(49)
$\displaystyle\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})_{+R}:=({{}^{\dagger}}\mathcal{E}_{B},\iota_{G*}(\nabla_{\odot})+R)$

forms a $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normal $G$-oper. The resulting assignment
$(\mathscr{E}^{\spadesuit}_{\odot},R)\mapsto\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})_{+R}$
defines an isomorphism of $k$-schemes

(50) $\displaystyle\mathcal{O}p_{G^{\odot}}\times^{H^{0}(X,\Omega^{\otimes
2})}H^{0}(X,\mathcal{V}_{G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}p_{G}.$

In particular, the natural $H^{0}(X,\mathcal{V}_{G})$-action on
$\mathcal{O}p_{G^{\odot}}\times^{H^{0}(X,\Omega^{\otimes
2})}H^{0}(X,\mathcal{V}_{G})$ yields, via (50), an affine structure on
$\mathcal{O}p_{G}$ modeled on that space (cf. [Wak4, Theorem 2.24]).

Moreover, if $\rho$ is an element of $\mathfrak{c}(k)^{\times r}$ (where
$\rho:=\emptyset$ if $r=0$), then this structure restrict to an affine
structure on $\mathcal{O}p_{G,\rho}\left(\subseteq\mathcal{O}p_{G}\right)$
modeled on $H^{0}(X,{{}^{c}}\mathcal{V}_{G})\left(\subseteq
H^{0}(X,\mathcal{V}_{G})\right)$ (cf. [Wak4, Theorem 2.36]).

###  3.4. The $\mathrm{SL}_{n}$-oper associated to an $\mathrm{SL}_{2}$-oper

Let $n$ be a positive integer with $1<n<p$ and
$\mathscr{F}^{\heartsuit}_{\odot}:=(\mathcal{F}_{\odot},\nabla_{\odot},\\{\mathcal{F}_{\odot}^{j}\\}_{j=0}^{2})$
an $\mathrm{SL}_{2}$-oper on $\mathscr{X}$ (cf. Remark 2.1). Write
$\varTheta:=\mathcal{F}^{1}_{\odot}$, which is a line bundle on $X$. Then, we
obtain a composite isomorphism

(51)
$\displaystyle\Omega\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\varTheta\otimes_{\mathcal{O}_{X}}(\mathcal{F}_{\odot}/\varTheta)^{\vee}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\varTheta^{\otimes
2},$

where the first arrow denotes the isomorphism given by the $1$-st Kodaira-
Spencer map
$\mathrm{KS}_{\mathscr{F}_{\odot}^{\heartsuit}}^{1}:\varTheta\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega\otimes_{\mathcal{O}_{X}}(\mathcal{F}_{\odot}/\varTheta)$
for $\mathscr{F}^{\heartsuit}$ (cf. (34)) and the second arrow arises from the
fixed isomorphism
$\left(\varTheta\otimes_{\mathcal{O}_{X}}(\mathcal{F}_{\odot}/\varTheta)=\right)\mathrm{det}(\mathcal{F}_{\odot})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{X}$.
Thus, the line bundle $\varTheta$ together with (51) specifies a theta
characteristic of $X^{\mathrm{log}}$ in the sense of [Wak4, Example 4.34]; we
shall refer to it as the theta characteristic associated to
$\mathscr{F}^{\heartsuit}_{\odot}$.

The $(n-1)$-st symmetric product $\mathrm{Sym}^{n-1}\mathcal{F}_{\odot}$ of
$\mathcal{F}_{\odot}$ over $\mathcal{O}_{X}$ forms a rank $n$ vector bundle on
$X$; it admits a log connection $\mathrm{Sym}^{n-1}\nabla_{\odot}$ induced
naturally by $\nabla_{\odot}$. Moreover,
$\mathrm{Sym}^{n-1}\mathcal{F}_{\odot}$ has an $n$-step decreasing filtration
$\\{(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})^{j}\\}_{j=0}^{n}$ defined in such
a way that
$(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})^{0}:=\mathrm{Sym}^{n-1}\mathcal{F}_{\odot}$,
$(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})^{n}=0$, and
$(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})^{j}$ (for each $j=1,\cdots,n-1$) is
defined as the image of $\varTheta^{\otimes
j}\otimes\mathcal{F}_{\odot}^{\otimes(n-1-j)}$ via the natural quotient
$\mathcal{F}_{\odot}^{\otimes(n-1)}\twoheadrightarrow\mathrm{Sym}^{n-1}\mathcal{F}_{\odot}$.
For each $j=0,\cdots,n-1$, there exists a canonical composite of isomorphisms

(52)
$\displaystyle(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})^{j}/(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})^{j+1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\varTheta^{\otimes
j}\otimes_{\mathcal{O}_{X}}(\mathcal{F}_{\odot}/\varTheta)^{\otimes(n-1-j)}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\varTheta^{\otimes(2j-n+1)}.$

Moreover, by the assumption $n<p$, the collection

(53)
$\displaystyle\mathrm{Sym}^{n-1}\mathscr{F}_{\odot}^{\heartsuit}:=(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot},\mathrm{Sym}^{n-1}\nabla_{\odot},\\{(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})^{j}\\}_{j=0}^{n})$

forms an $\mathrm{SL}_{n}$-oper on $\mathscr{X}$, called the $(n-1)$-st
symmetric product of $\mathscr{F}^{\heartsuit}_{\odot}$. The resulting
assignment
$\mathscr{F}^{\heartsuit}_{\odot}\mapsto\mathrm{Sym}^{n-1}\mathscr{F}^{\heartsuit}_{\odot}$
determines a morphism

(54)
$\displaystyle\mathcal{O}p_{\mathrm{SL}_{2}}\rightarrow\mathcal{O}p_{\mathrm{SL}_{n}}.$

This morphism makes the following square diagram commute:

(59)

where the right-hand and left-hand vertical arrows are obtained via
projectivization.

###  3.5. The $\mathrm{SL}_{2l+1}$-oper associated to a
$\mathrm{PGL}_{2}$-oper

This subsection discusses a construction of an $\mathrm{SL}_{n}$-oper for an
odd $n$ using a $G^{\odot}$-oper that is compatible with (44) and (54). The
advantage of considering this construction is that $\mathscr{X}$ always admits
$G^{\odot}$-opers in contrast to the case of $\mathrm{SL}_{2}$-opers. (In
fact, there is a nodal curve that does not have theta characteristics; this
implies $\mathcal{O}p_{\mathrm{PGL}_{2}}=\emptyset$ for such a curve.)

Let
$\mathscr{E}^{\spadesuit}_{\odot}:=(\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
be a $G^{\odot}$-oper on $\mathscr{X}$. Write
$\mathcal{E}_{\odot}:=\mathcal{E}_{B^{\odot}}\times^{B^{\odot}}G^{\odot}$ and
$\mathscr{E}_{\odot}:=(\mathcal{E}_{\odot},\nabla_{\odot})$. We can choose a
$\mathrm{GL}_{2}$-oper
$\mathscr{F}^{\heartsuit}_{\odot}:=(\mathcal{F}_{\odot},\nabla_{\mathcal{F}_{\odot}},\\{\mathcal{F}^{j}_{\odot}\\}_{j=0}^{2})$
on $\mathscr{X}$ whose projectivization is isomorphic to
$\mathscr{E}^{\spadesuit}_{\odot}$ (cf. [Wak4, Theorem 4.66]). For an integer
$l$ with $1\leq l<\frac{p-1}{2}$, the previous discussion can be applied to
obtain a $\mathrm{GL}_{2l+1}$-oper

(60)
$\displaystyle\mathrm{Sym}^{2l}\mathscr{F}^{\heartsuit}_{\odot}:=(\mathrm{Sym}^{2l}\mathcal{F}_{\odot},\mathrm{Sym}^{2l}\nabla_{\mathcal{F}_{\odot}},\\{(\mathrm{Sym}^{2l}\mathcal{F}_{\odot})^{j}\\}_{i=0}^{2l+1}).$

In particular, there exists a sequence of natural isomorphisms

(61) $\displaystyle\mathrm{det}(\mathrm{Sym}^{2l}\mathcal{F}_{\odot})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigotimes_{j=0}^{2l}(\mathrm{Sym}^{2l}\mathcal{F}_{\odot})^{j}/(\mathrm{Sym}^{2l}\mathcal{F}_{\odot})^{j+1}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigoplus_{j=0}^{2l}(\mathcal{F}^{1}_{\odot})^{\otimes
2l}\otimes_{\mathcal{O}_{X}}\mathcal{T}^{\otimes 2l-j}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathcal{F}^{1}_{\odot})^{\otimes
2l(2l+1)}\otimes_{\mathcal{O}_{X}}\mathcal{T}^{\otimes l(2l+1)}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}((\mathcal{F}^{1}_{\odot})^{\otimes
2l}\otimes_{\mathcal{O}_{X}}\mathcal{T}^{\otimes l})^{\otimes(2l+1)}.$

Since $p\nmid 2l+1$, there exists a log connection $\nabla_{\mathcal{L}}$ on
$\mathcal{L}:=(\mathcal{F}^{1})^{\otimes
2l}\otimes_{\mathcal{O}_{X}}\mathcal{T}^{\otimes l}$ whose $(2l+1)$-fold
tensor product corresponds to $\mathrm{Sym}^{2l}\nabla_{\odot}$ via (61) (cf.
[Wak4, Proposition 4.22, (i)]). Let us write

(62)
$\displaystyle\mathrm{Sym}^{2l}\mathcal{E}_{\odot}:=\mathcal{L}^{\vee}\otimes_{\mathcal{O}_{X}}\mathrm{Sym}^{2l}\mathcal{F}_{\odot}\
\ \ \text{and}\ \ \
(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})^{j}:=\mathcal{L}^{\vee}\otimes_{\mathcal{O}_{X}}(\mathrm{Sym}^{2l}\mathcal{F}_{\odot})^{j}$

($j=0,\cdots,2l+1$). Then, there exist canonical isomorphisms

(63)
$\displaystyle(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})^{j}/(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})^{j+1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega^{\otimes
j-l}$

($j=0,\cdots,2l$). If $\nabla^{\vee}_{\mathcal{L}}$ denotes the log connection
on the dual $\mathcal{L}^{\vee}$ induced naturally by $\nabla_{\mathcal{L}}$,
the log connection
$\mathrm{det}(\nabla_{\mathcal{L}}^{\vee}\otimes\mathrm{Sym}^{2l}\nabla_{\mathcal{F}_{\odot}})$
on the determinant $\mathrm{det}(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})$ is
compatible with the trivial connection $d$ on $\mathcal{O}_{X}$ via the
composite isomorphism

(64)
$\displaystyle\mathrm{det}(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigotimes_{j=0}^{2l}(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})^{j}/(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})^{j+1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigotimes_{j=0}^{2l}\Omega^{\otimes
j-l}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{X}.$

Write $\mathrm{Sym}^{2l}\nabla_{\odot}$ for the log connection
$\nabla_{\mathcal{L}}^{\vee}\otimes\mathrm{Sym}^{2l}\nabla_{\mathcal{F}_{\odot}}$
on $\mathrm{Sym}^{2l}\mathcal{E}_{\odot}$.

Then, the resulting pair
$\mathrm{Sym}^{2l}\mathscr{E}_{\odot}:=(\mathrm{Sym}^{2l}\mathcal{E}_{\odot},\mathrm{Sym}^{2l}\nabla_{\odot})$
specifies a flat bundle, and moreover, the collection of data

(65)
$\displaystyle\mathrm{Sym}^{2l}\mathscr{E}_{\odot}^{\spadesuit}:=(\mathrm{Sym}^{2l}\mathcal{E}_{\odot},\mathrm{Sym}^{2l}\nabla_{\odot},\\{(\mathrm{Sym}^{2l}\mathcal{E}_{\odot})^{j}\\}_{j=0}^{2l+1})$

has a structure of $\mathrm{SL}_{2l+1}$-oper. The isomorphism classes of
$\mathrm{Sym}^{2l}\mathscr{E}_{\odot}$ and
$\mathrm{Sym}^{2l}\mathscr{E}_{\odot}^{\spadesuit}$ depend only on
$\mathscr{E}^{\spadesuit}_{\odot}$ (i.e,, not on the choice of
$\mathscr{F}^{\heartsuit}_{\odot}$). We refer to
$\mathrm{Sym}^{2l}\mathscr{E}_{\odot}^{\spadesuit}$ as the $2l$-th symmetric
product of $\mathscr{E}_{\odot}^{\spadesuit}$.

The resulting assignment
$\mathscr{E}^{\spadesuit}_{\odot}\mapsto\mathrm{Sym}^{2l}\mathscr{E}^{\spadesuit}_{\odot}$
determines a well-defined morphism

(66)
$\displaystyle\mathcal{O}p_{G^{\odot}}\rightarrow\mathcal{O}p_{\mathrm{SL}_{2l+1}},$

which makes the following diagram commute:

(71)

##  4\. Infinitesimal deformations of opers

This section deals with infinitesimal deformations of a flat $G$-bundle, as
well as a $G$-oper. In particular, we verify dualities between certain
deformation spaces of a flat $G$-bundle with prescribed residues/radii. These
results are probably already known at least in the case of characteristic $0$
or in some restricted situations. However, the author could not find any
suitable references consistent with our situation, so we here discuss them.

###  4.1. De Rham cohomology/Parabolic de Rham cohomology

Let $\nabla^{\prime}:\mathcal{K}^{0}\rightarrow\mathcal{K}^{1}$ be a morphism
of sheaves on $X$. It may be regarded as a complex concentrated at degrees $0$
and $1$; this complex is denoted by $\mathcal{K}^{\bullet}[\nabla^{\prime}]$.
(In particular, we have $\mathcal{K}^{j}[\nabla^{\prime}]:=\mathcal{K}^{j}$
for $j=0,1$). For each $l\geq 0$, we shall write
$\mathbb{H}^{l}(X,\mathcal{K}^{\bullet}[\nabla^{\prime}])$ for the $l$-th
hypercohomology group of the complex $\mathcal{K}^{\bullet}[\nabla^{\prime}]$.

Given an integer $l$ and a sheaf $\mathcal{F}$ on $X$, we define the complex
$\mathcal{F}[l]$ to be $\mathcal{F}$ (considered as a complex concentrated at
degree $0$) shifted down by $l$, so that $\mathcal{F}[l]^{-l}=\mathcal{F}$ and
$\mathcal{F}[l]^{i}=0$ for $i\neq l$.

For a flat bundle $\mathscr{F}:=(\mathcal{F},\nabla)$ on $\mathscr{X}$, we
shall set $\Omega^{0}_{\mathrm{par}}(\mathcal{F}):=\mathcal{F}$ and

(72)
$\displaystyle\Omega^{1}_{\mathrm{par}}(\mathcal{F}):=\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}+\mathrm{Im}(\nabla)\left(\subseteq\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F}\right).$

Then, $\nabla$ restricts to a $k$-linear morphism

(73)
$\displaystyle\nabla_{\mathrm{par}}:\Omega^{0}_{\mathrm{per}}(\mathcal{F})\rightarrow\Omega^{1}_{\mathrm{per}}(\mathcal{F}).$

The $k$-vector space

(74) $\displaystyle
H^{l}_{\mathrm{dR}}(X,\mathscr{F}):=\mathbb{H}^{l}(X,\mathcal{K}^{\bullet}[\nabla])\
\left(\text{resp.,}\
H^{l}_{\mathrm{dR},\mathrm{par}}(X,\mathscr{F}):=\mathbb{H}^{l}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])\right)$

is called the $l$-th de Rham cohomology group (resp., the $l$-th parabolic de
Rham cohomology group) of $\mathscr{F}$. The $k$-linear morphism
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla])$
induced by the inclusion of complexes
$\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}]\rightarrow\mathcal{K}^{\bullet}[\nabla]$
is injective. By using this injection, we occasionally regard
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])$ as a subspace
of $\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla])$.

Here, we prove a comparison between the parabolic de Rham cohomology and the
hypercohomology of the restriction

(75)
$\displaystyle{{}^{c}}\nabla_{\mathrm{par}}:\Omega_{\mathrm{par}}^{0,c}(\mathcal{F})\rightarrow\Omega_{\mathrm{par}}^{1,c}(\mathcal{F})$

of $\nabla$, where
$\Omega_{\mathrm{par}}^{0,c}(\mathcal{F}):=\nabla^{-1}(\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F})$
and
$\Omega_{\mathrm{par}}^{1,c}(\mathcal{F}):=\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}$.

######  Proposition 4.1.

For each $l\geq 0$, the morphism of $k$-vector spaces

(76)
$\displaystyle\mathbb{H}^{l}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla_{\mathrm{par}}])\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{l}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])$

induced by the inclusion
$\mathcal{K}^{\bullet}[{{}^{c}}\nabla_{\mathrm{par}}]\hookrightarrow\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}]$
is an isomorphism.

###### Proof.

Let us choose $i\in\\{1,\cdots,r\\}$. Then,
$\sigma^{*}_{i}(\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F})$ may be identified
with $\sigma^{*}_{i}(\mathcal{F})$ via
$\sigma_{i}^{*}(\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F})\cong\sigma_{i}^{*}(\Omega)\otimes_{k}\sigma_{i}^{*}(\mathcal{F})\cong\sigma_{i}^{*}(\mathcal{F})$,
where the second “$\cong$” follows from the residue map
$\sigma^{*}_{i}(\Omega)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k$. Under
this identification, the log connection $\nabla$ specifies an endomorphism
$\nu_{i}:\sigma^{*}_{i}(\mathcal{F})\rightarrow\sigma_{i}^{*}(\mathcal{F})\left(=\sigma_{i}^{*}(\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F})\right)$
of $\sigma_{i}^{*}(\mathcal{F})$. By taking quotients of the both sides of the
equality
$\Omega_{\mathrm{par}}^{0}(\mathcal{F})/{{}^{c}}\mathcal{F}=\bigoplus_{i=1}^{r}\sigma_{i*}(\sigma_{i}^{*}(\mathcal{F}))$,
we obtain

(77)
$\displaystyle\Omega_{\mathrm{par}}^{0}(\mathcal{F})/\Omega_{\mathrm{par}}^{0,c}(\mathcal{F})=\bigoplus_{i=1}^{r}\sigma_{i*}(\sigma_{i}^{*}(\mathcal{F})/\mathrm{Ker}(\nu_{i})).$

Also, the identification
$(\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F})/\Omega_{\mathrm{par}}^{1,c}(\mathcal{F})=\left(\bigoplus_{i=1}^{r}\sigma_{i*}(\sigma^{*}_{i}(\Omega)\otimes_{k}\sigma^{*}_{i}(\mathfrak{g}_{\mathcal{E}}))=\right)\bigoplus_{i=1}^{r}\sigma_{i*}(\mathcal{F})$
restricts to

(78)
$\displaystyle\Omega_{\mathrm{par}}^{1}(\mathcal{F})/\Omega_{\mathrm{par}}^{1,c}(\mathcal{F})=\bigoplus_{i=1}^{r}\sigma_{i*}(\mathrm{Im}(\nu_{i})).$

By using (77) and (78), we obtain a morphism of short exact sequences

(83)

Since the right-hand vertical arrow in this diagram is an isomorphism, the
inclusion
$\mathcal{K}^{\bullet}[{{}^{c}}\nabla_{\mathrm{par}}]\hookrightarrow\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}]$
turns out to be a quasi-isomorphism. This means that the induced morphism
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla_{\mathrm{par}}])\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])$
is an isomorphism. ∎

###  4.2. Duality for de Rham cohomology

Now, let $(\mathcal{E},\nabla)$ be a flat $G$-bundle on $\mathscr{X}$. The log
connection $\nabla$ induces a log connection
$\nabla^{\mathrm{ad}}:\mathfrak{g}_{\mathcal{E}}\rightarrow\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}}$
on the adjoint bundle $\mathfrak{g}_{\mathcal{E}}$; it restricts to a log
connection

(84)
$\displaystyle{{}^{c}}\nabla^{\mathrm{ad}}:{{}^{c}}\mathfrak{g}_{\mathcal{E}}\rightarrow\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathfrak{g}_{\mathcal{E}}$

on
${{}^{c}}\mathfrak{g}_{\mathcal{E}}\left(=\mathfrak{g}_{\mathcal{E}}(-D)\right)$.
If ${{}^{c}}d$ denotes the restriction
${{}^{c}}\mathcal{O}_{X}\rightarrow\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{O}_{X}\left(=\omega_{X}\right)$
of the universal logarithmic derivation $d$, then the pairing
$({{}^{c}}\mathfrak{g}_{\mathcal{E}},{{}^{c}}\nabla^{\mathrm{ad}})\otimes_{\mathcal{O}_{X}}(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})\rightarrow({{}^{c}}\mathcal{O}_{X},{{}^{c}}d)$
arising from the Killing form $\kappa$ on $\mathfrak{g}$ yields a composite
$k$-linear pairing

(85)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])\times\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])\xrightarrow{\cup}\mathbb{H}^{2}(X,\mathcal{K}^{\bullet}[{{}^{c}}d])\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(X,\omega_{X})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k.$

The following assertion for the case of flat $G$-bundles with vanishing
$p$-curvature can be found in [Wak4, Corollary 6.16].

######  Proposition 4.2.

The pairing (85) is nondegenerate. That is to say, the $k$-linear morphism

(86)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])^{\vee}$

determined by (85) is an isomorphism.

###### Proof.

Let us consider the diagram

(91)

where the upper and lower horizontal sequences are the exact sequences arising
from the Hodge to de Rham spectral sequences
$H^{b}(X,\mathcal{K}^{a}[{{}^{c}}\nabla^{\mathrm{ad}}])\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])$
and
$H^{b}(X,\mathcal{K}^{a}[\nabla^{\mathrm{ad}}])\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])$,
respectively. The assumed condition on $G$ described in § 1.4 implies that the
Killing form $\kappa$ is nondegenerate, so it induces an isomorphism
$\mathfrak{g}_{\mathcal{E}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}_{\mathcal{E}}^{\vee}$.
All the vertical arrows in (91) except for the middle one are isomorphisms
because they come from the pairings given by Serre duality. It follows that
(86) is an isomorphism by the five lemma, and this completes the proof of this
assertion. ∎

Denote by $q$ the $k$-rational point of $\mathcal{C}onn_{G}$ classifying
$(\mathcal{E},\nabla)$. Also, let $T_{q}\mathcal{C}onn_{G}$ (resp.,
$T_{q}^{\vee}\mathcal{C}onn_{G}$) denote the tangent space (resp., the
cotangent space) of $\mathcal{C}onn_{G}$ at $q$. As discussed in [Wak4, §
6.1.4] (or, e.g., [Oss, Proposition 3.6] for the case where $\mathscr{X}$ is
unpointed and smooth), the underlying set of
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])$ may be
identified with the space of first-order deformations of
$(\mathcal{E},\nabla)$. That is to say, there exists a canonical isomorphism
of $k$-vector spaces

(92)
$\displaystyle\gamma:T_{q}\mathcal{C}onn_{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}]).$

If $(\mathcal{E},\nabla)$ is equipped with a structure of marking and its
residues are given by $\mu:=(\mu_{i})_{i=1}^{r}\in\mathfrak{g}^{\times r}$
(where $\mu:=\emptyset$ if $r=0$), then a similar discussion yields a
canonical isomorphism of $k$-vector spaces

(93)
$\displaystyle\gamma_{\mu}:T_{q}\mathcal{C}onn_{G,\mu}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}]),$

where we use the same notation “$q$” to denote the $k$-rational point of
$\mathcal{C}onn_{G,\mu}$ classifying $(\mathcal{E},\nabla)$ (with marking). In
particular, by using the isomorphisms (86), (92), and (93), we obtain an
isomorphism

(94) $\displaystyle
T_{q}\mathcal{C}onn_{G,\mu}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T_{q}^{\vee}\mathcal{C}onn_{G}.$

###  4.3. Duality for parabolic de Rham cohomology

Next, let us consider the case of parabolic de Rham cohomology.

We shall denote by

(95)
$\displaystyle\kappa^{\triangleright}:\left((\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathfrak{g}_{\mathcal{E}})\times\mathfrak{g}_{\mathcal{E}}=\right)\Omega^{1,c}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}})\times\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})\rightarrow{{}^{c}}\Omega$

the $\mathcal{O}_{X}$-bilinear pairing induced from the Killing form $\kappa$
of $\mathfrak{g}$. Since $\kappa$ is nondegenerate, we see that
$\kappa^{\triangleright}$ is nondegenerate. By applying Serre duality to
$\kappa^{\triangleright}$, we obtain a nondegenerate pairing of $k$-vector
spaces

(96) $\displaystyle
H^{1-l}(X,\Omega_{\mathrm{par}}^{1,c}(\mathfrak{g}_{\mathcal{E}}))\otimes_{k}H^{l}(X,\Omega^{0}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}}))\rightarrow\left(H^{1}(X,{{}^{c}}\Omega)=\right)k$

for $l=0,1$.

Next, we shall write $U:=X\setminus\bigcup\\{\sigma_{i}\\}_{i=1}^{r}$ and
write $\iota$ for the inclusion $U\hookrightarrow X$. The
$\mathcal{O}_{X}$-module $\iota_{*}(\iota^{*}(\mathfrak{g}_{\mathcal{E}}))$
(resp.,
$\iota_{*}(\iota^{*}(\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}}))$)
contains $\Omega^{0,c}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}})$ (resp.,
$\Omega^{1}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}})$) as an
$\mathcal{O}_{X}$-submodule. Here, let us consider the natural skew-symmetric
$\mathcal{O}_{X}$-bilinear pairing

(97)
$\displaystyle\widetilde{\kappa}^{\blacktriangleright}:\iota_{*}(\iota^{*}(\mathfrak{g}_{\mathcal{E}}))\times\iota_{*}(\iota^{*}(\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}}))\rightarrow\iota_{*}(\iota^{*}(\Omega))$

arising from $\kappa$.

######  Proposition 4.3.

Assume (cf. Remark 4.4) that there exists a $B$-reduction $\mathcal{E}_{B}$ of
$\mathcal{E}$ for which the pair
$\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ forms a $G$-oper of radii
$[0]^{\times r}:=([0],[0],\cdots,[0])\in\mathfrak{c}(k)^{\times r}$ (where
$[0]^{\times r}:=\emptyset$ if $r=0$). Then,
$\widetilde{\kappa}^{\blacktriangleright}$ restricts to a nondegenerate
$\mathcal{O}_{X}$-bilinear pairing

(98)
$\displaystyle\kappa^{\blacktriangleright}:\Omega^{0,c}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}})\times\Omega^{1}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}})\rightarrow{{}^{c}}\Omega.$

In particular, for $l=0,1$, the morphism

(99) $\displaystyle
H^{1-l}(X,\Omega^{0,c}_{\mathrm{par}}(\mathfrak{g}))\otimes_{k}H^{l}(X,\Omega^{1}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}}))\rightarrow\left(H^{1}(X,{{}^{c}}\Omega)=\right)k$

obtained by applying Serre duality to this pairing is nondegenerate.

###### Proof.

After replacing $\mathscr{E}^{\spadesuit}$ with its
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normalization (cf. [Wak4, Proposition 2.18]), we may assume that
$\mathscr{E}^{\spadesuit}$ is
$\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal. In
particular, since $\nabla$ is of radii $[0]^{\times r}$, its residue at
$\sigma_{i}$ coincides with $q_{-1}\in\mathfrak{g}$ for every $i=1,\cdots,r$.

We first consider the former assertion. Just as in the case of
$\kappa^{\triangleright}$, the restriction of
$\widetilde{\kappa}^{\blacktriangleright}$ to $U$ is nondegenerate. Thus, the
problem is reduced to examining the pairing
$\widetilde{\kappa}^{\blacktriangleright}$ around the marked points. Let us
choose $i\in\\{1,\cdots,r\\}$ and choose a local function
$t\in\mathcal{O}_{X}$ defining the closed subscheme
$\mathrm{Im}(\sigma_{i})\subseteq X$. Then, the formal neighborhood $Q$ of
$\sigma_{i}$ is naturally isomorphic to $\mathrm{Spec}(k[\\![t]\\!])$, and we
have ${{}^{c}}\Omega|_{Q}=k[\\![t]\\!]dt$. The choice of such a function $t$
specifies a canonical identification
$\mathfrak{g}_{\mathcal{E}}|_{Q}=k[\\![t]\\!]\otimes_{k}\mathfrak{g}$ (cf.
[Wak4, § 2.4.3]), which also gives
$\iota_{*}(\iota^{*}(\mathfrak{g}_{\mathcal{E}}))|_{Q}=k(\\!(t)\\!)\otimes_{k}\mathfrak{g}$.
Under these identifications, the restriction of
$\widetilde{\kappa}^{\blacktriangleright}$ to $Q$ coincides with the Killing
form $\kappa$ tensored with $k(\\!(t)\\!)$. Here, recall (cf. [Wak4, § 2.4.4
and § 6.2.3]) that there exist direct sum decompositions of $\mathfrak{g}$:

(100)
$\displaystyle\mathfrak{g}=\mathrm{Ker}(\mathrm{ad}(q_{-1}))\oplus\mathrm{Im}(\mathrm{ad}(q_{1}))=\mathrm{Ker}(\mathrm{ad}(q_{1}))\oplus\mathrm{Im}(\mathrm{ad}(q_{-1})),$

and that the isomorphism
$\mathfrak{g}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}^{\vee}$
arising from $\kappa$ restricts to
$\mathrm{Ker}(\mathrm{ad}(q_{-1}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ker}(\mathrm{ad}(q_{1}))^{\vee}$,
as well as to
$\mathrm{Im}(\mathrm{ad}(q_{1}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Im}(\mathrm{ad}(q_{-1}))^{\vee}$.
In particular, by the equality $\mu_{i}^{\nabla}=q_{-1}$, the first (resp.,
second) decomposition of (100) yields

(101)
$\displaystyle\Omega_{\mathrm{par}}^{0,c}(\mathfrak{g}_{\mathcal{E}})|_{Q}=k[\\![t]\\!]\otimes_{k}\mathrm{Ker}(\mathrm{ad}(q_{-1}))\oplus
k[\\![t]\\!]t\otimes_{k}\mathrm{Im}(\mathrm{ad}(q_{1}))\hskip 31.29802pt$
$\displaystyle\left(\text{resp.,}\
\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})|_{Q}=k[\\![t]\\!]dt\otimes_{k}\mathrm{Ker}(\mathrm{ad}(q_{1}))\oplus
k[\\![t]\\!]\frac{dt}{t}\otimes_{k}\mathrm{Im}(\mathrm{ad}(q_{-1}))\right).$

According to this description, each section $v$ (resp., $u$) of
$\Omega_{\mathrm{par}}^{0,c}(\mathfrak{g}_{\mathcal{E}})|_{Q}$ (resp.,
$\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})|_{Q}$) may be described
as the sum $v=v_{1}+t\cdot v_{2}$ (resp., $u=u_{1}+\frac{1}{t}\cdot u_{2}$)
for some $v_{1}\in k[\\![t]\\!]\otimes_{k}\mathrm{Ker}(\mathrm{ad}(q_{-1}))$,
$v_{2}\in k[\\![t]\\!]\otimes_{k}\mathrm{Im}(\mathrm{ad}(q_{1}))$ (resp.,
$u_{1}\in k[\\![t]\\!]dt\otimes_{k}\mathrm{Ker}(\mathrm{ad}(q_{1}))$,
$u_{2}\in k[\\![t]\\!]dt\otimes_{k}\mathrm{Im}(\mathrm{ad}(q_{-1}))$). Then,
we have

(102) $\displaystyle\widetilde{\kappa}^{\blacktriangleright}(v,u)$
$\displaystyle=\widetilde{\kappa}^{\blacktriangleright}(v_{1},u_{1})+\widetilde{\kappa}^{\blacktriangleright}(t\cdot
v_{2},\frac{1}{t}\cdot u_{2})$
$\displaystyle=\widetilde{\kappa}^{\blacktriangleright}(v_{1},u_{1})+\widetilde{\kappa}^{\blacktriangleright}(v_{2},u_{2})$
$\displaystyle=\widetilde{\kappa}^{\blacktriangleright}(v_{1}+v_{2},u_{1}+u_{2}).$

It follows that $\widetilde{\kappa}^{\blacktriangleright}(v,u)$ lies in
$k[\\![t]\\!]dt\left(={{}^{c}}\Omega|_{Q}\right)$, and the restriction of
$\widetilde{\kappa}^{\blacktriangleright}$ over $Q$ is nondegenerate. This
completes the proof of the former assertion.

The latter assertion follows directly from the former assertion. ∎

######  Remark 4.4 (Flat $G$-bundles with parabolic structure).

By the similarity of the proof, the same assertion as Proposition 4.3 remains
true after replacing the assumption (i.e., the existence of
“$\mathcal{E}_{B}$”) with the following condition (regarded as a kind of
parabolic structure): the flat $G$-bundle $(\mathcal{E},\nabla)$ is equipped
with a marking, and its residue at $\sigma_{i}$ coincides with
$q_{-1}\in\mathfrak{g}$ (via the fixed marking) for every $i=1,\cdots,r$.

Until the end of this subsection, we keep the assumption in the above
proposition. Let $\Box$ denote either the absence or presence of “$c\,$”. One
may calculate
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla_{\mathrm{par}}^{\mathrm{ad}}])$
as the total cohomology of the Čech double complex
$\mathrm{Tot}^{\bullet}(\check{C}(\mathscr{U},\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla_{\mathrm{par}}^{\mathrm{ad}}]))$
associated to
$\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla_{\mathrm{par}}^{\mathrm{ad}}]$, where
$\mathscr{U}:=\\{U_{\alpha}\\}_{\alpha\in I}$ is a finite affine open covering
of $X$. Denote by $I_{2}$ the set of pairs $(\alpha,\beta)\in I\times I$ with
$U_{\alpha\beta}:=U_{\alpha}\cap U_{\beta}\neq\emptyset$. Then, each element
$v$ of
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla_{\mathrm{par}}^{\mathrm{ad}}])$
may be given by a collection of data

(103) $\displaystyle
v=(\\{\partial_{\alpha\beta}\\}_{(\alpha,\beta)},\\{\delta_{\alpha}\\}_{\alpha})$

consisting of a Čech $1$-cocycle
$\\{\partial_{\alpha\beta}\\}_{(\alpha,\beta)\in
I_{2}}\in\check{C}^{1}(\mathscr{U},\Omega_{\mathrm{par}}^{0,\Box}(\mathfrak{g}_{\mathcal{E}}))$
with $\partial_{\alpha\beta}\in
H^{0}(U_{\alpha\beta},\Omega_{\mathrm{par}}^{0,\Box}(\mathfrak{g}_{\mathcal{E}}))$
and a Čech $0$-cochain $\\{\delta_{\alpha}\\}_{\alpha\in
I}\in\check{C}^{0}(\mathscr{U},\Omega^{1,\Box}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}}))$
with $\delta_{\alpha}\in
H^{0}(U_{\alpha},\Omega_{\mathrm{par}}^{1,\Box}(\mathfrak{g}_{\mathcal{E}}))$
which agree under ${{}^{\Box}}\nabla_{\mathrm{par}}^{\mathrm{ad}}$ and the
Čech coboundary map.

Using the description in terms of Čech double complexes, one can obtain a
skew-symmetric $k$-bilinear pairing

(104)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}_{\mathrm{par}}])\times\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])\rightarrow\left(H^{1}(X,{{}^{c}}\Omega)=\right)k$

given by assigning
$(\\{\partial_{\alpha\beta}\\}_{(\alpha,\beta)},\\{\delta_{\alpha}\\}_{\alpha})\otimes(\\{\partial^{\prime}_{\alpha\beta}\\}_{(\alpha,\beta)},\\{\delta^{\prime}_{\alpha}\\}_{\alpha})\mapsto\kappa^{\blacktriangleright}(\partial_{\alpha\beta},\delta^{\prime}_{\beta})-\kappa^{\triangleright}(\delta_{\alpha},\partial^{\prime}_{\alpha\beta})$.

######  Proposition 4.5.

Let us keep the assumption in Proposition 4.3 (or impose the condition
described in Remark 4.4 instead). Then, the pairing (104) is nondegenerate. In
particular, the morphism of $k$-vector spaces

(105)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}_{\mathrm{par}}])\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])^{\vee}$

induced by (104) is an isomorphism.

###### Proof.

Note that the morphism (105) fits into the morphism of exact sequences

(110)

where

  * •

the upper and lower horizontal sequences are the exact sequences induced from
the Hodge to de Rham spectral sequences
$E_{1}^{a,b}=H^{b}(X,\mathcal{K}^{a}[{{}^{c}}\nabla^{\mathrm{ad}}_{\mathrm{par}}])\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}_{\mathrm{par}}])$
and
$E_{1}^{a,b}=H^{b}(X,\mathcal{K}^{a}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])$,
respectively;

  * •

the second and fifth vertical arrows from the left are the isomorphisms
arising from the nondegenerate pairing (96) for $l=1$ and $0$, respectively;

  * •

the first and fourth vertical arrows from the left are the isomorphisms
arising from the nondegenerate pairing (99) for $l=1$ and $0$, respectively.

Thus, by applying the five lemma to this diagram, we see that the middle
vertical arrow (105) is an isomorphism. ∎

######  Corollary 4.6.

There exists a canonical nondegenerate skew-symmetric pairing

(111)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])\times\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])\rightarrow
k.$

In particular, we obtain an isomorphism of $k$-vector spaces

(112)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])^{\vee}.$

###### Proof.

The required pairing (111) is obtained by composing (104) and the inverse of
(76) in the case where “$\nabla$” is taken to be $\nabla^{\mathrm{ad}}$. ∎

Next, denote by $T_{q}\mathcal{C}onn_{G,[0]^{\times r}}$ (resp.,
$T_{q}^{\vee}\mathcal{C}onn_{G,[0]^{\times r}}$) the tangent space (resp., the
cotangent space) of $\mathcal{C}onn_{G,\rho}$ at $q$. In particular,
$T_{q}\mathcal{C}onn_{G,[0]^{\times r}}$ may be identified with the set of
deformations of $(\mathcal{E},\nabla)$ fixing the radii. Then, we can prove
the following assertion.

######  Proposition 4.7.

Let us keep the assumption in Proposition 4.3. Then, there exists a canonical
isomorphism of $k$-vector spaces

(113) $\displaystyle\gamma_{[0]^{\times r}}:T_{q}\mathcal{C}onn_{G,[0]^{\times
r}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}]).$

Moreover, the following diagram is commutative:

(118)

###### Proof.

After possibly replacing $\mathscr{E}^{\spadesuit}$ with its
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normalization, we may assume that $\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal.
In particular, the assumption implies that the residue of $\nabla$ at every
marked point coincides with $q_{-1}$ under the canonical local trivialization
of $\mathcal{E}\left(={{}^{\dagger}}\mathcal{E}_{G}\right)$ (cf. [Wak4, §
2.4.3]). Let us take a deformation
$(\mathcal{E}_{\varepsilon},\nabla_{\varepsilon})$ of $(\mathcal{E},\nabla)$
over $k_{\varepsilon}:=k[\varepsilon]/(\varepsilon^{2})$. It corresponds to an
element $v$ of $\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])$
via $\gamma$. This deformation is represented by a collection of data
$(\\{\partial_{\alpha\beta}\\}_{(\alpha,\beta)\in
I_{2}},\\{\delta_{\alpha}\\}_{\alpha\in I})$ (as displayed in (103)) in the
Čech double complex
$\mathrm{Tot}^{\bullet}(\check{C}(\mathscr{U},\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}]))$
associated to $\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}]$ and an affine open
covering $\mathscr{U}:=\\{U_{\alpha}\\}_{\alpha\in I}$ of $X$ indexed by a
finite set $I$. Denote by $\pi$ the projection
$X\times_{k}k_{\varepsilon}\rightarrow X$. Then, for each $\alpha\in I$, the
flat $G$-bundle
$(\mathcal{E}_{\varepsilon},\nabla_{\varepsilon})|_{U_{\alpha}}$ (i.e., the
restriction of $(\mathcal{E}_{\varepsilon},\nabla_{\varepsilon})$ to
$U_{\alpha}$) is isomorphic to
$(\pi^{*}(\mathcal{E})|_{U_{\alpha}},\pi^{*}(\nabla)|_{U_{\alpha}}+\delta_{\alpha}\cdot\varepsilon)$.
If $\sigma_{i}\in U_{\alpha}(k)$ for some $i\in\\{1,\cdots,r\\}$, then the
residue of $\nabla_{\varepsilon}$ at $\sigma_{i}$ coincides with
$q_{-1}+\overline{\delta}_{\alpha}\cdot\varepsilon\in\mathfrak{g}(k_{\varepsilon})$,
where $\overline{\delta}_{\alpha}$ denotes the image of $\delta_{\alpha}$ in
$\mathfrak{g}$ via the composite surjection
$\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}}\twoheadrightarrow\sigma^{*}_{i}(\Omega)\otimes_{k}\sigma_{i}^{*}(\mathfrak{g}_{\mathcal{E}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}$,
where the second morphism is given by both the residue map
$\sigma^{*}_{i}(\Omega)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k$ and the
isomorphism
$\sigma^{*}_{i}(\mathfrak{g}_{\mathcal{E}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}$
induced from the canonical trivialization
$\sigma_{i}^{*}(\mathcal{E})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G$.

Now, suppose that $v$ lies in
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])$.
After possibly replacing
$(\\{\partial_{\alpha\beta}\\}_{(\alpha,\beta)},\\{\delta_{\alpha}\\}_{\alpha})$
with another, we may assume that $\delta_{\alpha}\in
H^{0}(U_{\alpha},{{}^{c}}\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}})$
for every $\alpha\in I$. Hence, the equality $\overline{\delta}_{\alpha}=0$
holds, and the radius at each marked point coincides with
$\left(\chi(q_{-1}+0\cdot\varepsilon)=\right)[0]$. This means that
$(\mathcal{E}_{\varepsilon},\nabla_{\varepsilon})$ is classified by
$T_{q}\mathcal{C}onn_{G,\rho}$.

Conversely, suppose that $(\mathcal{E}_{\varepsilon},\nabla_{\varepsilon})$ is
of radii $[0]^{\times r}$. Let us choose $i\in\\{1,\cdots,r\\}$ and choose
$\alpha_{i}\in I$ with $\sigma_{i}\in U_{\alpha_{i}}(k)$. Recall from [Kos,
Theorem 2] (or [Ngô, Lemme 1.2.3]) that if $\mathfrak{g}^{\mathrm{reg}}$
denotes the set of regular elements in $\mathfrak{g}$, then the fiber of the
projection
$\chi|_{\mathfrak{g}^{\mathrm{reg}}}:\mathfrak{g}^{\mathrm{reg}}\rightarrow\mathfrak{c}$
over $[0]$ forms a homogenous space with respect to the adjoint $G$-action.
Since both $q_{-1}$ and
$q_{-1}+\overline{\delta}_{\alpha_{i}}\cdot\varepsilon$ belong to this fiber,
there exists an element in $G(k_{\varepsilon})$ of the form
$e+w_{i}\cdot\varepsilon$, where $e$ denotes the identity element of $G$ and
$w_{i}$ is an element of $\mathfrak{g}$, satisfying
$\mathrm{Ad}(e+w_{i}\cdot\varepsilon)(q_{-1})\left(=q_{-1}+[w_{i},q_{-1}]\cdot\varepsilon\right)=q_{-1}+\overline{\delta}_{\alpha}\cdot\varepsilon$.
Hence, we have $[w_{i},q_{-1}]=\overline{\delta}_{\alpha}$. Let us take a
section $\lambda_{i}\in H^{0}(U_{\alpha},\mathfrak{g}_{\mathcal{E}})$ whose
image in
$\left(H^{0}(U_{\alpha},\mathfrak{g}_{\mathcal{E}})/H^{0}(U_{\alpha},{{}^{c}}\mathfrak{g}_{\mathcal{E}})=H^{0}(U_{\alpha},\sigma^{*}_{i}(\mathfrak{g}_{\mathcal{E}}))=\right)\mathfrak{g}$
coincides with $w_{i}$, and replace $\delta_{\alpha_{i}}$ (resp.,
$\partial_{\alpha_{i}\beta}$) with the section
$\delta_{\alpha_{i}}-\nabla^{\mathrm{ad}}(w_{i})$ (resp.,
$\partial_{\alpha_{i}\beta}-w_{i}|_{U_{\alpha_{i}\beta}}$). Then, the
resulting collection
$(\\{\partial_{\alpha\beta}\\}_{(\alpha,\beta)},\\{\delta_{\alpha}\\}_{\alpha})$
still represents $v$ and furthermore lies in
$\mathrm{Tot}^{\bullet}(\check{C}(\mathscr{U},\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}]))$.
This means that $v$ lies in
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])$.

As a consequence, we conclude that $\gamma$ restricts to an isomorphism of
$k$-vector spaces $T_{q}\mathcal{C}onn_{G,\rho}$
$\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])$,
and this completes the proofs of the former and latter assertions of this
proposition. ∎

By applying the above proposition together with the isomorphism (112), we
obtain an isomorphism of $k$-vector spaces

(119) $\displaystyle T_{q}\mathcal{C}onn_{G,[0]^{\times
r}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T_{q}^{\vee}\mathcal{C}onn_{G,[0]^{\times
r}}.$

###  4.4. The deformation space of a $G$-oper I

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a $G$-oper on
$\mathscr{X}$, and write $\mathcal{E}:=\mathcal{E}_{B}\times^{B}G$.

######  Proposition 4.8.

Let $\Box$ denote either the absence or presence of “$c$”. Then, there exists
a short exact sequence

(120) $\displaystyle 0\longrightarrow
H^{0}(X,{{}^{\Box}}\mathcal{V}_{G})\xrightarrow{{}^{\prime}e_{\sharp,\Box}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla^{\mathrm{ad}}])\xrightarrow{{{}^{\prime}e}_{\flat,\Box}}H^{1}(X,\Omega\otimes_{\mathcal{O}_{X}}{{}^{\Box}}\mathcal{V}_{G}^{\vee})\longrightarrow
0.$

Moreover, the following diagram forms an isomorphism of short exact sequences:

(125)

where the left-hand and right-hand vertical arrows are the isomorphism arising
from Serre duality.

###### Proof.

After possibly replacing $\mathscr{E}^{\spadesuit}$ with its
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normalization, we may assume that $\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal.
The Hodge to de Rham spectral sequence

(126)
${{}^{\prime}E}^{a,b}_{1}:=H^{b}(X,\mathcal{K}^{a}[{{}^{\Box}}\nabla^{\mathrm{ad}}])\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla^{\mathrm{ad}}]),$

of the complex $\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla^{\mathrm{ad}}]$ yields
a short exact sequence

(127) $\displaystyle
0\longrightarrow\mathrm{Coker}(H^{0}({{}^{\Box}}\nabla^{\mathrm{ad}}))\longrightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla^{\mathrm{ad}}])\longrightarrow\mathrm{Ker}(H^{1}({{}^{\Box}}\nabla^{\mathrm{ad}}))\longrightarrow
0,$

where $H^{j}({{}^{\Box}}\nabla^{\mathrm{ad}})$ ($j=0,1$) denotes the morphism
$H^{j}(X,\mathcal{K}^{0}[{{}^{\Box}}\nabla^{\mathrm{ad}}])\rightarrow
H^{j}(X,\mathcal{K}^{1}[{{}^{\Box}}\nabla^{\mathrm{ad}}])$ induced by
${{}^{\Box}}\nabla^{\mathrm{ad}}$. (Namely, this is obtained from the dual of
the lower horizontal sequence in (91).)

Here, observe that the following square diagram is commutative:

(132)

where

  * •

$\varsigma^{\veebar}$ denotes the morphism
$\mathfrak{g}_{\mathcal{E}}^{\vee}\rightarrow\Omega\otimes_{\mathcal{O}_{X}}\mathcal{V}_{G}^{\vee}$
defined as the tensor product of the dual
$\varsigma^{\vee}:\Omega^{\vee}\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}}^{\vee}\rightarrow\mathcal{V}_{G}^{\vee}$
of $\varsigma$ and the identity morphism of $\Omega$;

  * •

the left-hand and middle vertical arrows denote the isomorphisms arising from
Serre duality;

  * •

the right-hand vertical arrow denotes the isomorphism induced from (91) (via
$\mathfrak{g}_{\mathcal{E}}\cong\mathfrak{g}_{\mathcal{E}}^{\vee}$).

It follows from [Wak4, Lemma 6.4, (ii)] that the composite of the lower
horizontal arrows is an isomorphism, so the composite of the upper horizontal
arrows turns out to be an isomorphism. Also, by a similar argument together
with [Wak4, Lemma 6.4, (i)], we obtain the following commutative square
diagram all of whose arrows are isomorphisms:

(137)

Under the identifications of $k$-vector spaces given by the isomorphisms in
(132) and (137), the short exact sequence (127) coincides with the desired
sequence (120). This completes the proof of the former assertion. Moreover,
the latter assertion follows from the various definitions of morphisms
involved. ∎

In the resp’d portion of the following discussion, we suppose that the
$G$-oper $\mathscr{E}^{\spadesuit}$ is of radii
$\rho\in\mathfrak{c}(k)^{\times r}$ (where $\rho:=\emptyset$ if $r=0$). Denote
by $q$ the $k$-rational point of $\mathcal{O}p_{G}$ (resp.,
$\mathcal{O}p_{G,\rho}$) classifying $\mathscr{E}^{\spadesuit}$; we use the
same notation “$q$” to denote the image of this point via the immersion
$\mathrm{Imm}_{G}:\mathcal{O}p_{G}\hookrightarrow\mathcal{C}onn_{G}$ (resp.,
$\mathrm{Imm}_{G,\widetilde{\rho}}:\mathcal{O}p_{G,\rho}\hookrightarrow\mathcal{C}onn_{G,\widetilde{\rho}}$).
The affine structure on $\mathcal{O}p_{G}$ (resp., $\mathcal{O}p_{G,\rho}$)
recalled in § 3.3 gives an identification between $H^{0}(X,\mathcal{V}_{G})$
(resp., $H^{0}(X,{{}^{c}}\mathcal{V}_{G})$) and the space of first-order
deformations of the $G$-oper $\mathscr{E}^{\spadesuit}$ (resp., the $G$-oper
$\mathscr{E}^{\spadesuit}$ fixing the radii). Hence, if
$T_{q}\mathcal{O}p_{G}$ (resp., $T_{q}\mathcal{O}p_{G,\rho}$) denotes the
tangent space of $\mathcal{O}p_{G}$ (resp., $\mathcal{O}p_{G,\rho}$) at $q$,
then there exists a canonical isomorphism of $k$-vector spaces

(138)
$\displaystyle\gamma_{\sharp}:T_{q}\mathcal{O}p_{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{0}(X,\mathcal{V}_{G})\
\left(\text{resp.,}\
\gamma_{\sharp,\rho}:T_{q}\mathcal{O}p_{G,\rho}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{0}(X,{{}^{c}}\mathcal{V}_{G})\right).$

Moreover, $\gamma$ and $\gamma_{\sharp}$ (resp., $\gamma_{\widetilde{\rho}}$
and $\gamma_{\sharp,\rho}$) make the following square diagram commute:

(147) $\displaystyle\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.44872pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{q}\mathcal{O}p_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.11707pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.42361pt\hbox{$\scriptstyle{\gamma_{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-22.9297pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-40.56499pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.11707pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{q}\mathcal{C}onn_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
90.32237pt\raise-22.85971pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\gamma}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.37793pt\raise-22.85971pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
90.32237pt\raise-40.42499pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{0}(X,\mathcal{V}_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
34.99126pt\raise-60.34163pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.97223pt\hbox{$\scriptstyle{{{}^{\prime}}e_{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
57.44872pt\raise-52.20274pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
57.44872pt\raise-52.20274pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])}$}}}}}}}\ignorespaces}}}}\ignorespaces}\
\left(\text{resp.,}\ \vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
25.66043pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T_{q}\mathcal{O}p_{G,\rho}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.54048pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.42361pt\hbox{$\scriptstyle{\gamma_{\sharp,\rho}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-23.70442pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-41.40665pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.54048pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{q}\mathcal{C}onn_{G,\widetilde{\rho}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
92.7458pt\raise-24.12221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.0625pt\hbox{$\scriptstyle{\gamma_{\widetilde{\rho}}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
84.80136pt\raise-24.12221pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\wr}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
92.7458pt\raise-41.26665pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{0}(X,{{}^{c}}\mathcal{V}_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
34.7819pt\raise-61.18329pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.97223pt\hbox{$\scriptstyle{{{}^{\prime}}e_{\sharp,c}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
58.66043pt\raise-53.0444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
58.66043pt\raise-53.0444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])}$}}}}}}}\ignorespaces}}}}\ignorespaces}\right),$

where the upper horizontal arrow denotes the differential at $q$ of
$\mathrm{Imm}_{G}$ (resp., $\mathrm{Imm}_{G,\widetilde{\rho}}$). Thus, we have
obtained the following assertion.

######  Theorem 4.9.

There exist canonical short exact sequences

(148) $\displaystyle 0\longrightarrow T_{q}\mathcal{O}p_{G}$
$\displaystyle\longrightarrow T_{q}\mathcal{C}onn_{G}\longrightarrow
T_{q}^{\vee}\mathcal{O}p_{G,\rho}\longrightarrow 0,$ $\displaystyle
0\longrightarrow T_{q}\mathcal{O}p_{G,\rho}$ $\displaystyle\longrightarrow
T_{q}\mathcal{C}onn_{G,\widetilde{\rho}}\longrightarrow
T_{q}^{\vee}\mathcal{O}p_{G}\longrightarrow 0,$

where the second arrows in the upper and lower sequences are the differentials
at $q$ of $\mathrm{Imm}_{G}$ and $\mathrm{Imm}_{G,\rho}$, respectively.
Moreover, the second sequence in (148) is compatible with the dual of the
first one via the isomorphism (94).

###### Proof.

The assertion follows from Proposition 4.8 together with the commutativity of
the diagrams displayed in (147). ∎

###  4.5. The deformation space of a $G$-oper II

Next, we prove the self-duality of the parabolic de Rham cohomology group
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])$.
To this end, let us first consider the following lemma.

######  Lemma 4.10.

Suppose that $\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
and of radii $[0]^{\times r}$. For each $j\in\mathbb{Z}$, we shall set

(149)
$\displaystyle\Omega_{\mathrm{par}}^{0,\Box}(\mathfrak{g}_{\mathcal{E}})^{j}:=\Omega_{\mathrm{par}}^{0,\Box}(\mathfrak{g}_{\mathcal{E}})\cap\mathfrak{g}_{\mathcal{E}}^{j}\
\ \text{and}\ \
\Omega_{\mathrm{par}}^{1,\Box}(\mathfrak{g}_{\mathcal{E}})^{j}:=\Omega_{\mathrm{par}}^{1,\Box}(\mathfrak{g}_{\mathcal{E}})\cap(\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}}^{j}),$

where $\Box$ denotes either the absence or presence of “$c$”. Also, denote by

(150)
$\displaystyle{{}^{\Box}}\nabla_{\mathrm{par}}^{\mathrm{ad}(j)}:\Omega_{\mathrm{par}}^{0,\Box}(\mathfrak{g}_{\mathcal{E}})^{j}\rightarrow\Omega_{\mathrm{par}}^{0,\Box}(\mathfrak{g}_{\mathcal{E}})^{j-1}$

the morphism obtained by restricting $\nabla_{\mathrm{par}}^{\mathrm{ad}}$.
Then, the following assertions hold:

  * (i)

For each integer $j$, the composite

(151) $\displaystyle H^{0}(X,{{}^{c}}\mathcal{V}_{G}^{j})\rightarrow
H^{0}(X,\Omega^{1,c}_{\mathrm{par}}(\mathfrak{g}_{\mathcal{E}})^{j})\xrightarrow{\mathrm{quotient}}\mathrm{Coker}(H^{0}({{}^{c}}\nabla_{\mathrm{par}}^{\mathrm{ad}(j+1)}))$

is an isomorphism, where the first arrow arises from the natural inclusion
${{}^{c}}\mathcal{V}_{G}^{j}\hookrightarrow\Omega_{\mathrm{par}}^{1,c}(\mathfrak{g}_{\mathcal{E}})^{j}$.
In particular, we have

(152) $\displaystyle
H^{0}(X,{{}^{c}}\mathcal{V}_{G})\cong\mathrm{Coker}(H^{0}({{}^{c}}\nabla_{\mathrm{par}}^{\mathrm{ad}})).$

  * (ii)

For each integer $j$, the composite

(153)
$\displaystyle\mathrm{Ker}(H^{1}(\nabla^{\mathrm{ad}(j)}_{\mathrm{par}}))\xrightarrow{\mathrm{inclusion}}H^{1}(X,\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j})\rightarrow
H^{1}(X,\Omega\otimes_{\mathcal{O}_{X}}\mathcal{V}_{G}^{\vee,j})$

is an isomorphism, where
$\mathcal{V}_{G}^{\vee,j}:=(\mathcal{V}_{G}/\mathcal{V}_{G}^{-j+1})^{\vee}\left(\subseteq\mathcal{V}_{G}^{\vee}\right)$
and the second arrow arises from the natural surjection
$\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j}\twoheadrightarrow\Omega\otimes_{\mathcal{O}_{X}}\mathcal{V}_{G}^{\vee,j}$.
In particular, we have

(154)
$\displaystyle\mathrm{Ker}(H^{1}(\nabla^{\mathrm{ad}}_{\mathrm{par}}))\cong
H^{1}(X,\Omega\otimes_{\mathcal{O}_{X}}\mathcal{V}_{G}^{\vee}).$

###### Proof.

We only consider assertion (ii) since assertion (i) follows from a similar
argument. For each integer $j$, we shall write
$\nabla_{\mathrm{par}}^{\mathrm{ad}(j/j+1)}$ for the $\mathcal{O}_{X}$-linear
morphism
$\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}}^{j})/\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}}^{j+1})\rightarrow\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j-1}/\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j}$
induced from $\nabla_{\mathrm{par}}^{\mathrm{ad}}$.

First, let us consider the case of $j\geq 0$. Since the residue of $\nabla$ at
every marked point coincides with $q_{-1}$ (by the assumption that
$\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
and of radii $[0]^{\times r}$), one may verify that the resp’d portion of
[Wak4, (768)] restricts to a split exact sequence

(155) $\displaystyle
0\longrightarrow\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j}/\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j+1}\xrightarrow{\nabla_{\mathrm{par}}^{\mathrm{ad}(j/j+1)}}\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j-1}/\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j}\longrightarrow\mathcal{V}_{G,j-1}\longrightarrow
0.$

In particular, the equality
$\mathrm{Ker}(H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j/j+1)}))=0$ holds (by
which we obtain
$\mathrm{Ker}(H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j)}))=0$). Hence, the
assertion for $j\geq 0$ follows from the fact that
$\mathcal{V}_{G}^{\vee,j}\left(=(\mathcal{V}_{G}/\mathcal{V}_{G}^{-j+1})^{\vee}\right)=0$.

Next, we shall prove the case of $j\leq-1$. Consider the following morphism of
sequences:

(160)

Since the residue of $\nabla$ at every marked point coincides with $q_{-1}$,
the line bundles
$\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}}^{j})/\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}}^{j+1})$
and
$\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j-1}/\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j}$
coincide with
$\mathfrak{g}_{\mathcal{E}}^{j}/\mathfrak{g}_{\mathcal{E}}^{j+1}$ and
$\Omega\otimes_{\mathcal{O}_{X}}(\mathfrak{g}_{\mathcal{E}}^{j-1}/\mathfrak{g}_{\mathcal{E}}^{j})$,
respectively. In particular, these are isomorphic to direct sums of finite
copies of $\Omega^{\otimes j}$ (cf. [Wak4, § 2.1.4]), so we have

(161)
$H^{0}(X,\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}}^{j})/\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}}^{j+1}))=H^{0}(X,\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j-1}/\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j})=0.$

By the above equalities together with the fact that $\mathrm{dim}(X)=1$ (which
implies $H^{2}(-)=0$), both the upper and lower horizontal sequences in (160)
turn out to be exact. Also, the right-hand vertical arrow in (160) is
surjective because of the long exact sequence arising from the non-resp’d
portion of [Wak4, (768)]. By descending induction on $j$, one can verify the
surjectivity of both the left-hand and middle vertical arrows in (160). Thus,
the snake lemma applied to (160) shows that the natural sequence

(162)
$0\rightarrow\mathrm{Ker}(H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j+1)}))\rightarrow\mathrm{Ker}(H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j)}))\rightarrow\mathrm{Ker}(H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j/j+1)}))\rightarrow
0$

is exact. Let us consider the morphism of short exact sequences

(167)

where the left-hand vertical arrow is (153) with $j$ replaced by $j+1$ and the
exactness of the lower horizontal sequence follows from the natural
decomposition
$\mathcal{V}_{G}^{\vee,j}=\mathcal{V}_{G}^{\vee,j+1}\oplus(\mathcal{V}_{G,-j})^{\vee}$.
The right-hand vertical arrow in this diagram is an isomorphism because it
coincides with the inverse of the isomorphism
$\mathbb{R}^{1}f_{*}(\Omega\otimes_{\mathcal{O}_{X}}(\mathcal{V}_{G,-j})^{\vee})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ker}(\mathbb{R}^{1}f_{*}(\nabla^{\mathrm{ad}(j/j+1)}))$
arising from the splitting of the non-resp’d portion of [Wak4, (768)]. Hence,
by descending induction on $j$, we see that the middle vertical arrow (153) is
an isomorphism. This completes the proof of the assertion. ∎

By using the above lemma, we obtain the following proposition.

######  Proposition 4.11.

Suppose that the $G$-oper $\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
and of radii $[0]^{\times r}$ (where $[0]^{\times r}:=\emptyset$ if $r=0$).
Then, the following assertions hold:

  * (i)

$\mathbb{H}^{0}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])=\mathbb{H}^{2}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])=0$.

  * (ii)

There exists a short exact sequence

(168) $\displaystyle 0\longrightarrow
H^{0}(X,{{}^{c}}\mathcal{V}_{G})\xrightarrow{{{}^{\prime}}e_{\sharp,\mathrm{par}}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])\xrightarrow{{{}^{\prime}}e_{\flat,\mathrm{par}}}H^{1}(X,\Omega\otimes_{\mathcal{O}_{X}}\mathcal{V}_{G}^{\vee})\longrightarrow
0,$

which fits into the following isomorphism of short exact sequences:

(173)

In particular,
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])$
is a $k$-vector space of dimension
$(2g-2+r)\cdot\mathrm{dim}(\mathfrak{g})-r\cdot\mathrm{rk}(\mathfrak{g})$.

###### Proof.

First, we shall prove assertion (i). To this end, it suffices to verify the
equality
$\mathbb{H}^{2}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])=0$
because the proof of the remaining one is similar. Since
$H^{1}(X,\mathcal{V}_{G,j-1})=0$ for every $j$, it follows from the exactness
of (155) that the morphism

(174) $\displaystyle
H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j/j+1)}):H^{1}(X,\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j}/\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j+1})\rightarrow
H^{1}(X,\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j-1}/\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{\mathcal{E}})^{j})$

induced from $\nabla_{\mathrm{par}}^{\mathrm{ad}(j/j+1)}$ is surjective when
$j\geq 0$. Next, consider the following morphism of short exact sequences
defined for each $j\in\mathbb{Z}$:

(179)

By descending induction on $j$, the surjectivity of
$H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j/j+1)})$ implies that of
$H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j)}):H^{1}(X,\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j})\rightarrow
H^{1}(X,\Omega_{\mathrm{par}}^{0}(\mathfrak{g}_{\mathcal{E}})^{j-1})$. Then,
the required equality
$\mathbb{H}^{2}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])=0$
follows from the surjectivity of
$H^{1}(\nabla_{\mathrm{par}}^{\mathrm{ad}(j)})$ for
$j=-\mathrm{rk}(\mathfrak{g})$ together with the Hodge to de Rham spectral
sequence
$E_{1}^{a,b}=H^{b}(X,\mathcal{K}^{a}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])$.

Assertion (ii) follows from the commutativity of the diagram (110), the
definition of (112), Lemma 4.10, (i) and (ii), and [Wak4, Proposition 2.23,
(ii)]. ∎

Let us keep the assumption in the above proposition. As mentioned at the end
of § 3.3, there exists a canonical affine structure on
$\mathcal{O}p_{G,[0]^{\times r}}$ modeled on
$H^{0}(X,{{}^{c}}\mathcal{V}_{G})$. This affine structure induces an
isomorphism

(180) $\displaystyle\gamma_{\sharp,[0]^{\times
r}}:T_{q}\mathcal{O}p_{G,[0]^{\times
r}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{0}(X,{{}^{c}}\mathcal{V}_{G}).$

It follows from the various constructions involved that the following square
diagram is commutative:

(185)

where the upper horizontal arrow denotes the differential at $q$ of
$\mathrm{Imm}_{G,[0]^{\times r}}$.

######  Theorem 4.12.

  * (i)

There exists a canonical short exact sequence

(186) $\displaystyle 0\longrightarrow T_{q}\mathcal{O}p_{G,[0]^{\times
r}}\longrightarrow T_{q}\mathcal{C}onn_{G,[0]^{\times r}}\longrightarrow
T_{q}^{\vee}\mathcal{O}p_{G,[0]^{\times r}}\longrightarrow 0,$

where the second arrow is the differential of $\mathrm{Imm}_{G,[0]^{\times
r}}$ at $q$. Moreover, this sequence is, in an evident sense, invariant under
taking the dual via (119).

  * (ii)

Let us consider $T_{q}\mathcal{O}p_{G,[0]^{\times r}}$ as a $k$-vector
subspace of $T_{q}\mathcal{C}onn_{G,[0]^{\times r}}$ via the second arrow in
the lower sequence of (148). Then, $T_{q}\mathcal{O}p_{G,[0]^{\times r}}$ is
Lagrangian with respect to the nondegenerate pairing
$T_{q}\mathcal{C}onn_{G,[0]^{\times r}}\times
T_{q}\mathcal{C}onn_{G,[0]^{\times r}}\rightarrow k$ induced by (119).

###### Proof.

Assertion (i) follows from Proposition 4.11, (ii), together with the
isomorphisms $\gamma_{[0]^{\times r}}$ and $\gamma_{\sharp,[0]^{\times r}}$.

Next, by the definition of ${{}^{\prime}}e_{\sharp,\mathrm{par}}$, we see that
$T_{q}\mathcal{O}p_{G,[0]^{\times r}}$ is isotropic. Hence, assertion (ii)
follows from the equality $\mathrm{dim}(T_{q}\mathcal{O}p_{G,[0]^{\times
r}})=\frac{1}{2}\cdot\mathrm{dim}(T_{q}\mathcal{C}onn_{G,[0]^{\times r}})$
resulting from assertion (i). ∎

##  5\. Cohomology of the symmetric products of opers

In this section, we examine the (parabolic) de Rham cohomology of symmetric
products of a $\mathrm{PGL}_{2}$-oper (or an $\mathrm{SL}_{2}$-oper). We will
show that, if a given $G$-oper comes from a $\mathrm{PGL}_{2}$-oper via change
of structure group, then the (parabolic) de Rham cohomology group of the
induced adjoint bundle decomposes into the direct sum of such cohomology
groups (cf. (236)).

###  5.1. The de Rham cohomology of an $\mathrm{SL}_{n}$-oper

We shall fix an integer $n$ with $1<n<p$, and let
$\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\\{\mathcal{F}^{j}\\}_{i=0}^{n})$
be an $\mathrm{SL}_{n}$-oper on $\mathscr{X}$ (cf. Remark 2.1). Write
$\mathscr{F}:=(\mathcal{F},\nabla)$ and
${{}^{c}}\mathscr{F}:=({{}^{c}}\mathcal{F},{{}^{c}}\nabla)$, where
${{}^{c}}\nabla$ denotes the log connection on ${{}^{c}}\mathcal{F}$ obtained
by restricting $\nabla$. Let $\Box$ denote either the absence or presence of
“$c$”. Then, the natural morphisms of complexes
$\Omega\otimes_{\mathcal{O}_{X}}{{}^{\Box}}\mathcal{F}^{n-1}[-1]\rightarrow\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla]$
and
$\mathcal{K}^{\bullet}[{{}^{\Box}}\nabla]\rightarrow{{}^{\Box}}\mathcal{F}/{{}^{\Box}}\mathcal{F}^{1}[0]$
together induce a sequence of $k$-vector spaces

(187) $\displaystyle 0\longrightarrow
H^{0}(X,\Omega\otimes_{\mathcal{O}_{X}}{{}^{\Box}}\mathcal{F}^{n-1})\longrightarrow
H_{\mathrm{dR}}^{1}(X,{{}^{\Box}}\mathscr{F})\longrightarrow
H^{1}(X,{{}^{\Box}}\mathcal{F}/{{}^{\Box}}\mathcal{F}^{1})\longrightarrow 0.$

Also, the natural morphisms
$\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1}[-1]\rightarrow\mathcal{K}^{\bullet}[\nabla_{\mathrm{per}}]$
and
$\mathcal{K}^{\bullet}[\nabla_{\mathrm{per}}]\rightarrow\mathcal{F}/\mathcal{F}^{1}[0]$
induce a sequence of $k$-vector spaces

(188) $\displaystyle 0\longrightarrow
H^{0}(X,\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1})\longrightarrow
H_{\mathrm{dR},\mathrm{per}}^{1}(X,\mathscr{F})\longrightarrow
H^{1}(X,\mathcal{F}/\mathcal{F}^{1})\longrightarrow 0.$

Then, the following assertion holds.

######  Proposition 5.1.

  * (i)

The sequence (187) is exact.

  * (ii)

Suppose that $\mathscr{F}^{\heartsuit}$ is of radii $[0]^{\times
r}\in\mathfrak{c}(k)^{\times r}$ (where $[0]^{\times r}:=\emptyset$ if $r=0$).
Then, the sequence (188) is exact.

###### Proof.

We only consider assertion (ii) because the proof of assertion (i) is
relatively simpler than that of assertion (ii). For each $j=0,\cdots,n$, we
shall set $\Omega_{\mathrm{par}}^{0}(\mathcal{F})^{j}:=\mathcal{F}^{j}$ and
$\Omega_{\mathrm{par}}^{1}(\mathcal{F})^{j}:=\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{j}+\nabla(\mathcal{F}^{j+1})$,
where $\mathcal{F}^{n+1}:=0$. Also, denote by $\nabla^{j}_{\mathrm{par}}$
($j=1,\cdots,n$) the morphism
$\Omega^{0}_{\mathrm{par}}(\mathcal{F})^{j}\rightarrow\Omega^{1}_{\mathrm{par}}(\mathcal{F})^{j-1}$
obtained by restricting $\nabla$. By descending induction on $j\geq 1$, we
shall prove the claim that the morphism

(189) $\displaystyle
H^{l}(X,\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1})\rightarrow\mathbb{H}^{l+1}(X,\mathcal{K}^{\bullet}[\nabla^{j}_{\mathrm{par}}])$

induced by the inclusion
$\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1}[-1]\hookrightarrow\mathcal{K}^{\bullet}[\nabla^{j}_{\mathrm{par}}]$
is an isomorphism for every $l$. The base step, i.e., the case of $j=n$ is
clear because $\nabla^{n}_{\mathrm{par}}$ coincides with the zero map
$0\rightarrow\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1}$. To
prove the induction step, we assume that the case of $j=j_{0}$ ($1<j_{0}\leq
n$) has been proved. Note that the Kodaira-Spencer map
$\mathrm{KS}_{\mathscr{F}^{\heartsuit}}^{j_{0}-1}$ (cf. (34)) decomposes into
the composite of morphisms between line bundles

(190)
$\displaystyle\left(\mathcal{F}^{j_{0}-1}/\mathcal{F}^{j_{0}}=\right)\Omega_{\mathrm{par}}^{0}(\mathcal{F})^{j_{0}-1}/\Omega_{\mathrm{par}}^{0}(\mathcal{F})^{j_{0}}$
$\displaystyle\xrightarrow{\mathrm{KS}_{\mathscr{F}^{\heartsuit},\mathrm{par}}^{j_{0}-1}}\Omega_{\mathrm{par}}^{1}(\mathcal{F})^{j_{0}-2}/\Omega_{\mathrm{par}}^{1}(\mathcal{F})^{j_{0}-1}$
$\displaystyle\xrightarrow{\mathrm{inclusion}}\Omega\otimes_{\mathcal{O}_{X}}(\mathcal{F}^{j_{0}-2}/\mathcal{F}^{j_{0}-1}),$

where the first arrow, i.e.,
$\mathrm{KS}_{\mathscr{F}^{\heartsuit},\mathrm{par}}^{j_{0}-1}$, is the
morphism induced from $\nabla^{j}_{\mathrm{par}}$. Since
$\mathrm{KS}_{\mathscr{F}^{\heartsuit}}^{j_{0}-1}$ is an isomorphism by
assumption, the morphism
$\mathrm{KS}_{\mathscr{F}^{\heartsuit},\mathrm{par}}^{j_{0}-1}$ is verified to
be an isomorphism. The morphism
$\mathbb{H}^{l+1}(X,\mathcal{K}^{\bullet}[\nabla^{j_{0}}_{\mathrm{par}}])\rightarrow\mathbb{H}^{l+1}(X,\mathcal{K}^{\bullet}[\nabla^{j_{0}-1}_{\mathrm{par}}])$
induced by the inclusion
$\mathcal{K}^{\bullet}[\nabla^{j_{0}}_{\mathrm{par}}]\hookrightarrow\mathcal{K}^{\bullet}[\nabla^{j_{0}-1}_{\mathrm{par}}]$
is an isomorphism because the following diagram forms a morphism of short
exact sequences:

(195)

Hence, the induction hypothesis implies that the morphism
$H^{l}(X,\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1})\rightarrow\mathbb{H}^{l+1}(X,\mathcal{K}^{\bullet}[\nabla^{j_{0}-1}_{\mathrm{par}}])$
induced by the composite of natural inclusions

(196)
$\displaystyle\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1}[-1]\hookrightarrow\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{j_{0}}]\hookrightarrow\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{j_{0}-1}]$

is an isomorphism. This completes the proof of the claim. In particular, for
every $l$, we obtain an isomorphism

(197) $\displaystyle
H^{l}(X,\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{l+1}(X,\mathcal{K}^{\bullet}[\nabla^{1}_{\mathrm{par}}]).$

We go back to the proof of assertion (ii). According to [Wak4, Proposition
4.55], we may assume that the $\mathrm{PGL}_{n}$-oper induced by
$\mathscr{F}^{\heartsuit}$ via projectivization is normal in the sense of
[Wak4, Definition 4.53]. Let us fix $i\in\\{1,\cdots,r\\}$, and moreover,
choose a local function $t$ defining $\sigma_{i}\in X(k)$. Then, around the
marked point $\sigma_{i}$, the log connection $\nabla$ may be described as

(198)
$\displaystyle\nabla=d+\frac{dt}{t}\otimes\begin{pmatrix}0&0&0&\cdots&0&a_{n}\\\
1&0&0&\cdots&0&a_{n-1}\\\ 0&1&0&\cdots&0&a_{n-2}\\\ 0&0&1&\cdots&0&a_{n-3}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&1&a_{1}\end{pmatrix}$

for some local functions $a_{1},\cdots,a_{n}$ (cf. [Wak4, Remark 4.30]). Then,
the assumption in (ii) together with [Wak4, Theorem 4.49] implies that
$a_{j}\equiv 0$ mod $(t)$ for every $j=1,\cdots,n$. This implies
$\Omega^{1}_{\mathrm{par}}(\mathcal{F})^{0}=\Omega^{1}_{\mathrm{par}}(\mathcal{F})$.
Hence, the diagram

(203)

forms a short exact sequence of morphisms, which induces the long exact
sequence of $k$-vector spaces

(204) $\displaystyle H^{0}(X,\mathcal{F}/\mathcal{F}^{1})$
$\displaystyle\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{1}_{\mathrm{par}}])\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])$
$\displaystyle\rightarrow
H^{1}(X,\mathcal{F}/\mathcal{F}^{1})\rightarrow\mathbb{H}^{2}(X,\mathcal{K}^{\bullet}[\nabla^{1}_{\mathrm{par}}]).$

Since $\mathrm{deg}(\mathcal{F}/\mathcal{F}^{1})<0$, the equality
$H^{0}(X,\mathcal{F}/\mathcal{F}^{1})=0$ holds. On the other hand, by the
claim proved above, the $k$-vector space
$\mathbb{H}^{2}(X,\mathcal{K}^{\bullet}[\nabla^{1}_{\mathrm{par}}])\left(\cong
H^{1}(X,\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathcal{F}^{n-1})\cong
H^{0}(X,(\mathcal{F}^{n-1})^{\vee})^{\vee}\right)$ is verified to be zero
because of the fact that $\mathrm{deg}(\mathcal{F}^{n-1})>0$. Hence, (204)
together with (197) for $l=0$ implies the exactness of (188), thus completing
the proof of this proposition. ∎

###  5.2. Duality for symmetric products of opers

Next, let us describe the short exact sequences (and their dualities)
discussed in the previous subsection for the $\mathrm{SL}_{n}$-oper arising
from an $\mathrm{SL}_{2}$-oper/a $\mathrm{PGL}_{2}$-oper via change of
structure group. If the underlying space is a compact Riemann surface, then
the assertion corresponding to the following theorem can be found in [Gun,
Theorem 6] (for $n=3$) and [Wen, § 4.4.4]. The case of the uniformizing
$\mathrm{SL}_{2}$-opers on modular curves was already proved in [Sch, Theorem
2.7]. (Note that, by the discussion at the beginning of § 2.4 in loc. cit.,
the radius of the uniformizing $\mathrm{SL}_{2}$-oper at every marked point
coincides with $[0]$; hence, the following Theorem 5.2, (ii), can be regarded