# Opers with real monodromy
and Eichler-Shimura isomorphism

Yasuhiro Wakabayashi

###### Abstract.

The purpose of the present paper is to investigate $G$-opers on pointed
Riemann surfaces (for a simple algebraic group $G$ of adjoint type) and their
monodromy maps. In the first part, we review some general facts on $G$-opers,
or more generally, principal $G$-bundles with holomorphic connection having
simple poles along marked points, including the correspondence with
$G$-representations of the fundamental group. One of the main results, proved
in the second part, asserts that the space of certain $G$-opers with real
monodromy forms a discrete set. This fact generalizes the discreteness theorem
for real projective structures, already proved by G. Faltings. As an
application, we establish the Eichler-Shimura isomorphism for each
$\mathrm{PSL}_{2}$-oper with real monodromy. The resulting decomposition of
the (parabolic) de Rham cohomology group of its symmetric product defines a
polarized real Hodge structure.

††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 30F30, Secondary 34M03;††footnotetext: Key
words: Riemann surface, connection, oper, Fuchsian group, character variety,
Eichler-Shimura isomorphism

###### Contents

  1. 1 Introduction
    1. 1.1 Theme of the present paper
    2. 1.2 Statement of results
    3. 1.3 Organization of the present paper
  2. 2 Flat $G$-bundles on a pointed Riemann surface
    1. 2.1 Flat $G$-bundles
    2. 2.2 Log connections on the trivial $G$-bundle
    3. 2.3 Gauge transformation
    4. 2.4 Residue of a flat $G$-bundle
    5. 2.5 Local description of flat $G$-bundles
  3. 3 $G$-representations of the fundamental group
    1. 3.1 $G$-representations
    2. 3.2 From $G$-representations to flat $G$-bundles
    3. 3.3 From flat $G$-bundles to $G$-representations
    4. 3.4 Local monodromy
    5. 3.5 Group cohomology
    6. 3.6 Comparison with sheaf cohomology
    7. 3.7 Representation variety
    8. 3.8 Flat $G$-bundles with prescribed nilpotent residue
  4. 4 $G$-opers on a pointed Riemann surface
    1. 4.1 $G$-opers
    2. 4.2 Change of structure group
    3. 4.3 Canonical $B$-bundle
    4. 4.4 Affine structure on $\mathrm{Op}_{P}(\mathscr{X},G)$
    5. 4.5 Permissible connections
    6. 4.6 Correspondence with $\mathrm{PGL}_{2}$-opers
    7. 4.7 Permissible $G$-opers
  5. 5 Monodromy of $G$-opers
    1. 5.1 Parabolic de Rham cohomology
    2. 5.2 Monodromy of a $G$-oper
  6. 6 Proofs of the main theorems
    1. 6.1 Bilinear form on the parabolic cohomology
    2. 6.2 Hermitian inner product on the maximal torus
    3. 6.3 Discreteness of the intersection
    4. 6.4 Eichler-Shimura decompositions for $G^{\odot}$-opers
    5. 6.5 Hodge structures associated to $G^{\odot}$-opers
    6. 6.6 Acknowledgements

##  1\. Introduction

###  1.1. Theme of the present paper

The purpose of the present paper is to study the monodromy of flat connections
on principal $G$-bundles for a linear algebraic group $G$ over the field of
complex numbers $\mathbb{C}$.

In order to state the framework and our results, let us take a collection
$\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i=1}^{r})$ consisting of a compact Riemann
surface $X$ of genus $g\geq 0$ and a (possibly empty) set of mutually distinct
points $\sigma_{1},\cdots,\sigma_{r}$ in $X$ with $2g-2+r>0$. The fundamental
group $\Gamma$ of $X^{*}:=X\setminus\bigcup_{i=1}^{r}\\{\sigma_{i}\\}$ at a
fixed base point $x_{0}$ is isomorphic to the group finitely generated by
elements $A_{1},\cdots,A_{g}$, $B_{1},\cdots,B_{g}$, and $T_{1},\cdots,T_{r}$
(where each $T_{i}$ is represented by a loop around $\sigma_{i}$) with
relation $\prod_{j=1}^{g}[A_{j},B_{j}]\cdot\prod_{i=1}^{r}T_{i}=e$. This
explicit description helps us to investigate the category
$\mathcal{R}ep(\Gamma,G(\mathbb{C}))$ of representations of $\Gamma$ valued in
the group $G(\mathbb{C})$ of $\mathbb{C}$-rational points of $G$ (cf. (29)).

The Riemann-Hilbert correspondence (in the most basic form) can be generalized
to the relationship between objects in $\mathcal{R}ep(\Gamma,G(\mathbb{C}))$
and flat $G$-bundles, i.e., principal $G$-bundles equipped with a flat
holomorphic connection. In fact, each flat $G$-bundle on $X^{*}$ has a flat
trivialization on some local neighborhood of every point, which we call a
fundamental framing (cf. Definition 2.2). If $\gamma$ is a path beginning and
ending at $x_{0}$, then there is an element of $G(\mathbb{C})$ which expresses
the transformation effected on a fixed fundamental framing at $x_{0}$, by the
process of analytic continuation around $\gamma$. In this way, each flat
$G$-bundle associates a $G(\mathbb{C})$-representation of $\Gamma$, which is
well-defined up to post-composition with an inner automorphism of
$G(\mathbb{C})$ and uniquely determined after fixing a trivialization of the
fiber of the underlying principal $G$-bundle over $x_{0}$. The resulting
functor

(1)
$\displaystyle\mathcal{C}onn(X^{*},G)\rightarrow\mathcal{R}ep(\Gamma,G(\mathbb{C})),$

where $\mathcal{C}onn(X^{*},G)$ denotes the category of flat $G$-bundles on
$X^{*}$ (cf. (13)), is verified to be an equivalence of categories (cf.
Theorem 3.1), as in the classical case of linear differential equations, i.e.,
$G=\mathrm{GL}_{n}$; see §§ 3.1-3.3 for the detailed discussion concerning
this equivalence.

We can also consider flat $G$-bundles on the pointed Riemann surface
$\mathscr{X}$, which are by definition principal $G$-bundles on $X$ equipped
with a meromorphic connection with simple poles along the marked points. The
residue of such a flat $G$-bundle at each marked point $\sigma_{i}$
($i=1,\cdots,r$) determines an element of the adjoint quotient
$\mathfrak{c}:=\mathfrak{g}/\\!/G$ of the Lie algebra
$\mathfrak{g}:=\mathrm{Lie}(G)$; this element is called the radius at
$\sigma_{i}$ (cf. § 2.4).

Under the assumption that $G$ is a simple algebraic group of adjoint type (or
a somewhat more general algebraic group), there is an important class of flat
$G$-bundles, called $G$-opers, that appear in some areas of mathematics,
including the geometric Langlands program. Roughly speaking, a $G$-oper on
$\mathscr{X}$ is a flat $G$-bundle on $\mathscr{X}$ equipped with a Borel
reduction satisfying a strict form of Griffiths transversality (cf. § 4.1 for
the precise definition). One obtains the moduli space

(2) $\displaystyle\mathrm{Op}_{P}(\mathscr{X},G)$

(cf. (76)) parametrizing $G$-opers on $\mathscr{X}$ with certain parabolic
conditions on its radii. It is known that $\mathrm{Op}_{P}(\mathscr{X},G)$ can
be represented by a complex manifold whose dimension is equal to the value

(3) $\displaystyle
a(G):=\frac{2g-2+r}{2}\cdot\mathrm{dim}(G)-\frac{r}{2}\cdot\mathrm{rk}(G)$

(cf. [BeDr, Proposition 3.1.10], [Wak1, Theorem A]), where $\mathrm{rk}(G)$
denotes the rank of $G$. For the study of $G$-opers and their moduli space, we
refer the reader to, e.g., [BeDr], [Fre1], [Fre2], [Fre3], [FrBe], and [Wak1].

We shall set

(4) $\displaystyle\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$

(cf. (59)) to be the set of irreducible representations $\Gamma\rightarrow
G(\mathbb{C})$ up to conjugation (i.e., isomorphism classes of
$\mathrm{ob}(\mathcal{R}ep(\Gamma,G(\mathbb{C})))$) satisfying certain
conditions on the images of the elements $T_{i}$; it forms a complex manifold
of dimension $2\cdot a(G)$ (cf. [Wei]). The study of local monodromy (cf.
Theorem 3.2, Corollary 3.3) leads us to show that taking the monodromy maps
yields, via (1), an injection between complex manifolds

(5)
$\displaystyle\mathrm{Op}_{P}(\mathscr{X},G)\hookrightarrow\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$

(cf. Proposition 5.3). By this injection, $\mathrm{Op}_{P}(\mathscr{X},G)$ can
be regarded as a submanifold of $\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$.

On the other hand, since $G$ admits a real form $G_{\mathbb{R}}$ (by which it
makes sense to speak of the subgroup $G(\mathbb{R})$ of $G(\mathbb{C})$
consisting of elements invariant under the complex conjugation), we have a
real submanifold

(6) $\displaystyle\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))$

of $\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ parametrizing representations
whose images are contained in $G(\mathbb{R})$ (up to conjugation). Since the
sum of the real dimensions of $\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))$ and
$\mathrm{Op}_{P}(\mathscr{X},G)$ equals that of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$, it is natural to ask whether or
not these real submanifolds intersect transversally, or equivalently, the
intersection

(7)
$\displaystyle\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))\cap\mathrm{Op}_{P}(\mathscr{X},G)$

(which classifies certain $G$-opers with real monodromy) forms a discrete set.
So we are mainly interested in knowing what the deformation space of $G$-opers
with real monodromy looks like.

###  1.2. Statement of results

To the best of the author’s knowledge, there is only one explicit result on
this subject, dealing with $\mathrm{PSL}_{2}$-opers. In fact, G. Faltings
introduced permissible connections (cf. [Fal, § 5]) on a certain rank $2$
vector bundle, and proved that their moduli space is transversal to the
submanifold $\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{2}(\mathbb{R}))$ (cf.
[Fal, Theorem 12]). Since permissible connections are equivalent to objects in
$\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{2})$ after a certain adjustment at
the marked points (cf. Proposition 4.3), this result can be regarded as
asserting about the discreteness of the intersection
$\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{2}(\mathbb{R}))\cap\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{2})$.
This case is of particular interest as it contains the canonical
$\mathrm{PSL}_{2}$-opers arising from the Fuchsian uniformization. For topics
on $\mathrm{PSL}_{2}$-opers (= projective structures) with real monodromy, we
refer the reader to [BHS], [Hel], [Tak], and [Wil].

In the present paper, we partially generalize Faltings’ result to a general
$G$. To this end, we begin by reviewing some basic things about flat
$G$-bundles on pointed Riemann surfaces, including their monodromy maps and
parabolic cohomology groups. Some of the facts described here with proofs
(e.g., Corollary 2.7, Theorems 3.1, 3.2, and Proposition 3.6) may be well-
known to experts in this field. In fact, when $G$ is, say, a matrix group, the
assertions would directly follow from the classical case of linear
differential equations, i.e., $G=\mathrm{GL}_{n}$. At any rate, however, the
author could not find any reasonable references to them, so decided to write
the proofs with some care.

After studying some properties on flat $G$-bundles and $G$-opers, we prove the
following assertion; this is the first main theorem of the present paper.

######  Theorem A (= Theorem 6.3).

Let us keep the above notation. Then, the two submanifolds
$\mathrm{Rep}^{0}_{P}(\Gamma,G(\mathbb{R}))$, $\mathrm{Op}_{P}(\mathscr{X},G)$
of $\mathrm{Rep}^{0}_{P}(\Gamma,G(\mathbb{C}))$ transversally intersect at the
points classifying permissible $G$-opers (cf. Definition 4.5 for the
definition of a permissible $G$-oper). In particular, the space of permissible
$G$-opers with real monodromy forms a discrete set.

######  Remark 1.1.

  * (i)

Permissible $G$-opers are, by definition, essentially equivalent to
$\mathrm{PSL}_{2}$-opers via change of structure group by the morphism
$\iota_{G}:\mathrm{PSL}_{2}\rightarrow G$ induced from a suitable choice of an
$\mathfrak{s}\mathfrak{l}_{2}$-triple in $\mathfrak{g}$. However, Theorem A
does not follow immediately from Faltings’ result (dealing with
$\mathrm{PSL}_{2}$-opers) and is essentially stronger than it. This is because
we need to consider, in the present situation, the deformation spaces having
many more parameters not derived from deformations as a
$\mathrm{PSL}_{2}$-oper.

  * (ii)

While it is expected that the above theorem still be true after removing the
assumption of being “permissible”, we have no idea how to prove it at the time
of writing the present paper (cf. Conjecture 6.4).

Next, we shall describe an application of Theorem A concerning a
generalization of the Eichler-Shimura isomorphism. Let
$\mathscr{E}^{\spadesuit}_{\odot}$ be a $\mathrm{PSL}_{2}$-oper classified by
$\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{2}(\mathbb{R}))\cap\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{2})$;
it has a real monodromy map
$\mu:\Gamma\rightarrow\mathrm{PSL}_{2}(\mathbb{R})$. For $N\geq 2$, $\mu$
(resp., $\mathscr{E}^{\spadesuit}_{\odot}$) induces a
$\mathrm{PSL}_{N}(\mathbb{R})$-representation $\mu_{N}$ of $\Gamma$ (resp., a
$\mathrm{PSL}_{N}$-oper $\mathscr{E}^{\spadesuit}$ on $\mathscr{X}$) via
change of structure group using the embedding
$\mathrm{PSL}_{2}\hookrightarrow\mathrm{PSL}_{N}$ obtained by integrating a
principal $\mathfrak{s}\mathfrak{l}_{2}$-triple in $\mathfrak{g}$. Denote by

(8) $\displaystyle
T_{[\mu_{N}]}\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{N}(\mathbb{C}))\
\left(\text{resp.,}\
T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})\right)$

the tangent space of
$\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{N}(\mathbb{C}))$ (resp.,
$\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})$) at the point classifying
$\mu_{N}$ (resp., $\mathscr{E}^{\spadesuit}$).

Then, Theorem A says that the subspace of
$T_{[\mu_{N}]}\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{N}(\mathbb{C}))$
consisting of elements invariant under the complex conjugation is disjoint
from
$T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})$.
By this fact,
$T_{[\mu_{N}]}\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{N}(\mathbb{C}))$
decomposes into a direct sum

(9) $\displaystyle
T_{[\mu_{N}]}\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{N}(\mathbb{C}))=T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})\oplus\overline{T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})}.$

of $T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})$
and its complex conjugate
$\overline{T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})}$.
Regarding the left-hand side of this decomposition, we recall that
$T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{N})$
forms a structure of affine space modeled on the $\mathbb{C}$-vector space
$\bigoplus_{j=1}^{N-1}H^{0}(X,\Omega^{\otimes(j+1)}(-D))$, where $\Omega$
denotes the sheaf of holomorphic $1$-forms on $X$ and $D$ denotes the divisor
on $X$ defined by the union of the $\sigma_{i}$’s.

On the other hand, the irreducible decomposition of the
$\mathbb{C}[\Gamma]$-module
$\mathfrak{s}\mathfrak{l}_{N}\left(=\mathrm{Lie}(\mathrm{PSL}_{N})\right)$
determined by $\mu_{N}$ gives rise to a decomposition

(10) $\displaystyle
T_{[\mu_{N}]}\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{N}(\mathbb{C}))=\bigoplus_{j=1}^{N-1}H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu}),$

where $H_{P}^{1}(-)$ denotes the $1$-st parabolic cohomology (cf. (41)) and
$V_{2j,\mathbb{C},\mu}$ denotes the $\mathbb{C}$-vector space of homogenous
degree $2j$ polynomials in $\mathbb{C}[x,y]$ equipped with a natural
$\Gamma$-action induced from $\mu$ (cf. (158)). Since (9) preserves the direct
sum decomposition, we obtain the Eichler-Shimura isomorphism for the
$\mathrm{PSL}_{2}$-oper $\mathscr{E}^{\spadesuit}_{\odot}$ by considering its
$j$-th factor. The assertion is described as follows. (Analogous results for
opers in positive characteristic can be found in [Wak2].)

######  Theorem B (cf. Theorems 6.6, 6.8 for the precise statements).

Let $\mathscr{E}^{\spadesuit}_{\odot}$ be a $\mathrm{PSL}_{2}$-oper classified
by
$\mathrm{Rep}_{P}^{0}(\Gamma,\mathrm{PSL}_{2}(\mathbb{R}))\cap\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{2})$
with monodromy map $\mu:\Gamma\rightarrow\mathrm{PSL}_{2}(\mathbb{R})$. Then,
for each positive integer $j$, there exists a canonical decomposition

(11) $\displaystyle
H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})=H^{0}(X,\Omega^{\otimes(j+1)}(-D))\oplus\overline{H^{0}(X,\Omega^{\otimes(j+1)}(-D))}$

of the $1$-st parabolic cohomology group
$H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})$ of the $\mathbb{C}[\Gamma]$-module
$V_{2j,\mathbb{C},\mu}$. Moreover, the real form
$H^{1}_{P}(\Gamma,V_{2j,\mathbb{R},\mu})$ of
$H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})$ admits a polarized real Hodge
structure of weight $1$ whose Hodge decomposition is given by (11).

######  Remark 1.2.

As far as the author knows, the Eichler-Shimura isomorphisms established in
the previous work are essentially the case of Fuchsian representations. On the
other hand, the above theorem actually involves new situations, since the
representations associated to $\mathrm{PSL}_{2}$-opers are not only the
Fuchsian ones. In fact, if $r=0$, then it is known that every non-elementary
liftable $\mathrm{PSL}_{2}(\mathbb{R})$-representation of $\Gamma$ arises from
the monodromy map of a projective structure, i.e., $\mathrm{PSL}_{2}$-oper
(cf. [GKM, Theorem 1.1.1] or [GGP]). Hence, we can apply Theorem B to any such
representation.

###  1.3. Organization of the present paper

First of all, we note that many of the arguments and notational conventions
follow [Wak1] (and [Wak2]). Although [Wak1] only deals with pointed stable
curves in a purely algebraic treatment, various arguments discussed there can
also be applied to our analytic situation.

In § 2, we consider flat $G$-bundles on a pointed Riemann surface and some
relevant basic things, including an explicit description of gauge
transformation using the Maurer-Cartan form (cf. (20)). The local triviality
of a flat $G$-bundle (cf. Proposition 2.6, Corollary 2.7) asserted in that
section generalizes classical results for linear differential equations; in
fact, the proofs are entirely similar to the classical case.

In § 3, we examine $G$-representations on a surface group and describe
infinitesimal deformations of such a representation in terms of parabolic
cohomology groups. Also, two versions of the (most basic form of) Riemann-
Hilbert correspondence for flat $G$-bundles are formulated in Theorems 3.1,
3.7.

In § 4, we discuss $G$-opers on a pointed Riemann surface $\mathscr{X}$, as
well as the moduli space $\mathrm{Op}_{P}(\mathscr{X},G)$ parametrizing those
having specific radii. Some of the facts on $G$-opers proved in [Wak1] (for
pointed stable curves) or [BeDr] (for compact Riemann surfaces) are
reformulated in order to use in the subsequent discussion. After that, we
recall the notion of a permissible connection defined by Faltings in [Fal],
and construct a bijective correspondence between permissible connections and
objects in $\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{2})$ (cf. Proposition
4.3). The definition of a permissible $G$-oper is described at the end of that
section (cf. Definition 4.5).

In § 5, we deal with the parabolic de Rham cohomology of a flat vector bundle
and the comparison with the parabolic sheaf cohomology of the corresponding
local system (cf. Proposition 5.2). Moreover, it is shown that
$\mathrm{Op}_{P}(\mathscr{X},\mathrm{PSL}_{2})$ may be embedded, via taking
the monodromy groups, into the representation variety
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ (cf. (131)).

The first part of § 6 provides a proof of Theorem A (cf. Theorem 6.3) by
investigating a canonical bilinear form on the tangent space of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$. An essential ingredient in the
proof is an explicit description of that form restricted to the tangent space
of $\mathrm{Op}_{P}(\mathscr{X},G)$ (cf. Theorem 6.1); this description shows
the transversality of the intersection required in Theorem A. In the second
part, the classical Eichler-Shimura isomorphism is generalized to the case of
general $\mathrm{PSL}_{2}$-opers with real monodromy by applying Theorem A
(cf. Theorem 6.6), and we formulate it in terms of Hodge structures (cf.
Theorem 6.8). This proves Theorem B.

##  2\. Flat $G$-bundles on a pointed Riemann surface

This first section discusses flat $G$-bundles on a pointed Riemann surface and
some relevant notions used in the subsequent discussion. At the end of this
section, we prove the local triviality of a flat $G$-bundle (cf. Proposition
2.6, Corollary 2.7), which generalizes the existence assertion of local
solutions for a linear differential equation.

###  2.1. Flat $G$-bundles

Let $\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i=1}^{r})$ (where $r\in\mathbb{Z}_{\geq
0}$) be a pointed Riemann surface, i.e., a Riemann surface $X$ (that is not
necessarily compact) together with a possibly empty set of mutually distinct
marked points $\sigma_{1},\cdots,\sigma_{r}\in X$. Write $\mathcal{O}_{X}$ for
the sheaf of holomorphic functions on $X$, $\Omega_{X}$ for the sheaf of
holomorphic $1$-forms on $X$, and
$\mathcal{T}_{X}\left(:=\Omega_{X}^{\vee}\right)$ for the dual of
$\Omega_{X}$, i.e., the sheaf of holomorphic vector fields on $X$. Also, we
shall set $\Omega_{X^{\mathrm{log}}}:=\Omega_{X}(D)$ and
$\mathcal{T}_{X^{\mathrm{log}}}:=\mathcal{T}_{X}(-D)\left(=\Omega_{X^{\mathrm{log}}}^{\vee}\right)$,
where $D$ denotes the divisor on $X$ defined as the union of the
$\sigma_{i}$’s.

Let $G$ be a connected linear algebraic group over $\mathbb{C}$ with identity
element $e_{G}$. Denote by $\mathfrak{g}$ the Lie algebra of $G$ and by
$\mathrm{Ad}_{G}:G\rightarrow\mathrm{GL}(\mathfrak{g})$ the adjoint
representation of $G$. By a $G$-bundle on $X$, we shall mean a holomorphic
principal $G$-bundle on $X$, where the elements of $G$ act from the right.

Let us fix a $G$-bundle $\pi:\mathcal{E}\rightarrow X$ on $X$. The pull-back
$D_{\mathcal{E}}$ of $D$ via $\pi$ specifies a divisor on $\mathcal{E}$. The
$G$-action on $\mathcal{E}$ induces a $G$-action on the direct image
$\pi_{*}(\mathcal{T}_{\mathcal{E}}(D_{\mathcal{E}}))$ of the holomorphic
vector bundle $\mathcal{T}_{\mathcal{E}}(D_{\mathcal{E}})$. The subsheaf
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}:=\pi_{*}(\mathcal{T}_{\mathcal{E}}(D_{\mathcal{E}}))^{G}$
of $G$-invariant sections of
$\pi_{*}(\mathcal{T}_{\mathcal{E}}(D_{\mathcal{E}}))$ fits into the following
short exact sequence of $\mathcal{O}_{X}$-modules induced by differentiating
$\pi$:

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

(cf. [Wak1, § 1.2.5]), where $\mathfrak{g}_{\mathcal{E}}$ denotes the adjoint
bundle associated to $\mathcal{E}$, i.e., the vector bundle obtained from
$\mathcal{E}$ via change of structure group by $\mathrm{Ad}_{G}$.

Recall that a log connection on $\mathcal{E}$ is, by definition, an
$\mathcal{O}_{X}$-linear morphism
$\nabla:\mathcal{T}_{X^{\mathrm{log}}}\rightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}$
satisfying
$d_{\mathcal{E}}\circ\nabla=\mathrm{id}_{\mathcal{T}_{X^{\mathrm{log}}}}$.
When $r=0$ (i.e., $\\{\sigma_{i}\\}_{i}=\emptyset$), any log connection is
referred to as a non-logarithmic connection, or simply, a connection.

Since $\Omega_{X^{\mathrm{log}}}$ is a holomorphic line bundle, any log
connection is automatically flat, i.e., has vanishing curvature (cf. [Wak1,
Definition 1.23] for the definition of curvature). By a flat $G$-bundle on
$\mathscr{X}$, we shall mean a pair $\mathscr{E}:=(\mathcal{E},\nabla)$
consisting of a $G$-bundle $\mathcal{E}$ on $X$ and a (flat) log connection
$\nabla$ on $\mathcal{E}$. One can define the notion of an isomorphism between
flat $G$-bundles. Thus, we obtain the category

(13) $\displaystyle\mathcal{C}onn(\mathscr{X},G)$

consisting of flat $G$-bundles on $\mathscr{X}$ and isomorphisms between them;
it is verified to forms a $\mathbb{C}$-stack by considering families of such
objects.

###  2.2. Log connections on the trivial $G$-bundle

The trivial $G$-bundle on $X$ is the $G$-bundle on $X$ defined as the product
$X\times G$, where $G$ acts only on the second factor and the first projection
$X\times G\rightarrow X$ is considered as its structure morphism.

For each morphism $h:X\rightarrow G$, the left-translation
$\mathrm{L}_{h}:X\times G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X\times
G$ by $h$ (cf. [Wak1, § 1.1.2]) determines an automorphism of the trivial
$G$-bundle $X\times G$. Conversely, any automorphism of the trivial $G$-bundle
on $X$ coincides with the left-translation by some morphism $X\rightarrow G$.

The differentials of the first and second projections $\mathrm{pr}_{1}$,
$\mathrm{pr}_{2}$ from $X\times G$ yield an $\mathcal{O}_{X\times G}$-linear
isomorphism

(14) $\displaystyle\mathcal{T}_{(X\times
G)^{\mathrm{log}}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{pr}_{1}^{*}(\mathcal{T}_{X^{\mathrm{log}}})\oplus\mathrm{pr}_{2}^{*}(\mathcal{T}_{G}).$

By applying the functor $\mathrm{pr}_{1}^{*}(-)^{G}$ to this isomorphism, we
obtain a decomposition

(15) $\displaystyle\widetilde{\mathcal{T}}_{(X\times
G)^{\mathrm{log}}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{T}_{X^{\mathrm{log}}}\oplus\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}$

of $\widetilde{\mathcal{T}}_{(X\times G)^{\mathrm{log}}}$. The direct summand
$\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}$ of the codomain is naturally
isomorphic to the adjoint bundle $\mathfrak{g}_{X\times G}$ of the trivial
$G$-bundle $X\times G$. The surjection $d_{X\times
G}:\widetilde{\mathcal{T}}_{(X\times
G)^{\mathrm{log}}}\twoheadrightarrow\mathcal{T}_{X^{\mathrm{log}}}$ (cf. (12))
may be identified, via (15), with the first projection
$\mathcal{T}_{X^{\mathrm{log}}}\oplus\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}\twoheadrightarrow\mathcal{T}_{X^{\mathrm{log}}}$.

Given a log connection $\nabla$ on the trivial $G$-bundle $X\times G$, we
shall write $\nabla^{\mathfrak{g}}$ for the composite

(16)
$\displaystyle\nabla^{\mathfrak{g}}:\mathcal{T}_{X^{\mathrm{log}}}\xrightarrow{\nabla}\widetilde{\mathcal{T}}_{(X\times
G)^{\mathrm{log}}}\xrightarrow{(\ref{Eq123})}\mathcal{T}_{X^{\mathrm{log}}}\oplus\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}\xrightarrow{\text{second
projection}}\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}.$

This morphism characterizes the log connection $\nabla$ because the equality
$\nabla=(\mathrm{id}_{\mathcal{T}_{X^{\mathrm{log}}}},\nabla^{\mathfrak{g}})$
holds under the identification $\widetilde{\mathcal{T}}_{(X\times
G)^{\mathrm{log}}}=\mathcal{T}_{X^{\mathrm{log}}}\oplus\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}$
given by (15). The assignment $\nabla\mapsto\nabla^{\mathfrak{g}}$ determines
a bijective correspondence between the set of log connections on $X\times G$
and the set of $\mathcal{O}_{X}$-linear morphisms
$\mathcal{T}_{X^{\mathrm{log}}}\rightarrow\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}$.
The log connection

(17)
$\displaystyle\nabla^{\mathrm{triv}}:\mathcal{T}_{X^{\mathrm{log}}}\rightarrow\widetilde{\mathcal{T}}_{(X\times
G)^{\mathrm{log}}}$

corresponding to the zero map
$\mathcal{T}_{X^{\mathrm{log}}}\rightarrow\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}$
via this bijection is called the trivial (log) connection.

###  2.3. Gauge transformation

We recall the Maurer-Cartan form on the algebraic group $G$, which is used to
describe gauge transformations of a log connection. Note that the Lie algebra
$\mathfrak{g}$ can be identified with the space
$\Gamma(G,\mathcal{T}_{G})^{G}$ of vector fields on $G$ invariant under the
various left-translations. That is to say, given any element
$v\in\mathfrak{g}=\mathcal{T}_{G}|_{e_{G}}$, we can use left-translation to
produce a global vector field $v^{\dagger}$ on $G$ with
$v^{\dagger}|_{h}=d\mathrm{L}_{h}|_{e_{G}}(v)$ for every $h\in G(\mathbb{C})$,
where $d\mathrm{L}_{h}|_{e_{G}}$ denotes the $\mathbb{C}$-linear morphism
$\mathcal{T}_{G}|_{e_{G}}\rightarrow\mathcal{T}_{G}|_{h}$ obtained as the
differential of $\mathrm{L}_{h}$ at $e_{G}$.

Then, there exists an $\mathcal{O}_{G}$-linear isomorphism
$\omega_{G}^{\triangleright}:\mathcal{T}_{G}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{G}\otimes_{\mathbb{C}}\mathfrak{g}$
determined uniquely by the condition that
$\omega_{G}^{\triangleright}(v^{\dagger})=1\otimes v$ for any
$v\in\mathfrak{g}$. The $\mathfrak{g}$-valued $1$-form

(18)
$\displaystyle\omega_{G}\in\Gamma(G,\Omega_{G}\otimes_{\mathbb{C}}\mathfrak{g})\left(\cong\mathrm{Hom}_{G}(\mathcal{T}_{G},\mathcal{O}_{G}\otimes_{\mathbb{C}}\mathfrak{g})\right)$

corresponding to this morphism is called the (left-invariant) Maurer-Cartan
form on $G$ (cf. [Wak1, Definition 1.1]).

Let $\eta:\mathcal{E}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}$
be an automorphism of a $G$-bundle $\mathcal{E}$ on $X$. For a log connection
$\nabla$ on $\mathcal{E}$, the composite

(19)
$\displaystyle\eta_{*}(\nabla):\mathcal{T}_{X^{\mathrm{log}}}\xrightarrow{\nabla}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}\xrightarrow{d\eta^{G}}\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}},$

where $d\eta^{G}$ denotes the automorphism of
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}$ obtained by
differentiating $\eta$, forms another log connection. We shall refer to
$\eta_{*}(\nabla)$ as the gauge transformation of $\nabla$ by $\eta$ (cf.
[Wak1, Definition 1.20]).

If $\mathcal{E}=X\times G$ and $\eta=\mathrm{L}_{h}$ for a morphism
$h:X\rightarrow G$, then the gauge transformation $\mathrm{L}_{h*}(\nabla)$
satisfies the equality

(20)
$\displaystyle\mathrm{L}_{h*}(\nabla)^{\mathfrak{g}}=(h^{-1})^{*}(\omega_{G})+\mathrm{Ad}_{G}(h)\circ\nabla^{\mathfrak{g}}$

of morphisms
$\mathcal{T}_{X^{\mathrm{log}}}\rightarrow\mathcal{O}_{X}\otimes\mathfrak{g}$
(cf. [Wak1, Proposition 1.22]).

######  Remark 2.1 (Gauge transformation for a matrix group).

Suppose that $G$ is a matrix group, i.e., equipped with an injective morphism
$\iota:G\hookrightarrow\mathrm{GL}_{N}$ for some $N\in\mathbb{Z}_{>0}$. In
particular, each element of $G$ may be considered as an $N\times N$ matrix.
Keeping the above notation, we obtain the matrix $d(h^{-1})$ (resp., $dh$)
with coefficients in $\Gamma(X,\Omega_{X^{\mathrm{log}}})$ obtained by
differentiating every component of the $N\times N$ matrix $h^{-1}$ (resp.,
$h$), and (20) reads

(21) $\displaystyle\mathrm{L}_{h*}(\nabla)^{\mathfrak{g}}=h\cdot
d(h^{-1})+h\cdot\nabla^{\mathfrak{g}}\cdot h^{-1}\left(=-(dh)\cdot
h^{-1}+h\cdot\nabla^{\mathfrak{g}}\cdot h^{-1}\right)$

(cf. [Wak1, Example 1.5]).

A framing of a $G$-bundle $\mathcal{E}$ on $X$ is a $G$-equivariant
isomorphism $\theta:X\times
G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}$, which determines a
trivialization of $\mathcal{E}$. Choosing a framing $\theta$ is equivalent to
choosing a global section $h:X\rightarrow\mathcal{E}$ of the projection
$\mathcal{E}\rightarrow X$; this equivalence is given in such a way that
$\theta(x,a)=h(x)\cdot a$ and $h(x)=\theta(x,e_{G})$ (for $x\in X$, $a\in G$).
The following definition generalizes the notion of a fundamental matrix for a
linear differential equation (cf. [vdPS, Definition 1.9]).

######  Definition 2.2.

Let $\mathscr{E}:=(\mathcal{E},\nabla)$ be a flat $G$-bundle on $\mathscr{X}$.
A fundamental framing for $\mathscr{E}$ is a framing $\theta:X\times
G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}$ of $\mathcal{E}$ via
which $\nabla^{\mathrm{triv}}$ is compatible with $\nabla$. In other words, a
fundamental framing is an isomorphism of flat $G$-bundles $\theta:(X\times
G,\nabla^{\mathrm{triv}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}$.

######  Remark 2.3 (Equivalent definition of a fundamental framing).

Let $\mathscr{E}:=(\mathcal{E},\nabla)$ be a flat $G$-bundle on $\mathscr{X}$
with $\mathcal{E}=X\times G$, and let $h$ be a morphism $X\rightarrow G$. As
discussed above, the graph $X\rightarrow X\times G$ of $h$ (regarded as a
global section of the structure morphism of $\mathcal{E}$) corresponds to a
framing $\theta:X\times
G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}$. Then, $\theta$
defines a fundamental framing for $\mathscr{E}$ if and only if the morphism
$\nabla^{\mathfrak{g}}$ coincides with the composite

(22)
$\displaystyle\mathcal{T}_{X^{\mathrm{log}}}\xrightarrow{dh}h^{*}(\mathcal{T}_{G})\xrightarrow{h^{*}(\omega^{\triangleright}_{G})}h^{*}(\mathcal{O}_{G}\otimes_{\mathbb{C}}\mathfrak{g})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g},$

where the last arrow arises from the natural isomorphism
$h^{*}(\mathcal{O}_{G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{X}$.

###  2.4. Residue of a flat $G$-bundle

Let $\mathscr{E}:=(\mathcal{E},\nabla)$ be a flat $G$-bundle on $\mathscr{X}$,
and suppose that $r>0$. Let us choose $i\in\\{1,\cdots,r\\}$, and moreover
choose a local holomorphic function $t\in\mathcal{O}_{X}$ defining
$\sigma_{i}$. Then, the element

(23)
$\displaystyle\mathrm{Res}_{\sigma_{i}}(\nabla):=\overline{\nabla\left(t\frac{d}{dt}\right)}\in\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}|_{\sigma_{i}},$

where $(-)|_{\sigma_{i}}$ denotes the fiber of $(-)$ over $\sigma_{i}$, lies
in $\mathfrak{g}_{\mathcal{E}}|_{\sigma_{i}}$ and does not depend on the
choice of $t$ (cf. the discussion in [Wak1, § 1.6.3]). We shall refer to
$\mathrm{Res}_{\sigma_{i}}(\nabla)$ as the residue (or, the monodromy
operator, in the terminology of [Wak1, Definition 1.46]) of $\nabla$ at
$\sigma_{i}$. Each log connection $\nabla$ with
$\mathrm{Res}_{\sigma_{i}}(\nabla)=0$ for every $i=1,\cdots,r$ may be
identified with a non-logarithmic connection.

Next, denote by $\mathfrak{c}$ the GIT quotient of $\mathfrak{g}$ by the
adjoint $G$-action; it admits a natural projection
$\chi:\mathfrak{g}\twoheadrightarrow\mathfrak{c}$. If we choose a
$G$-equivariant trivialization
$\mathcal{E}|_{\sigma_{i}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G$
(inducing
$\mathfrak{g}_{\mathcal{E}}|_{\sigma_{i}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}$),
then $\mathrm{Res}_{\sigma_{i}}(\nabla)$ specifies an element of
$\mathfrak{g}$ via this trivialization and induces

(24)
$\displaystyle\mathrm{Rad}_{\sigma_{i}}(\nabla):=\chi(\mathrm{Res}_{\sigma_{i}}(\nabla))\in\mathfrak{c}(\mathbb{C}).$

This element does not depend on the choice of
$\mathcal{E}|_{\sigma_{i}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G$,
i.e., depends only on $\nabla$. We refer to
$\mathrm{Rad}_{\sigma_{i}}(\nabla)$ as the radius of $\nabla$ at $\sigma_{i}$
(cf. [Wak1, Definition 2.29]).

###  2.5. Local description of flat $G$-bundles

Denote by $\mathbb{C}[\\![t]\\!]$ the ring of formal power series in the
variable $t$ and with complex coefficients, and set
$Q:=\mathrm{Spec}(\mathbb{C}[\\![t]\\!])$. The divisor given by the closed
point $0\in Q$ determines a log structure on $Q$; we denote the resulting log
scheme by $Q^{\mathrm{log}}$. Note that $\mathbb{C}[\\![t]\\!]$ has a
structure of differential ring over $\mathbb{C}$ by considering the derivation
$\partial:=\frac{d}{dt}$ (resp., $\partial^{\mathrm{log}}:=t\frac{d}{dt}$) on
$\mathbb{C}[\\![t]\\!]$. The tangent bundle $\mathcal{T}_{Q}$ of $Q$ (resp.,
the logarithmic tangent bundle $\mathcal{T}_{Q^{\mathrm{log}}}$ of
$Q^{\mathrm{log}}$) over $\mathbb{C}$ is topologically generated by the global
section corresponding to $\partial$ (resp., $\partial^{\mathrm{log}}$). We
will consider the pair $\mathscr{Q}:=(Q,\\{0\\})$ as a formal neighborhood of
a marked point in a pointed Riemann surface.

The various definitions and arguments discussed so far are applicable even
when “$X$” and “$\mathscr{X}$” are replaced by $Q$ and $\mathscr{Q}$,
respectively. In particular, of course, one can define the notion of a flat
$G$-bundle on $\mathscr{Q}$, as well as its gauge transformation. Each element
$v$ of $\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ associates a flat $G$-bundle
$\mathscr{E}_{v}:=(Q\times G,\nabla_{v})$ on $\mathscr{Q}$, where $\nabla_{v}$
denotes the log connection on the trivial $G$-bundle $Q\times G$ uniquely
determined by the equality $\nabla_{v}^{\mathfrak{g}}=v$; the resulting
assignment $v\mapsto\mathscr{E}_{v}$ defines a bijection between the set
$\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ and the set of flat $G$-bundles on
$\mathscr{Q}$ whose underlying $G$-bundles are trivial.

Moreover, two elements $v_{1}(t)$, $v_{2}(t)$ of
$\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ are called equivalent if there exists an
element $h\in G(\mathbb{C}[\\![t]\\!])$ such that the element

(25)
$\displaystyle\mathrm{L}_{h*}(v_{1}(t))^{\mathfrak{g}}:=(h^{-1})^{*}(\omega_{G})(\partial^{\mathrm{log}})+\mathrm{Ad}_{G}(h)(v_{1}(t))$

of $\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ coincides with $v_{2}(t)$. (Similarly
to the comment in Remark 2.1, this condition reads the equality
$v_{2}(t)=h\cdot d(h^{-1})+h\cdot v_{1}(t)\cdot h^{-1}$ after fixing an
injective morphism of algebraic groups $G\hookrightarrow\mathrm{GL}_{N}$.)
This binary relation in $\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ specifies an
equivalence relation, and hence we obtain the quotient set
$\overline{\mathfrak{g}(\mathbb{C}[\\![t]\\!])}$ of
$\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ by this equivalence relation. Since any
$G$-bundle on $Q$ is isomorphic to the trivial one, the assignment
$v\mapsto\mathscr{E}_{v}$ induces a bijection of sets

(26)
$\displaystyle\overline{\mathfrak{g}(\mathbb{C}[\\![t]\\!])}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\left\\{\begin{matrix}\text{isomorphism
classes}\\\ \text{of flat $G$-bundles on $\mathscr{Q}$}\end{matrix}\right\\}.$

For each
$v(t):=\sum_{i=0}^{\infty}t^{i}v_{i}\in\mathfrak{g}(\mathbb{C}[\\![t]\\!])$
(where $v_{i}\in\mathfrak{g}$), the element

(27) $\displaystyle\mathrm{Res}(v(t)):=v_{0}\in\mathfrak{g}$

is called the residue of $v(t)$; it coincides with the residue of the flat
$G$-bundle $\mathscr{E}_{v}$ at the unique marked point $0\in Q$.

######  Definition 2.4.

We shall say that an element $v$ of $\mathfrak{g}$ has weakly prepared
eigenvalues if there exists an injective morphism of algebraic groups
$\iota:G\hookrightarrow\mathrm{GL}_{N}$ (where $N\in\mathbb{Z}_{>0}$) such
that the distinct eigenvalues of $d\iota(v)\in\mathfrak{g}\mathfrak{l}_{N}$,
where $d\iota:\mathfrak{g}\hookrightarrow\mathfrak{g}\mathfrak{l}_{N}$ denotes
the differential of $\iota$, do not differ by integers (i.e., $d\iota(v)$ has
weakly prepared eigenvalues, in the sense of [Ked, Definition 7.3.3]).

######  Lemma 2.5.

Let $n$ be a positive integer and let $v_{0},v_{n},v_{n+1},v_{n+2},\cdots$ be
elements of $\mathfrak{g}$. We shall set
$v(t):=v_{0}+\sum_{i=n}^{\infty}t^{i}v_{i}\in\mathfrak{g}(\mathbb{C}[\\![t]\\!])$,
and suppose that $v_{0}\left(=\mathrm{Res}(v(t))\right)$ has weakly prepared
eigenvalues. Then, there exists an element $h$ of $G(\mathbb{C}[\\![t]\\!])$
such that $h\equiv e_{G}$ (mod $t^{n}$) and
$\mathrm{L}_{h*}(v(t))^{\mathfrak{g}}\equiv v_{0}$ (mod $t^{n+1}$).

###### Proof.

Let us fix an injective morphism $\iota:G\hookrightarrow\mathrm{GL}_{N}$
(where $N\in\mathbb{Z}_{>0}$) such that the distinct eigenvalues of
$d\iota(v_{0})$ do not differ by integers. Then, since $d\iota(v_{0})$ and
$nE_{N}+d\iota(v_{0})$ (where $E_{N}$ denotes the identity matrix in
$\mathfrak{g}\mathfrak{l}_{N}$) have no eigenvalues in common, the
$\mathbb{C}$-linear endomorphism $\alpha^{\prime}$ of
$\mathfrak{g}\mathfrak{l}_{N}$ given by $w\mapsto nw+[d\iota(v_{0}),w]$ is
injective (cf. the proof of [vdPS, Proposition 3.12]). Hence, the endomorphism
$\alpha$ of $\mathfrak{g}$ given by $w\mapsto nw+[v_{0},w]$ (i.e., obtained by
restricting $\alpha^{\prime}$) is verified to be bijective. This implies that
there exists an element $u\in\mathfrak{g}$ with
$\alpha(u)\left(=nu+[v_{0},u]\right)=-v_{n}$. Next, let us take an element $h$
of $G(\mathbb{C}[\\![t]\\!])$ with $h\equiv e_{G}+t^{n}u$ (mod $t^{n+1}$).
Then, we have

(28) $\displaystyle\ \ \
\,\mathrm{L}_{h*}\left(v_{0}+\sum_{i=n}^{\infty}t^{i}v_{i}\right)^{\mathfrak{g}}$
$\displaystyle=(h^{-1})^{*}(\omega_{G})+\mathrm{Ad}_{G}(h)\left(v_{0}+\sum_{i=n}^{\infty}t^{i}v_{i}\right)$
$\displaystyle\equiv(e_{G}+t^{n}u)\partial^{\mathrm{log}}(e_{G}-t^{n}u)+(e_{G}+t^{n}u)(v_{0}+t^{n}v_{n})(e_{G}-t^{n}u)\
(\text{mod}\ t^{n+1})$ $\displaystyle\equiv v_{0}-t^{n}(nu+[v_{0},u]+v_{n})\
(\text{mod}\ t^{n+1})$ $\displaystyle\equiv v_{0}\ (\text{mod}\ t^{n+1})\
(\text{mod}\ t^{n+1}).$

It follows that $h$ satisfies the required properties. This completes the
proof of this lemma. ∎

By applying successively the above lemma, we obtain the following assertion.

######  Proposition 2.6.

Let $v(t):=\sum_{i=0}^{\infty}t^{i}v_{i}$ (where $v_{i}\in\mathfrak{g}$) be an
element of $\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ such that
$v_{0}\left(=\mathrm{Res}(v(t))\right)$ has weakly prepared eigenvalues. Then,
the following assertions hold:

  * (i)

The element $v(t)$ is equivalent to $v_{0}$, i.e., there exists an element $h$
of $G(\mathbb{C}[\\![t]\\!])$ with $v_{0}=\mathrm{L}_{h*}(v(t))$. In
particular, any element $v(t)$ of $t\mathfrak{g}(\mathbb{C}[\\![t]\\!])$
(i.e., any element $v(t)$ of $\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ with
$\mathrm{Res}(v(t))=0$) is equivalent to $0$.

  * (ii)

If, moreover, $v(t)$ lies in the subring $\mathbb{C}\\{t\\}$ of
$\mathbb{C}[\\![t]\\!]$ consisting of all convergent power series, then the
element $h$ asserted in (i) can be taken as an element of
$G(\mathbb{C}\\{t\\})$.

###### Proof.

First, let us consider assertion (i). One can choose a collection of pairs
$\\{(h_{[i]},v_{[i]}(t))\\}_{i\in\mathbb{Z}_{\geq 0}}$, where $h_{[i]}\in
G(\mathbb{C}[\\![t]\\!])$ and
$v_{[i]}(t)\in\mathfrak{g}(\mathbb{C}[\\![t]\\!])$ ($i=0,1,2,\cdots$),
constructed inductively by the following rules:

  * •

$(h_{[0]},v_{[0]}(t)):=(e_{G},v(t))$;

  * •

For each $i=0,1,2,\cdots$, we set $h_{[i+1]}$ to be an element “$h$” obtained
by applying Lemma 2.5 to the case where “$n$” and “$v(t)$” are taken to be
$i+1$ and $v_{[i]}(t)$, respectively, and we set
$v_{[i+1]}(t):=\mathrm{L}_{h_{[i+1]}*}(v_{[i]}(t))^{\mathfrak{g}}$.

Then, the product $h:=\prod_{i\in\mathbb{Z}_{\geq 0}}h_{i}\left(=\cdots
h_{2}h_{1}h_{0}\right)$ converges to an element of $G(\mathbb{C}[\\![t]\\!])$
and satisfies the required property. This completes the proof of assertion
(i).

Next, we shall consider assertion (ii). Since $v_{0}$ has weakly prepared
eigenvalues, there exists an injective morphism
$\iota:G\hookrightarrow\mathrm{GL}_{N}$ (where $N\in\mathbb{Z}_{>0}$) such
that the distinct eigenvalues of $d\iota(v_{0})$ do not differ by integers.
Using this injection, we can reduce the problem to the case of
$G=\mathrm{GL}_{N}$. Then, the assertion follows from an argument entirely
similar to the proof of [vdPS, Lemma 3.42]. ∎

The following assertion, being a corollary of the above proposition, may be a
classical fact at least in the case of a matrix group $G$; see, e.g., [Lab,
Proposition 3.3.20]. (However, the author could not find any references that
were consistent with our setting and contained its proof without omission.)

######  Corollary 2.7.

Let $X$ be a Riemann surface and $\mathscr{E}$ a flat $G$-bundle on $X$. Then,
each point $x\in X$ has an open neighborhood $U$ in $X$ such that the
restriction $\mathscr{E}|_{U}$ of $\mathscr{E}$ to $U$ admits a fundamental
framing.

###### Proof.

The assertion follows from Proposition 2.6, (the latter assertion of) (i) and
(ii), together with the comment at the end of the first paragraph in § 2.4. ∎

##  3\. $G$-representations of the fundamental group

This section discusses the monodromy maps of flat $G$-bundles, which specify
$G$-representations of the fundamental group $\Gamma$ of the underlying
Riemann surface. The (most basic form of the) Riemann-Hilbert correspondence
for flat $G$-bundles will be described in Theorems 3.1 and 3.7. Also, we
recall the parabolic cohomology of a local system and the relationship with
the parabolic group cohomology of the corresponding
$\mathbb{C}[\Gamma]$-module (cf. Proposition 3.6).

###  3.1. $G$-representations

Let $X$ be a Riemann surface, and fix a base point $x_{0}$ in $X$. Also, let
$\widetilde{X}$ be the space of homotopy classes $[\gamma]$ of paths
$\gamma:[0,1]\rightarrow X$ in $X$ starting at $x_{0}$ (i.e.,
$\gamma(0)=x_{0}$). In a usual manner, we equip $\widetilde{X}$ with a
topology by which $\widetilde{X}$ forms the universal covering of $X$. We
shall denote by $\varpi:\widetilde{X}\twoheadrightarrow X$ the natural
projection. There exists a structure of Riemann surface on $\widetilde{X}$ so
that $\varpi$ is locally biholomorphic.

Denote by $\pi_{1}(X,x_{0})$ the fundamental group of $X$ at $x_{0}$. The
Riemann surface $\widetilde{X}$ admits a natural (properly discontinuous)
right $\pi_{1}(X,x_{0})$-action as deck transformations $[\delta]\mapsto
a_{[\delta]}\in\mathrm{Aut}(\widetilde{X})$ (for
$[\delta]\in\pi_{1}(X,x_{0})$). Here, $a_{[\delta]}$ is given by
$a_{[\delta]}([\gamma]):=[\gamma\delta]$ for any $[\gamma]\in\widetilde{X}$,
where $\gamma\delta$ denotes the product of the paths $\gamma$ and $\delta$,
or more precisely, the path $\gamma\delta$ first follows the loop $\delta$
with “twice the speed” and then follows $\gamma$ with “twice the speed”. The
projection $\varpi$ induces a biholomorphic map
$\widetilde{X}/\pi_{1}(X,x_{0})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X$
from the resulting quotient $\widetilde{X}/\pi_{1}(X,x_{0})$ to $X$.

We shall denote by

(29) $\displaystyle\mathcal{R}ep(\pi_{1}(X,x_{0}),G(\mathbb{C}))$

the category defined in such a way that the objects are homomorphisms from
$\pi_{1}(X,x_{0})$ to $G(\mathbb{C})$, and the morphisms from
$\mu_{1}:\pi_{1}(X,x_{0})\rightarrow G(\mathbb{C})$ to
$\mu_{2}:\pi_{1}(X,x_{0})\rightarrow G(\mathbb{C})$ are inner automorphisms
$\zeta$ of $G(\mathbb{C})$ with $\zeta\circ\mu_{1}=\mu_{2}$. The set

(30) $\displaystyle\mathrm{Rep}(\pi_{1}(X,x_{0}),G(\mathbb{C}))$

of isomorphism classes of objects in this category coincides with the set of
outer homomorphisms from $\pi_{1}(X,x_{0})$ to $G(\mathbb{C})$. To be precise,
it consists of equivalence classes $[\mu]$ of homomorphisms
$\mu:\pi_{1}(X,x_{0})\rightarrow G(\mathbb{C})$, where two such homomorphisms
are equivalent if they coincide with each other up to post-composition with an
inner automorphism of $G(\mathbb{C})$.

In what follows, we recall the construction of an equivalence between
$\mathcal{R}ep(\pi_{1}(X,x_{0}),G(\mathbb{C}))$ and
$\mathcal{C}onn(X,G)\left(:=\mathcal{C}onn((X,\emptyset),G)\right)$.

###  3.2. From $G$-representations to flat $G$-bundles

Let $\mu:\pi_{1}(X,x_{0})\rightarrow G(\mathbb{C})$ be a group homomorphism,
i.e., an object in $\mathcal{R}ep(\pi_{1}(X,x_{0}),G(\mathbb{C}))$. The
product $\widetilde{X}\times G$ admits a right $\pi_{1}(X,x_{0})$-action
$[\delta]\mapsto\widetilde{a}_{\mu,[\delta]}\in\mathrm{Aut}(\widetilde{X}\times
G)$ (for $[\delta]\in\pi_{1}(X,x_{0})$) given by
$([\gamma],h)\mapsto\widetilde{a}_{\mu,[\delta]}([\gamma],h):=([\gamma\circ\delta],\mu([\delta])^{-1}\cdot
h)$ for $[\gamma]\in\widetilde{X}$ and $h\in G$. The action
$\widetilde{a}_{\mu,(-)}$ is compatible with the $\pi_{1}(X,x_{0})$-action
$a_{(-)}$ on $X$ via $\varpi$, and hence, the trivial $G$-bundle
$\widetilde{X}\times G$ descends to a $G$-bundle $\mathcal{E}_{\mu}$ on $X$.
The trivial connection $\nabla^{\mathrm{triv}}$ on $\widetilde{X}\times G$ is
compatible with these actions; this means that, for any
$[\delta]\in\pi_{1}(X,x_{0})$, the following square diagram is commutative:

(35)

Hence, $\nabla^{\mathrm{triv}}$ descends to a connection $\nabla_{\mu}$ on
$\mathcal{E}_{\mu}$, and we have obtained a flat $G$-bundle

(36) $\displaystyle\mathscr{E}_{\mu}:=(\mathcal{E}_{\mu},\nabla_{\mu})$

on $X$.

Moreover, let $\mu_{i}:\pi_{1}(X,x_{0})\rightarrow G(\mathbb{C})$ ($i=1,2$) be
objects in $\mathcal{R}ep(\pi_{1}(X,x_{0}),G(\mathbb{C}))$ and
$h:G(\mathbb{C})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G(\mathbb{C})$ a
morphism from $\mu_{1}$ to $\mu_{2}$. The $\pi_{1}(X,x_{0})$-action
$\widetilde{a}_{\mu_{1},(-)}$ is compatible with $\widetilde{a}_{\mu_{2},(-)}$
via the automorphism $\mathrm{L}_{h}:(\widetilde{X}\times
G,\nabla^{\mathrm{triv}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\widetilde{X}\times
G,\nabla^{\mathrm{triv}})$. It follows that this automorphism descends to an
isomorphism
$\overline{\mathrm{L}}_{h}:\mathscr{E}_{\mu_{1}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}_{\mu_{2}}$.

The assignments $\mu\mapsto\mathscr{E}_{\mu}$ and
$h\mapsto\overline{\mathrm{L}}_{h}$ together determine a functor

(37)
$\displaystyle\mathcal{R}ep(\pi_{1}(X,x_{0}),G(\mathbb{C}))\rightarrow\mathcal{C}onn(X,G).$

###  3.3. From flat $G$-bundles to $G$-representations

Next, we consider the inverse direction. Let
$\mathscr{E}:=(\mathcal{E},\nabla)$ be a flat $G$-bundle on $X$, and fix an
identification
$G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}|_{x_{0}}$ preserving
the $G$-action.

Let us choose two points $p$, $q$ in $X$ and choose a path $\gamma$ from $p$
to $q$ lying in $X$. It follows from Corollary 2.7 that, for each point
$\gamma(t)$ on this path, there exist an open set $U_{\gamma(t)}\subseteq X$
containing $\gamma(t)$ and a fundamental framing
$\tau_{t}:(U_{\gamma(t)}\times
G,\nabla^{\mathrm{triv}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}|_{U_{\gamma(t)}}$
for $\mathscr{E}$ restricted to $U_{\gamma(t)}$. By compactness of $[0,1]$, we
can cover the image $\mathrm{Im}(\gamma)\left(\subseteq X\right)$ of $\gamma$
with a finite number of these open sets
$U_{\gamma(t_{0})},U_{\gamma(t_{1})},\cdots,U_{\gamma(t_{m})}$ with
$t_{0}=0<t_{1}<t_{1}<\cdots<t_{m}=1$. In particular, the set of fundamental
framings $\\{\tau_{t_{i}}\\}_{i=0}^{n}$ can be used to define the analytic
continuation along $\gamma$; the resulting $G$-equivariant automorphism
$G\left(\stackrel{{\scriptstyle\tau_{0}|_{p}}}{{\cong}}\mathcal{E}|_{p}\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G\left(\stackrel{{\scriptstyle\tau_{1}|_{q}}}{{\cong}}\mathcal{E}|_{q}\right)$
of $G$ arises from the left-translation $\mathrm{L}_{h}$ by some element $h\in
G(\mathbb{C})$.

Consider the case where $\gamma(0)=\gamma(1)=x_{0}$ and $\tau_{0}$ is taken to
satisfy the condition that its fiber over $x_{0}$ coincides with the fixed
identification
$G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}|_{x_{0}}$. Then, the
element “$h$” constructed above is well-defined, and we denote it by
$\mathrm{Mod}_{\mathscr{E}}([\gamma])$. The assignment

(38) $\displaystyle\mathrm{Mon}_{\mathscr{E}}:\pi_{1}(X,x_{0})\rightarrow
G(\mathbb{C})$

given by $[\gamma]\mapsto\mathrm{Mon}_{\mathscr{E}}([\gamma])$ forms a group
homomorphism, and it is called the monodromy map of $\mathscr{E}$.

Moreover, let $\mathscr{E}_{i}:=(\mathcal{E}_{i},\nabla_{i})$ ($i=1,2$) be
flat $G$-bundles on $X$ (equipped with a choice of an identification
$w_{i}:G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}_{i}|_{x_{0}}$)
and
$\alpha:\mathscr{E}_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}_{2}$
an isomorphism between them. Then, the fiber of $\alpha$ over $x_{0}$
determines, via the choices of identifications $w_{i}$, the left-translation
by some element $h_{\alpha}\in G(\mathbb{C})$. The assignments
$\mathscr{E}\mapsto\mathrm{Mon}_{\mathscr{E}}$ and $\alpha\mapsto h_{\alpha}$
together give a functor

(39)
$\displaystyle\mathcal{C}onn(X,G)\rightarrow\mathcal{R}ep(\pi_{1}(X,x_{0}),G(\mathbb{C})).$

The following assertion, generalizing the most basic form of the Riemann-
Hilbert correspondence for linear differential equations, follows immediately
from the constructions of (37) and (39).

######  Theorem 3.1.

The functors (37) and (39) are inverse to each other up to natural
isomorphism. In particular, there is an equivalence of categories

(40)
$\displaystyle\mathcal{C}onn(X,G)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{R}ep(\pi_{1}(X,x_{0}),G(\mathbb{C})).$

###  3.4. Local monodromy

We shall fix $a\in\mathbb{R}_{>0}$, and write
$Q_{a}:=\left\\{z\in\mathbb{C}\,|\,|z|<a\right\\}$. In particular, by putting
the origin $0$ as a marked point of $Q_{a}$, we obtain a pointed Riemann
surface $\mathscr{Q}_{a}:=(Q_{a},\\{0\\})$. Also, write
$Q_{a}^{*}:=Q_{a}\setminus\\{0\\}\left(=\left\\{z\in\mathbb{C}\,|\,0<|z|<a\right\\}\right)$.
The fundamental group of $Q_{a}^{*}$ is generated by the circle around $0$,
say through $a_{0}\in\mathbb{R}$ with $0<a_{0}<a$ and in the positive
direction. Let us write $\gamma_{0}$ for this generator. There are no
relations and thus the fundamental group $\pi_{1}(Q_{a}^{*},a_{0})$ is
isomorphic to the group $\mathbb{Z}$. For each flat $G$-bundle $\mathscr{E}$
on $Q_{a}^{*}$, the element $\mathrm{Mod}_{\mathscr{E}}([\gamma_{0}])\in
G(\mathbb{C})$ is called the local monodromy of $\mathscr{E}$.

######  Theorem 3.2.

Let $\mathscr{E}:=(\mathcal{E},\nabla)$ be a flat $G$-bundle on
$\mathscr{Q}_{a}$ such that $C:=\mathrm{Res}_{0}(\nabla)$ has weakly prepared
eigenvalues after choosing an identification
$\mathfrak{g}_{\mathcal{E}}|_{0}=\mathfrak{g}$. Then, the local monodromy of
the flat $G$-bundle $\mathscr{E}|_{Q_{a}^{*}}$ on $Q_{a}^{*}$ is conjugate to
$\mathrm{exp}(-2\pi\sqrt{-1}\cdot C)\in G(\mathbb{C})$, where $\mathrm{exp}$
denotes the exponential map $\mathfrak{g}\rightarrow G$.

###### Proof.

By Proposition 2.6, (i) and (ii), we may assume, after possibly replacing $a$
with a smaller positive value, that $\mathscr{E}$ is of the form $(Q_{a}\times
G,\nabla_{C})$, where $\nabla_{C}$ denotes the log connection on the trivial
$G$-bundle $Q_{a}\times G$ uniquely determined by
$\nabla_{C}^{\mathfrak{g}}=C$. Then, the (locally defined) $G$-valued function
$\mathrm{exp}(-\log z\cdot C)$ specifies a fundamental framing for
$\mathscr{E}$ (cf. Remark 2.3). Hence, the assertion follows from the fact
that analytic continuation around the generator of $\pi_{1}(Q_{a}^{*},a_{0})$
maps $\log z$ to $\log z+2\pi\sqrt{-1}$. ∎

######  Corollary 3.3.

Let us keep the notation in the above theorem. Then, the residue
$\mathrm{Res}_{0}(\nabla)$ at $0$ is nilpotent if and only if the local
monodromy of $\mathscr{E}|_{Q_{a}^{*}}$ is unipotent.

###  3.5. Group cohomology

Let us fix a pair of nonnegative integers $(g,r)$ with $2g-2+r>0$ and an
$r$-pointed compact Riemann surface
$\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i=1}^{r})$ of genus $g$. We shall write
$X^{*}:=X\setminus\bigcup_{i=1}^{r}\\{\sigma_{i}\\}_{i}$ and write $\Gamma$
for the fundamental group of $X^{*}$ at a fixed base point $x_{0}$, i.e.,
$\Gamma:=\pi_{1}(X^{*},x_{0})$. By the Fuchsian uniformization, $\Gamma$ can
be realized as a discrete subgroup of
$\mathrm{PSL}_{2}(\mathbb{R})\left(\subseteq\mathrm{PGL}_{2}(\mathbb{C})\right)$
that is finitely generated and admits an explicit presentation in terms of
$2g$ hyperbolic generators $A_{1},\cdots,A_{g},B_{1},\cdots,B_{g}$ and $r$
parabolic generators $T_{1},\cdots,T_{r}$ (where each $T_{i}$ is represented
by a loop around $\sigma_{i}$) with relation
$\prod_{i=1}^{g}[A_{i},B_{i}]\cdot\prod_{i=1}^{r}T_{i}=e_{\mathrm{PSL}_{2}(\mathbb{R})}$
(cf. [Hej], [Leh]).

Let $K$ be either $\mathbb{R}$ or $\mathbb{C}$. Recall that, for a group
$\Gamma_{0}$ and a $K[\Gamma_{0}]$-module $\mathfrak{a}$, a $1$-cocycle
(resp., a $1$-coboundary) is defined to be a map
$z:\Gamma_{0}\rightarrow\mathfrak{a}$ satisfying
$z(\alpha\beta)=z(\alpha)+\alpha z(\beta)$ for any $\alpha,\beta\in\Gamma_{0}$
(resp., $z(\alpha)=(\alpha-1)x_{z}$ for any $\alpha\in\Gamma_{0}$ with an
element $x_{z}\in\mathfrak{a}$ independent of $\alpha$). Write
$Z^{1}(\Gamma_{0},\mathfrak{a})$ (resp., $B^{1}(\Gamma_{0},\mathfrak{a})$) for
the $K$-vector space consisting of $1$-cocycles (resp., $1$-coboundaries), and
write
$H^{1}(\Gamma_{0},\mathfrak{a}):=Z^{1}(\Gamma_{0},\mathfrak{a})/B^{1}(\Gamma_{0},\mathfrak{a})$,
i.e., the $1$-st cohomology group of the $K[\Gamma_{0}]$-module
$\mathfrak{a}$.

In what follows, let us consider the case where $\Gamma_{0}=\Gamma$. Then, one
can define a parabolic $1$-cocycle (cf. [Shi2, § 8.1]) to be a $1$-cocycle
$z:\Gamma\rightarrow\mathfrak{a}$ satisfying
$z(T_{i})\in(T_{i}-1)\mathfrak{g}$ for every $T_{i}$ ($i=1,\cdots,r$). The
$K$-vector subspace of $Z^{1}(\Gamma,\mathfrak{a})$ consisting of parabolic
$1$-cocycles will be denoted by $Z_{P}^{1}(\Gamma,\mathfrak{a})$. Since
$B^{1}(\Gamma,\mathfrak{a})$ is a subspace of
$Z^{1}_{P}(\Gamma,\mathfrak{a})$, we obtain the quotient space

(41) $\displaystyle
H^{1}_{P}(\Gamma,\mathfrak{a}):=Z^{1}_{P}(\Gamma,\mathfrak{a})/B^{1}(\Gamma,\mathfrak{a}),$

which is called the $1$-st parabolic cohomology group of $\mathfrak{a}$. In
particular, $H^{1}_{P}(\Gamma,\mathfrak{a})$ may be thought of as a subspace
of $H^{1}(\Gamma,\mathfrak{a})$.

For each $i=1,\cdots,r$, let us choose a sufficiently small disc $U_{i}$ in
$X$ with center $\sigma_{i}$ (such that $U_{1},\cdots,U_{r}$ are pairwise
disjoint). If $\Gamma_{i}$ denotes the fundamental group of
$U_{i}^{*}:=U_{i}\setminus\\{\sigma_{i}\\}$ at a fixed base point $x_{i}\in
U_{i}^{*}$, then its generator can be taken as a circle $T^{\prime}_{i}$
around $\sigma_{i}$ in the positive direction. One can choose a suitable path
$\delta_{i}$ from $x_{0}$ to $x_{i}$ such that the induced homomorphism
$\Gamma_{i}\rightarrow\Gamma;[\gamma]\mapsto[\delta_{i}^{-1}(\nu_{i}\circ\gamma)\delta_{i}]$
maps the generator $T^{\prime}_{i}$ to $T_{i}$, where $\nu_{i}$ denotes the
inclusion $U_{i}^{*}\hookrightarrow X^{*}$. By this homomorphism,
$\mathfrak{a}$ is equipped with a $K[\Gamma_{i}]$-module structure, and it
gives rise to a morphism

(42) $\displaystyle\kappa_{i}:H^{1}(\Gamma,\mathfrak{a})\rightarrow
H^{1}(\Gamma_{i},\mathfrak{a}).$

######  Proposition 3.4.

Let us keep the above notation. Then, the following sequence is exact:

(43) $\displaystyle 0\longrightarrow
H_{P}^{1}(\Gamma,\mathfrak{a})\xrightarrow{\mathrm{inclusion}}H^{1}(\Gamma,\mathfrak{a})\xrightarrow{\bigoplus_{i=1}^{r}\kappa_{i}}\bigoplus_{i=1}^{r}H^{1}(\Gamma_{i},\mathfrak{a}).$

###### Proof.

Since $\Gamma_{i}\cong\mathbb{Z}$, a well-known fact of group cohomology says
that $H^{1}(\Gamma_{i},\mathfrak{a})$ is isomorphic to
$\mathrm{Coker}(\mathfrak{a}\xrightarrow{a\mapsto(T^{\prime}_{i}-1)a}\mathfrak{a})$.
Hence, the assertion follows from the definition of
$H^{1}_{P}(\Gamma,\mathfrak{a})$. ∎

###  3.6. Comparison with sheaf cohomology

Denote by $\mathbb{L}$ the $K$-local system on $X^{*}$ corresponding to the
$K[\Gamma]$-module $\mathfrak{a}$. That is to say, $\mathbb{L}$ is the
quotient of the constant $\mathfrak{a}$-valued local system on the universal
covering by the natural diagonal $\Gamma$-action using the $\Gamma$-action on
$\mathfrak{a}$. For $l\in\mathbb{Z}_{\geq 0}$, the $l$-th parabolic cohomology
of $\mathbb{L}$ is defined as the sheaf cohomology

(44) $\displaystyle H_{P}^{l}(X^{*},\mathbb{L}):=H^{l}(X,\nu_{*}(\mathbb{L}))$

of $\nu_{*}(\mathbb{L})$, where $\nu$ denotes the natural inclusion
$X^{*}\hookrightarrow X$.

######  Proposition 3.5.

  * (i)

There exists a canonical exact sequence

(45) $\displaystyle 0\longrightarrow
H_{P}^{1}(X^{*},\mathbb{L})\longrightarrow
H^{1}(X^{*},\mathbb{L})\xrightarrow{\bigoplus_{i=1}^{r}\nu_{i}^{*}}\bigoplus_{i=1}^{r}H^{1}(U_{i}^{*},\nu^{*}_{i}(\mathbb{L})).$

In particular, we can regard $H_{P}^{1}(X^{*},\mathbb{L})$ as a $K$-vector
subspace of $H^{1}(X^{*},\mathbb{L})$ via the second arrow in this sequence.

  * (ii)

Suppose further that the natural morphism
$H_{c}^{2}(X^{*},\mathbb{L})\rightarrow H^{2}(X^{*},\mathbb{L})$, where
$H_{c}^{*}$ denotes the cohomology with compact support, is injective. Then,
the last arrow in (45) becomes surjective.

###### Proof.

Let us consider the long exact sequence

(46) $\displaystyle\cdots\rightarrow
H^{l}_{D}(X,\nu_{!}(\mathbb{L}))\rightarrow
H_{c}^{l}(X^{*},\mathbb{L})\rightarrow H^{l}(X^{*},\mathbb{L})\rightarrow
H^{l+1}_{D}(X,\nu_{!}(\mathbb{L}))\rightarrow\cdots$

obtained by applying [Ive, Chap. II, Proposition 9.2] to $\nu_{!}(\mathbb{L})$
under the identifications
$H^{l}(X^{*},\mathbb{L})=H^{l}(X^{*},\nu^{*}(\nu_{!}(\mathbb{L})))$ and
$H_{c}^{l}(X^{*},\mathbb{L})=H^{l}(X,\nu_{!}(\mathbb{L}))$ (cf. [Ive, Chap.
III, Corollary 7.3]). Since $H_{P}^{1}(X^{*},\mathbb{L})$ coincides with the
image of the natural morphism $H^{1}_{c}(X^{*},\mathbb{L})\rightarrow
H^{1}(X^{*},\mathbb{L})$ (cf. [Loo, Lemma 5.3]), this sequence gives an exact
sequence

(47) $\displaystyle 0$ $\displaystyle\longrightarrow
H_{P}^{1}(X^{*},\mathbb{L})\longrightarrow H^{1}(X^{*},\mathbb{L})\rightarrow
H^{2}_{D}(X,\nu_{!}(\mathbb{L}))\left(=\bigoplus_{i=1}^{r}H^{2}_{\\{\sigma_{i}\\}}(X,\nu_{!}(\mathbb{L}))\right)$
$\displaystyle\longrightarrow H^{2}_{c}(X^{*},\mathbb{L})\longrightarrow
H^{2}(X^{*},\mathbb{L})\longrightarrow\cdots.$

On the other hand, by an argument similar to the proof of [DeWe, Lemma 1.2],
$H^{2}_{\\{\sigma_{i}\\}}(X,\nu_{!}(\mathbb{L}))$ may be identified with
$H^{1}(U_{i}^{*},\mathbb{L}|_{U_{i}^{*}})$, and the morphism
$H^{1}(X^{*},\mathbb{L})\rightarrow
H^{2}_{\\{\sigma_{i}\\}}(X,\nu_{!}(\mathbb{L}))$ constituting the third arrow
in (47) coincides with the pull-back
$\nu_{i}^{*}:H^{1}(X^{*},\mathbb{L})\rightarrow
H^{1}(U^{*}_{i},\nu_{i}^{*}(\mathbb{L}))$ under this identification. This
completes the proofs of assertions (i) and (ii). ∎

Next, we recall a natural isomorphism

(48) $\displaystyle
H^{1}(\Gamma,\mathfrak{a})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(X^{*},\mathbb{L})$

induced by the Cartan-Leray spectral sequence together with the fact that the
universal covering of $X^{*}$ is contractible (cf. [McC, Theorem
$8^{\mathrm{bis}}$.9]).

######  Proposition 3.6.

The isomorphism (48) restricts to an isomorphism of $K$-vector spaces

(49) $\displaystyle
H^{1}_{P}(\Gamma,\mathfrak{a})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{P}(X^{*},\mathbb{L}).$

###### Proof.

Note that (48) and the isomorphisms
$H^{1}(\Gamma_{i},\mathfrak{a})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(U_{i}^{*},\mathbb{L}|_{U_{i}^{*}})$
($i=1,\cdots,r$) obtained by the same manner make the following diagram
commute:

(54)

where the upper and lower horizontal sequences are (43) and (45),
respectively. Hence, the desired isomorphism can be obtained by restricting
(48) to the kernels of the rightmost arrows in the upper and lower sequences
in this diagram. ∎

###  3.7. Representation variety

In the rest of the present paper, let us fix a split simple algebraic group
$(G_{\mathbb{R}},T_{\mathbb{R}})$ over $\mathbb{R}$ of adjoint type, where
$T_{\mathbb{R}}$ is a maximal torus of $G_{\mathbb{R}}$. Also, fix a Borel
subgroup $B_{\mathbb{R}}$ of $G_{\mathbb{R}}$ defined over $\mathbb{R}$
containing $T_{\mathbb{R}}$. By base-changing by means of
$\mathbb{R}\hookrightarrow\mathbb{C}$, we obtain algebraic $\mathbb{C}$-groups

(55) $\displaystyle G:=G_{\mathbb{R}}\times^{\mathbb{R}}\mathbb{C},\ \ \
T:=T_{\mathbb{R}}\times^{\mathbb{R}}\mathbb{C},\ \ \ \text{and}\ \ \
B:=B_{\mathbb{R}}\times^{\mathbb{R}}\mathbb{C}.$

Write $\mathfrak{g}_{\mathbb{R}}$ (resp., $\mathfrak{g}$),
$\mathfrak{b}_{\mathbb{R}}$ (resp., $\mathfrak{b}$), and
$\mathfrak{t}_{\mathbb{R}}$ (resp., $\mathfrak{t}$) for the Lie algebras over
$\mathbb{R}$ (resp., $\mathbb{C}$) of $G_{\mathbb{R}}$ (resp., $G$),
$B_{\mathbb{R}}$ (rep., $B$), and $T_{\mathbb{R}}$ (resp., $T$), respectively.
Note that the elements of $G(\mathbb{C})$ invariant under the complex
conjugation form a subgroup, which we denote by $G(\mathbb{R})$.

Denote by $\Pi$ the set of simple roots in $B_{\mathbb{R}}$ with respect to
$T_{\mathbb{R}}$. For each character $\beta$ of $T_{\mathbb{R}}$, we set

(56)
$\displaystyle\mathfrak{g}^{\beta}_{\mathbb{R}}:=\left\\{x\in\mathfrak{g}_{\mathbb{R}}\,|\,\mathrm{Ad}_{G_{\mathbb{R}}}(t)(x)=\beta(t)\cdot
x\ \text{for all}\ t\in T_{\mathbb{R}}\right\\}.$

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

Let us fix a collection
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt:=\\{x_{\alpha}\\}_{\alpha\in\Pi}$ consisting of generators
$x_{\alpha}\in\mathfrak{g}^{\alpha}_{\mathbb{R}}$, and set
$q_{1}:=\sum_{\alpha\in\Pi}x_{\alpha}$. (If we take two such collections, then
they are conjugate to each other by an element of $T_{\mathbb{R}}$. For that
reason, the results obtained in the subsequent discussion are essentially
independent of the choice of
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$.) By
regarding the fundamental coweight $\check{\omega}_{\alpha}$ of $\alpha$ as an
element of $\mathfrak{t}$ via differentiation, we obtain
$\check{\rho}:=\sum_{\alpha\in\Pi}\check{\omega}_{\alpha}\in\mathfrak{t}_{\mathbb{R}}$.
Then, there is a unique collection $\\{y_{\alpha}\\}_{\alpha\in\Pi}$ of
generators $y_{\alpha}\in\mathfrak{g}^{-\alpha}_{\mathbb{R}}$ such that the
set $\\{q_{-1},2\check{\rho},q_{1}\\}$, where
$q_{-1}:=\sum_{\alpha\in\Pi}y_{\alpha}\in\mathfrak{g}_{\mathbb{R},-1}$, forms
an $\mathfrak{s}\mathfrak{l}_{2}$-triple. If $(\mathrm{PSL}_{2})_{\mathbb{R}}$
denotes the projective special linear group of degree two over $\mathbb{R}$
with Lie algebra $(\mathfrak{s}\mathfrak{l}_{2})_{\mathbb{R}}$, then the
resulting injection
$\iota_{\mathfrak{g}}:(\mathfrak{s}\mathfrak{l}_{2})_{\mathbb{R}}\hookrightarrow\mathfrak{g}_{\mathbb{R}}$
can be integrated to a morphism of algebraic groups

(57) $\displaystyle\iota_{G}:(\mathrm{PSL}_{2})_{\mathbb{R}}\rightarrow
G_{\mathbb{R}}$

over $\mathbb{R}$. We will use the same notation “$\iota_{G}$” to denote its
base-change
$\left((\mathrm{PSL}_{2})_{\mathbb{R}}\times_{\mathbb{R}}\mathbb{C}=\right)\mathrm{PSL}_{n}\rightarrow
G$ over $\mathbb{C}$.

Now, let $K$ be either $\mathbb{R}$ or $\mathbb{C}$. Given a representation
$\mu:\Gamma\rightarrow G(K)$, we shall write $\mu_{\mathbb{C}}$ for the
composite representation
$\mu_{\mathbb{C}}:\Gamma\xrightarrow{\mu}G(K)\hookrightarrow G(\mathbb{C})$
(hence $\mu=\mu_{\mathbb{C}}$ if $K=\mathbb{C}$). Denote by
$\mathrm{Hom}_{P}(\Gamma,G(K))$ the algebraic variety over $K$ parametrizing
all $G(K)$-representations $\mu:\Gamma\rightarrow G(K)$ of $\Gamma$ such that,
for each $i=1,\cdots,r$, the element $\mu(T_{i})$ is conjugate to
$\iota_{G}(T_{i})$ (which is also conjugate to
$\mathrm{exp}(-2\pi\sqrt{-1}\cdot q_{-1})$; see [Fal]). It has a natural
$G(K)$-action by conjugation and contains the Zariski open subset
$\mathrm{Hom}_{P}^{0}(\Gamma,G(K))$ consisting of representations $\mu$ such
that $\mu_{\mathbb{C}}$ are irreducible and the images
$\mathrm{Im}(\mu_{\mathbb{C}})$ have trivial centralizer. The representations
in $\mathrm{Hom}_{P}^{0}(\Gamma,G(K))$ define a full subcategory

(58) $\displaystyle\mathcal{R}ep_{P}^{0}(\Gamma,G(K))$

of $\mathcal{R}ep(\Gamma,G(K))$.

Moreover, we know (cf. [Men, § 3], [Gol]) that the quotient set

(59)
$\displaystyle\mathrm{Rep}_{P}^{0}(\Gamma,G(K)):=\mathrm{Hom}_{P}^{0}(\Gamma,G(K))/G(K)\left(\subseteq\mathrm{Rep}(\Gamma,G(K))\right)$

forms a real (resp., a complex) manifold if $K=\mathbb{R}$ (resp.,
$K=\mathbb{C}$). Note that $\iota_{G}(T_{i})$ (for each $i$) is a regular
element of $G(K)$, so the conjugacy class of $\iota_{G}(T_{i})$ has dimension
$\mathrm{dim}(G)-\mathrm{rk}(G)$. Hence, the $K$-dimension of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(K))$ coincides with the value
$2(g-1)\cdot\mathrm{dim}(G)+r\cdot(\mathrm{dim}(G)-\mathrm{rk}(G))=2\cdot
a(G)$ (cf. (3)); see [Wei] for a computation of the dimension. Since the image
of the representation $\mu_{\mathbb{C}}$ for each
$\mu\in\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))$ has trivial centralizer,
the inclusion $G(\mathbb{R})\hookrightarrow G(\mathbb{C})$ induces an
injective morphism between real manifolds
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))\hookrightarrow\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$.
By using this injection, we shall regard
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))$ as a real submanifold of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ of half-dimension.

Next, let $\mu$ be a $G(K)$-representation classified by
$\mathrm{Hom}_{P}^{0}(\Gamma,G(K))$. The composite
$\Gamma\xrightarrow{\mu}G(K)\xrightarrow{\mathrm{Ad}_{G}}\mathrm{GL}(\mathfrak{g}(K))$
determines a $K[\Gamma]$-module (resp., a local system)

(60) $\displaystyle\mathfrak{g}(K)_{\mu}\ \left(\text{resp.,}\
\mathbb{L}_{\mu,K}\right)$

whose underlying $K$-vector space (resp., whose fiber over every point) is
isomorphic to $\mathfrak{g}_{\mathbb{R}}\otimes_{\mathbb{R}}K$. In particular,
we have
$\mathfrak{g}(\mathbb{C})_{\mu}=\mathfrak{g}(\mathbb{R})_{\mu}\otimes_{\mathbb{R}}\mathbb{C}$
and
$\mathbb{L}_{\mu,\mathbb{C}}=\mathbb{L}_{\mu,\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}$.

The infinitesimal deformations of $\mu$ in $\mathrm{Hom}_{P}^{0}(\Gamma,G(K))$
are determined by the elements of $Z^{1}_{P}(\Gamma,\mathfrak{g}(K)_{\mu})$
(cf. [LuMa, § 2], [Rag, Chap. VI]). Also, the infinitesimal deformations
tangential to the $G(K)$-orbit of $\mu$ in $\mathrm{Hom}_{P}^{0}(\Gamma,G(K))$
are precisely the $1$-coboundary maps. It follows that the tangent space
$T_{[\mu]}\mathrm{Rep}_{P}^{0}(\Gamma,G(K))$ of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(K))$ at the point classifying the equivalence
class $[\mu]$ admits an isomorphism of $K$-vector spaces

(61) $\displaystyle
T_{[\mu]}\mathrm{Rep}_{P}^{0}(\Gamma,G(K))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{P}(\Gamma,\mathfrak{g}(K)_{\mu}).$

Moreover, the following square diagram is commutative:

(66)

where the upper horizontal arrow arises from the differential of the inclusion
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))\hookrightarrow\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$.

###  3.8. Flat $G$-bundles with prescribed nilpotent residue

A flat $G$-bundle $\mathscr{E}:=(\mathcal{E},\nabla)$ on $\mathscr{X}$ is
called irreducible if there are no pairs $(P,\mathcal{E}_{P})$ such that $P$
is a parabolic subgroup of $G$ and $\mathcal{E}_{P}$ is a $P$-reduction of
$\mathcal{E}|_{X^{*}}$ preserving the connection $\nabla|_{X^{*}}$ (i.e., the
image of $\nabla|_{X^{*}}$ is contained in
$\widetilde{\mathcal{T}}_{\mathcal{E}_{P}}\subseteq\widetilde{\mathcal{T}}_{(\mathcal{E}|_{X^{*}})^{\mathrm{log}}}$).
Thus, we obtain the full subcategory ($\mathbb{C}$-substack)

(67) $\displaystyle\mathcal{C}onn_{P}^{0}(\mathscr{X},G)$

of $\mathcal{C}onn(\mathscr{X},G)$ consisting of flat $G$-bundles
$\mathscr{E}:=(\mathcal{E},\nabla)$ satisfying the following three conditions:

  * (a)

$\mathscr{E}$ is irreducible;

  * (b)

$\mathscr{E}$ has no nontrivial automorphisms;

  * (c)

For each $i=1,\cdots,r$, the residue $\mathrm{Res}_{\sigma_{i}}(\nabla)$ at
$\sigma_{i}$ coincides with $q_{-1}$ under the identification
$\mathfrak{g}=\mathfrak{g}_{\mathcal{E}}|_{\sigma}$ induced from some
trivialization
$G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}|_{\sigma_{i}}$.

We shall prove the following assertion, generalizing [Con, Theorem 4.2] to
flat $G$-bundles.

######  Theorem 3.7.

The composite

(68)
$\displaystyle\mathcal{C}onn_{P}^{0}(\mathscr{X},G)\xrightarrow{\mathrm{restriction}}\mathcal{C}onn(X^{*},G)\xrightarrow{(\ref{Eq2000})}\mathcal{R}ep(\Gamma,G(\mathbb{C}))$

restricts to an equivalence of categories

(69)
$\displaystyle\mathcal{C}onn_{P}^{0}(\mathscr{X},G)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{R}ep_{P}^{0}(\Gamma,G(\mathbb{C})).$

###### Proof.

First, let $\mathscr{E}:=(\mathcal{E},\nabla)$ be a flat $G$-bundle classified
by $\mathcal{C}onn_{P}^{0}(\mathscr{X},G)$. By assumption, the representation
$\mathrm{Mon}_{\mathscr{E}|_{X^{*}}}:\Gamma\rightarrow G(\mathbb{C})$ is
irreducible and its image has trivial centralizer. Moreover, since the
equality $\mathrm{Res}_{\sigma_{i}}(\nabla)=q_{-1}$ holds (for each
$i=1,\cdots,r$) under a suitable trivialization
$\mathfrak{g}=\mathfrak{g}_{\mathcal{E}}|_{\sigma_{i}}$, the local monodromy
of $\mathscr{E}|_{X^{*}}$ around $\sigma_{i}$ is conjugate to
$\mathrm{exp}(-2\pi\sqrt{-1}\cdot q_{-1})$ (cf. Theorem 3.2), which is also
conjugate to $\iota_{G}(T_{i})$. It follows that
$\mathrm{Mon}_{\mathscr{E}|_{X^{*}}}$ belongs to
$\mathcal{R}ep_{P}^{0}(\Gamma,G(\mathbb{C}))$. Thus, the assignment
$\mathscr{E}\mapsto\mathrm{Mod}_{\mathscr{E}|_{X^{*}}}$ (which coincides with
the composite (68)) defines a functor

(70)
$\displaystyle\mathcal{C}onn_{P}^{0}(\mathscr{X},G)\rightarrow\mathcal{R}ep_{P}^{0}(\Gamma,G(\mathbb{C})).$

The problem is to prove that this functor is essentially surjective and fully
faithful. Let $\mu:\Gamma\rightarrow G(\mathbb{C})$ be a homomorphism
classified by $\mathcal{R}ep_{P}^{0}(\Gamma,G(\mathbb{C}))$; it corresponds to
a flat $G$-bundle $\mathscr{E}^{*}:=(\mathcal{E}^{*},\nabla^{*})$ on $X^{*}$.
For each $i=1,\cdots,r$, we shall choose a sufficiently small open
neighborhood $U_{i}$ of $\sigma_{i}$ in $X$. Then, we obtain the flat
$G$-bundle $\mathscr{E}_{i}:=(U_{i}\times G,\nabla_{q_{-1}})$ on the pointed
Riemann surface $(U_{i},\\{\sigma_{i}\\})$, where $\nabla_{q_{-1}}$ denotes
the log connection on the trivial $G$-bundle $U_{i}\times G$ determined by the
equality $\nabla_{q_{-1}}^{\mathfrak{g}}=q_{-1}$. The local monodromy of this
flat $G$-bundle around $\sigma_{i}$ is conjugate to
$\mathrm{exp}(-2\pi\sqrt{-1}\cdot q_{-1})$, so there exists an isomorphism
$\alpha_{i}:\mathscr{E}_{i}|_{U_{i}^{*}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}^{*}|_{U^{*}_{i}}$
(cf. Theorem 3.1). The isomorphism $\alpha_{i}$ ($i=1,\cdots,r$) can be used
to glue together $\mathscr{E}^{*}$ and $\mathscr{E}_{i}$’s to obtain a flat
$G$-bundle $\mathscr{E}$ on $\mathscr{X}$. By construction, $\mathscr{E}$
specifies an object in $\mathcal{C}onn_{P}^{0}(\mathscr{X},G)$ and its image
via (70) is isomorphic to $\mu$. This implies the essential surjectivity of
(70).

The remaining portion is the fully-faithfulness of (70). Let us take two flat
$G$-bundles $\mathscr{E}_{i}:=(\mathcal{E}_{i},\nabla_{i})$ ($i=1,2$)
classified by $\mathcal{C}onn_{P}^{0}(\mathscr{X},G)$. The map

(71)
$\displaystyle\mathrm{Isom}(\mathscr{E}_{1},\mathscr{E}_{2})\rightarrow\mathrm{Isom}(\mathscr{E}_{1}|_{X^{*}},\mathscr{E}_{2}|_{X^{*}})$

between the sets of isomorphisms obtained via restriction is immediately
verified to be injective. Moreover, let us take an isomorphism
$\alpha:\mathscr{E}_{1}|_{X^{*}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}_{2}|_{X^{*}}$.
Since $G$ is affine, there exists a closed immersion
$\iota:G\hookrightarrow\mathrm{GL}_{N}$ ($N\in\mathbb{Z}_{>0}$), so we obtain
a flat vector bundle $\iota_{*}(\mathscr{E}_{i})$ (for each $i=1,2$) induced
from $\mathscr{E}_{i}$ via change of structure group by $\iota$. The
isomorphism $\alpha$ induces an isomorphism
$\iota_{*}(\alpha)_{X^{*}}:\iota_{*}(\mathscr{E}_{1})|_{X^{*}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\iota_{*}(\mathscr{E}_{2})|_{X^{*}}$.
It follows from [Con, Theorem 4.2] that $\iota_{*}(\alpha)_{X^{*}}$ extends to
an isomorphism
$\iota_{*}(\alpha)_{\mathscr{X}}:\iota_{*}(\mathscr{E}_{1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\iota_{*}(\mathscr{E}_{2})$
over $X$. Since $\iota$ is a closed immersion,
$\iota_{*}(\alpha)_{\mathscr{X}}$ lies in
$\mathrm{Isom}(\mathscr{E}_{1},\mathscr{E}_{2})$. This implies the
surjectivity of (71), and hence completes the proof of the assertion. ∎

######  Remark 3.8 (Extending the equivalence of categories).

By an argument entirely similar to the proof of the above theorem, we see that
(69) can be extended to an equivalence of categories between the category of
flat $G$-bundles on $\mathscr{X}$ having nilpotent residues and the full
subcategory of $\mathcal{R}ep(\Gamma,G(\mathbb{C}))$ consisting of
representations $\mu:\Gamma\rightarrow G(\mathbb{C})$ such that $\mu(T_{i})$
is unipotent for every $i$.

##  4\. $G$-opers on a pointed Riemann surface

This section deals with $G$-opers on a pointed Riemann surface and their
moduli space. The corresponding discussions for pointed stable curves (in a
purely algebraic treatment) have already been made in [Wak1]. Due to the
similarity of the arguments, it will be sufficient to refer to that reference
for the facts described here. Therefore, we will not provide their proofs in
this text. For the case of unpointed Riemann surfaces, we refer the reader to
[BeDr]. In the latter half of this section, we recall the notion of a
permissible connection, introduced by G. Faltings, and consider the
relationship with $\mathrm{PSL}_{2}$-opers having specific radii (cf.
Proposition 4.3).

We keep the notation and assumptions concerning the algebraic group $G$
described in § 3.7.

###  4.1. $G$-opers

Let us fix a pair of nonnegative integers $(g,r)$ with $2g-2+r>0$ and an
$r$-pointed compact Riemann surface
$\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i=1}^{r})$ of genus $g$. To simplify
slightly the notation, we write $\Omega:=\Omega_{X^{\mathrm{log}}}$ and
$\mathcal{T}:=\mathcal{T}_{X^{\mathrm{log}}}$. Also, denote by $D$ the divisor
on $X$ defined as the union of the $\sigma_{i}$’s.

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

(72)
$\displaystyle\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{j}:=\widetilde{\mathcal{T}}_{\mathcal{E}_{B}^{\mathrm{log}}}+\mathfrak{g}_{\mathcal{E}_{G}}^{j}\left(\subseteq\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}\right).$

The inclusion
$\mathfrak{g}_{\mathcal{E}_{G}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}$
gives an isomorphism

(73)
$\displaystyle\mathfrak{g}_{\mathcal{E}_{G}}^{j}/\mathfrak{g}_{\mathcal{E}_{G}}^{j+1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{j}/\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{j+1}.$

On the other hand, since each
$\mathfrak{g}^{-\alpha}:=\mathfrak{g}_{\mathbb{R}}^{-\alpha}\otimes_{\mathbb{R}}\mathbb{C}$
($\alpha\in\Pi$) is closed under the adjoint $B$-action,
$\mathfrak{g}^{-1}/\mathfrak{g}^{0}=\bigoplus_{\alpha\in\Pi}\mathfrak{g}^{-\alpha}$
yields a canonical decomposition
$\mathfrak{g}^{-1}_{\mathcal{E}_{G}}/\mathfrak{g}^{0}_{\mathcal{E}_{G}}=\bigoplus_{\alpha\in\Pi}\mathfrak{g}_{\mathcal{E}_{G}}^{-\alpha}$.
By combining it with the inverse of (73) for $j=-1$, we obtain a decomposition

(74)
$\displaystyle\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{-1}/\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}=\bigoplus_{\alpha\in\Pi}\mathfrak{g}_{\mathcal{E}_{G}}^{-\alpha}.$

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 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. [Wak1, Definition 2.1, (i)] for the definition in the case
of pointed stable curves) if
$\nabla(\mathcal{T}_{X^{\mathrm{log}}})\subseteq\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{-1}$
and the composite

(75)
$\displaystyle\mathcal{T}\xrightarrow{\nabla}\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}^{-1}\twoheadrightarrow\widetilde{\mathcal{T}}^{-1}_{\mathcal{E}_{G}^{\mathrm{log}}}/\widetilde{\mathcal{T}}^{0}_{\mathcal{E}_{G}^{\mathrm{log}}}\twoheadrightarrow\mathfrak{g}^{-\alpha}_{\mathcal{E}_{B}}$

defined for each $\alpha\in\Pi$ is an isomorphism, where the last arrow
denotes the natural projection with respect to the decomposition (74). In a
natural manner, one can define the notion of an isomorphism between $G$-opers.

Denote by $[0]_{G}$ (or simply, $[0]$, if there is no fear of confusion) the
element of $\mathfrak{c}(\mathbb{C})$ determined by the image of the zero
section $0\in\mathfrak{g}$ via
$\chi:\mathfrak{g}\twoheadrightarrow\mathfrak{c}$. Then, we obtain the
category

(76) $\displaystyle\mathrm{Op}_{P}(\mathscr{X},G)$

whose objects are $G$-opers
$\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ on $\mathscr{X}$ with
$\mathrm{Rad}_{\sigma_{i}}(\nabla)=[0]$ for every $i=1,\cdots,r$, and whose
morphisms are isomorphisms between such $G$-opers. (In the case of $r=0$, we
do not impose any condition on the radii of $G$-opers.) Since each $G$-oper
does not have non-trivial automorphisms (cf. [Wak1, Proposition 2.9]),
$\mathrm{Op}_{P}(\mathscr{X},G)$ may be regarded as a set.

###  4.2. Change of structure group

In what follows, we shall set

(77) $\displaystyle G^{\odot}:=\mathrm{PSL}_{2}$

for simplicity. This group contains the Borel subgroup $B^{\odot}$ (resp., the
maximal torus $T^{\odot}$) consisting of the images of upper triangular
matrices (resp., diagonal matrices) in $\mathrm{SL}_{2}$ via the natural
surjection $\mathrm{SL}_{2}\twoheadrightarrow G^{\odot}$. Write
$\mathfrak{g}^{\odot}$ and $\mathfrak{b}^{\odot}$ for the Lie algebras of
$G^{\odot}$ and $B^{\odot}$, respectively (hence
$\mathfrak{g}^{\odot}=\mathfrak{s}\mathfrak{l}_{2}$). In particular, we have
$\iota_{G}(B^{\odot})\subseteq B$ and
$\iota_{\mathfrak{g}}(\mathfrak{b}^{\odot})\subseteq\mathfrak{b}$.

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

(78)
$\displaystyle\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot}):=(\mathcal{E}_{B},\iota_{G*}(\nabla_{\odot}))$

is verified to form a $G$-oper on $\mathscr{X}$ (cf. [Wak1, § 2.4.2]).
Moreover, the equality
$\mathrm{Rad}_{\sigma_{i}}(\iota_{G*}(\nabla_{\odot}))=[0]_{G}$ holds if
$\mathrm{Rad}_{\sigma_{i}}(\nabla_{\odot})=[0]_{G^{\odot}}$ ($i=1,\cdots,r$).
This implies that the assignment
$\mathscr{E}_{\odot}^{\spadesuit}\mapsto\iota_{G*}(\mathscr{E}_{\odot}^{\spadesuit})$
determines a map of sets

(79)
$\displaystyle\mathrm{Op}_{P}(\mathscr{X},G^{\odot})\rightarrow\mathrm{Op}_{P}(\mathscr{X},G)$

(cf. [Wak1, Remark 2.37]).

###  4.3. Canonical $B$-bundle

Just as in the discussion of [Wak1, § 2.4.3], one can construct a canonical
$B$-bundle ${{}^{\dagger}}\mathcal{E}_{B}$ (as well as a canonical $G$-bundle
${{}^{\dagger}}\mathcal{E}_{G}$) on $X$ that is isomorphic to the underlying
$B$-bundle of any $G$-oper; this $B$-bundle is constructed as follows.

First, we shall consider the case of $G=G^{\odot}$ (and $B=B^{\odot}$). Let
$U$ be an open subset of $X$, and assume that there exists a holomorphic line
bundle $\varTheta$ on $U$ equipped with an isomorphism $\varTheta^{\otimes
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega|_{U}$. (Such a line
bundle always exists locally, and we refer to it as a theta characteristic of
$U$.) We set ${{}^{\dagger}}\mathcal{E}_{G^{\odot},U}$ to be the
$G^{\odot}$-bundle on $U$ defined as the projectivization of the rank $2$
vector bundle $\mathcal{D}_{U,<j}\otimes\varTheta$, where for each
$j\in\mathbb{Z}_{\geq 0}$ we denote by $\mathcal{D}_{U,<j}$ the sheaf of
holomorphic logarithmic differential operators on $U$ (with respect to the log
structure obtained by restricting that on $X^{\mathrm{log}}$) of order $<j$.
Then, the line subbundle
$\mathcal{D}_{U,<1}\otimes\varTheta\left(\subseteq\mathcal{D}_{U,<2}\otimes\varTheta\right)$
determines a $B^{\odot}$-reduction ${{}^{\dagger}}\mathcal{E}_{B^{\odot},U}$
of ${{}^{\dagger}}\mathcal{E}_{G^{\odot},U}$. Since
${{}^{\dagger}}\mathcal{E}_{B^{\odot},U}$ does not depend on the choice of
$\varTheta$, we can glue together the $B$-bundles
${{}^{\dagger}}\mathcal{E}_{B^{\odot},U}$ defined for various open subsets $U$
(admitting a theta characteristic) to obtain a globally defined $B$-bundle
${{}^{\dagger}}\mathcal{E}_{B^{\odot}}$. This is the desired
$B^{\odot}$-bundle.

For a general $G$, the desired $B$-bundle (resp., $G$-bundle) is defined as

(80)
$\displaystyle{{}^{\dagger}}\mathcal{E}_{B}:={{}^{\dagger}}\mathcal{E}_{B^{\odot}}\times^{B^{\odot},\iota_{G}}B\
\left(\text{resp.,}\
{{}^{\dagger}}\mathcal{E}_{G}:={{}^{\dagger}}\mathcal{E}_{B^{\odot}}\times^{B^{\odot},\iota_{G}}G\right).$

When we want to clarify the underlying pointed Riemann surface $\mathscr{X}$,
we write ${{}^{\dagger}}\mathcal{E}_{G,\mathscr{X}}$ and
${{}^{\dagger}}\mathcal{E}_{B,\mathscr{X}}$ (or
${{}^{\dagger}}\mathcal{E}_{G,X}$ and ${{}^{\dagger}}\mathcal{E}_{B,X}$, if
$r=0$) instead of ${{}^{\dagger}}\mathcal{E}_{G}$ and
${{}^{\dagger}}\mathcal{E}_{B}$, respectively.

As mentioned in [Wak1, § 2.4.3], the $B$-bundle
${{}^{\dagger}}\mathcal{E}_{B}$ satisfies the following properties.

  * •

Let us take an arbitrary pair $(V,\partial)$ consisting of an open subset $V$
of $X$ and a section $\Gamma(V,\mathcal{T})$ with
$\mathcal{T}|_{V}=\mathcal{O}_{V}\partial$. (Such a pair will be called a log
chart on $\mathscr{X}$.) Then, there exists a specific trivialization (i.e.,
framing)

(81) $\displaystyle\mathrm{triv}_{B,(V,\partial)}:V\times
B\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{{}^{\dagger}}\mathcal{E}_{B}|_{V}$

of the $B$-bundle ${{}^{\dagger}}\mathcal{E}_{B}$ restricted to $V$. (In fact,
if $V$ admits a theta characteristic $\varTheta$, then the section $\partial$
gives $(\mathcal{D}_{V,<2}\otimes\varTheta)=\varTheta^{\oplus 2}$, which
yields a trivialization of ${{}^{\dagger}}\mathcal{E}_{B^{\odot}}|_{V}$.)
Moreover, the formations of both ${{}^{\dagger}}\mathcal{E}_{B}$ and
$\mathrm{triv}_{B,(V,\partial)}$ are functorial with respect to pull-back to
locally biholomorphic maps.

  * •

If $r>0$, then for each $i=1,\cdots,r$ there exists a canonical trivialization

(82)
$\displaystyle\mathrm{triv}_{B,\sigma_{i}}:B\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{{}^{\dagger}}\mathcal{E}_{B}|_{\sigma_{i}}$

of the fiber ${{}^{\dagger}}\mathcal{E}_{B}|_{\sigma_{i}}$ of
${{}^{\dagger}}\mathcal{E}_{B}$ over the point $\sigma_{i}$. Moreover, this
trivialization is compatible with $\mathrm{triv}_{(B,(V,\partial))}$ in an
evident sense.

Next, denote by
$\mathrm{ad}:\mathfrak{g}\hookrightarrow\mathrm{End}(\mathfrak{g})$ (resp.,
$\mathrm{ad}_{\mathbb{R}}:\mathfrak{g}_{\mathbb{R}}\hookrightarrow\mathrm{End}(\mathfrak{g}_{\mathbb{R}})$)
the adjoint operator induced by differentiating
$\mathrm{Ad}_{G}:G\rightarrow\mathrm{GL}(\mathfrak{g})$ (resp.,
$\mathrm{Ad}_{G_{\mathbb{R}}}:G_{\mathbb{R}}\rightarrow\mathrm{GL}(\mathfrak{g}_{\mathbb{R}})$).
Consider the complex (resp., real) vector space of
$\mathrm{ad}(q_{1})$-invariants

(83)
$\displaystyle\mathfrak{g}^{\mathrm{ad}(q_{1})}:=\left\\{x\in\mathfrak{g}\,|\,\mathrm{ad}(q_{1})(x)=0\right\\}\
\left(\text{resp.,}\
\mathfrak{g}_{\mathbb{R}}^{\mathrm{ad}(q_{1})}:=\left\\{x\in\mathfrak{g}_{\mathbb{R}}\,|\,\mathrm{ad}(q_{1})(x)=0\right\\}\right),$

which has dimension equal to the rank $\mathrm{rk}(G)$ of $G$. The Cartan
decomposition on $\mathfrak{g}$ (resp., $\mathfrak{g}_{\mathbb{R}}$) restricts
to a decomposition
$\mathfrak{g}^{\mathrm{ad}(q_{1})}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})}$
(resp.,
$\mathfrak{g}_{\mathbb{R}}^{\mathrm{ad}(q_{1})}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad}(q_{1})}$).

Since $\mathfrak{g}^{\mathrm{ad}(q_{1})}$ is closed under the
$B^{\odot}$-action via $\iota_{G}$, one can construct a rank $\mathrm{rk}(G)$
vector bundle
$(\mathfrak{g}^{\mathrm{ad}(q_{1})})_{{{}^{\dagger}}\mathcal{E}_{B^{\odot}}}$
on $X$ obtained from $\mathfrak{g}^{\mathrm{ad}(q_{1})}$ twisted by using
${{}^{\dagger}}\mathcal{E}_{B^{\odot}}$. In particular, we obtain

(84)
$\displaystyle\mathcal{V}_{G}:=\Omega\otimes(\mathfrak{g}^{\mathrm{ad}(q_{1})})_{{{}^{\dagger}}\mathcal{E}_{B^{\odot}}}$

(cf. [Wak1, § 2.4.5]). For example, $\mathcal{V}_{G^{\odot}}$ is canonically
isomorphic to $\Omega^{\otimes 2}$. The decomposition
$\mathfrak{g}^{\mathrm{ad}(q_{1})}=\bigoplus_{j\in\mathbb{Z}}\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})}$
yields
$\mathcal{V}_{G}=\bigoplus_{j\in\mathbb{Z}}\mathcal{V}_{G,j}\left(=\bigoplus_{j=1}^{\mathrm{rk}(G)}\mathcal{V}_{G,j}\right)$.
Regarding the direct summands of this decomposition, we have

(85)
$\displaystyle\mathcal{V}_{G,j}\cong\Omega^{\otimes(j+1)}\otimes_{\mathbb{C}}\mathfrak{g}^{\mathrm{ad}(q_{1})}_{j}\left(=\Omega^{\otimes(j+1)}\otimes_{\mathbb{R}}\mathfrak{g}^{\mathrm{ad}(q_{1})}_{\mathbb{R},j}\right)$

for every $j\in\mathbb{Z}$.

###  4.4. Affine structure on $\mathrm{Op}_{P}(\mathscr{X},G)$

The $B^{\odot}$-equivariant inclusion
$\mathfrak{g}^{\mathrm{ad}(q_{1})}\hookrightarrow\mathfrak{g}$ yields an
$\mathcal{O}_{X}$-linear injection
$\varsigma:\mathcal{V}_{G}\hookrightarrow\Omega\otimes\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{G}}$.
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
$\Omega\otimes\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{G}}$ and
$\mathcal{V}_{G}$, respectively.

We here recall the following definition (cf. [Wak1, Definition 2.14]).

######  Definition 4.1.

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a $G$-oper on
$\mathscr{X}$. We shall say that $\mathscr{E}^{\spadesuit}$ is
$\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal if
$\mathcal{E}_{B}={{}^{\dagger}}\mathcal{E}_{B}$ and, for any log chart
$(V,\partial)$ on $\mathscr{X}$, the pull-back
$\mathrm{triv}_{G,(V,\partial)}^{*}(\nabla)$ (where
$\mathrm{triv}_{G,(V,\partial)}$ denotes the isomorphism $V\times
G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{{}^{\dagger}}\mathcal{E}_{G}|_{V}$
extending $\mathrm{triv}_{B,(V,\partial)}$) satisfies the equality

(86)
$\displaystyle\mathrm{triv}_{G,(V,\partial)}^{*}(\nabla)^{\mathfrak{g}}(\partial)=1\otimes
q_{-1}+v\in\Gamma(V,\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g})$

for some
$v\in\Gamma(V,\mathcal{O}_{X}\otimes_{\mathbb{C}}\mathfrak{g}^{\mathrm{ad}(q_{1})})$.

######  Remark 4.2 ($\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normality).

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

  * (i)

By an argument entirely similar to the proof of [Wak1, Proposition 2.19], we
can find a unique 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 ${{}^{\dagger}}\mathscr{E}^{\spadesuit}$ on
$\mathscr{X}$ and an isomorphism of $G$-opers
$\mathrm{nor}_{\mathscr{E}^{\spadesuit}}:{{}^{\dagger}}\mathscr{E}^{\spadesuit}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{E}^{\spadesuit}$.
We refer to
$({{}^{\dagger}}\mathscr{E}^{\spadesuit},\mathrm{nor}_{\mathscr{E}^{\spadesuit}})$
as the $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normalization of $\mathscr{E}^{\spadesuit}$.

  * (ii)

Suppose that $\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal.
The composite

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

is an isomorphism, which maps the zero element
$0\in\mathfrak{g}^{\mathrm{ad}(q_{1})}$ to $[0]$. Hence, if the radius
$\mathrm{Rad}_{\sigma_{i}}(\nabla)$ ($i\in\\{1,\cdots,r\\}$) coincides with
$[0]$, then the equality $\mathrm{Res}_{\sigma_{i}}(\nabla)=q_{-1}$ holds
under the identification
$\mathfrak{g}=\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{G}}|_{\sigma_{i}}$
induced by $\mathrm{triv}_{B,\sigma_{i}}$.

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

(88)
$\displaystyle\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})_{+v}:=({{}^{\dagger}}\mathcal{E}_{B},\iota_{G*}(\nabla_{\odot})+v)$

forms a $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normal $G$-oper.

If, moreover, the equality
$\mathrm{Rad}_{\sigma_{i}}(\nabla_{\odot})=[0]_{G^{\odot}}$ holds for every
$i$ and $v$ lies in $H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))$, then we have
$\mathrm{Rad}_{\sigma_{i}}(\iota_{G*}(\nabla_{\odot})+v)=[0]_{G}$ (cf. Remark
4.2, (ii)). Hence, as proved in [Wak1, Theorem 2.24], the assignment

(89)
$\displaystyle(\mathscr{E}^{\spadesuit}_{\odot},v)\mapsto\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})_{+v}$

yields an affine structure on $\mathrm{Op}_{P}(\mathscr{X},G)$ modeled on
$H^{0}(X,\mathcal{V}_{G}(-D))$. In particular,
$\mathrm{Op}_{P}(\mathscr{X},G)$ has a structure of complex manifold of
dimension $a(G)$. The map (79) preserves the affine structure, meaning that
the morphism

(90)
$\displaystyle\mathrm{Op}_{P}(\mathscr{X},G^{\odot})\times^{H^{0}(X,\Omega^{\otimes
2}(-D))}H^{0}(X,\mathcal{V}_{G}(-D))\rightarrow\mathrm{Op}_{P}(\mathscr{X},G)$

given by (89) is an isomorphism.

###  4.5. Permissible connections

In §§ 4.5-4.6, we assume that $r$ is even. Then, since $\mathrm{deg}(\Omega)$
is even, $\mathscr{X}$ has a globally defined theta characteristic $\varTheta$
(cf. § 4.3). Recall that extensions of $\varTheta^{\vee}(D)$ by $\varTheta$
can be classified by $\mathrm{Ext}^{1}(\varTheta^{\vee}(D),\varTheta)$, which
admits a sequence of canonical isomorphisms

(91) $\displaystyle\mathrm{Ext}^{1}(\varTheta^{\vee}(D),\varTheta)\cong
H^{1}(X,\varTheta^{\otimes 2}(-D))\cong H^{1}(X,\Omega(-D))\cong\mathbb{C}.$

Denote by ${{}^{\ddagger}}\mathcal{V}$ the holomorphic vector bundle on $X$ of
rank $2$ defined as the extension whose class corresponds to the value
$(2g-2+r)/2\in\mathbb{C}$ via (91). In particular, $\varTheta$ may be regarded
as a line subbundle of ${{}^{\ddagger}}\mathcal{V}$. We refer to
${{}^{\ddagger}}\mathcal{V}$ as the uniformization-bundle associated to
$\varTheta$ (cf. [Fal, § 5]).

Next, let $\nabla$ be a log connection on ${{}^{\ddagger}}\mathcal{V}$, i.e.,
a $\mathbb{C}$-linear morphism
${{}^{\ddagger}}\mathcal{V}\rightarrow\Omega\otimes{{}^{\ddagger}}\mathcal{V}$
with $\nabla(av)=da\otimes v+a\cdot\nabla(v)$ for any local sections
$a\in\mathcal{O}_{X}$, $v\in{{}^{\ddagger}}\mathcal{V}$. We shall say that
$\nabla$ is a permissible connection on ${{}^{\ddagger}}\mathcal{V}$ (cf.
[Fal, § 5]) if it satisfies the following three conditions:

  * •

The log connection $\mathrm{det}(\nabla)$ on the determinant
$\mathrm{det}({{}^{\ddagger}}\mathcal{V})$ of ${{}^{\ddagger}}\mathcal{V}$
induced from $\nabla$ is compatible with the trivial connection on
$\mathcal{O}_{X}(D)$ via the composite isomorphism

(92)
$\displaystyle\mathrm{det}({{}^{\ddagger}}\mathcal{V})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\varTheta^{\vee}(D)\otimes\varTheta\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{X}(D);$

  * •

For each $i=1,\cdots,r$, the residue (i.e., the monodromy operator, in the
sense of [Wak1, Definition 4.42]) of the connection $\nabla$ at $\sigma_{i}$
restricts to the zero map on the line
$\varTheta|_{\sigma_{i}}\subseteq{{}^{\ddagger}}\mathcal{V}|_{\sigma_{i}}$;

  * •

The composite

(93)
$\displaystyle\varTheta\xrightarrow{\mathrm{inclusion}}{{}^{\ddagger}}\mathcal{V}\xrightarrow{\nabla}\Omega\otimes{{}^{\ddagger}}\mathcal{V}\xrightarrow{\mathrm{quotient}}\left(\Omega\otimes({{}^{\ddagger}}\mathcal{V}/\varTheta)=\right)\Omega\otimes\varTheta^{\vee}(D)\cong\varTheta(D),$

i.e., the Kodaira-Spencer map of $\nabla$ with respect to
$\varTheta\subseteq{{}^{\ddagger}}\mathcal{V}$ (cf. [Wak1, Example 4.20]),
coincides with the inclusion $\varTheta\hookrightarrow\varTheta(D)$.

Also, two permissible connections on ${{}^{\ddagger}}\mathcal{V}$ are
equivalent if they can be obtained from each other by an automorphism of the
extension
$0\rightarrow\varTheta\rightarrow{{}^{\ddagger}}\mathcal{V}\rightarrow\varTheta^{\vee}(D)\rightarrow
0$. Thus, we obtain the set

(94) $\displaystyle\mathrm{Per}(\mathscr{X},\varTheta)$

of equivalence classes of permissible connections on
${{}^{\ddagger}}\mathcal{V}$.

###  4.6. Correspondence with $\mathrm{PGL}_{2}$-opers

Let $\nabla$ be a permissible connection on ${{}^{\ddagger}}\mathcal{V}$.
Denote by ${{}^{\ddagger}}\widetilde{\mathcal{V}}$ the rank $2$ vector bundle
on $X$ defined as the push-forward of the following diagram:

(99)

In particular, $\varTheta(D)$ may be regarded as a line subbundle of
${{}^{\ddagger}}\widetilde{\mathcal{V}}$, and we have
$\mathrm{det}({{}^{\ddagger}}\widetilde{\mathcal{V}})\cong\varTheta(D)\otimes\varTheta^{\vee}(D)\cong\mathcal{O}_{X}(2D)$.
Then, $\nabla$ extends uniquely to a log connection $\widetilde{\nabla}$ on
${{}^{\ddagger}}\widetilde{\mathcal{V}}$. The Kodaira-Spencer map of
$\widetilde{\nabla}$ with respect to
$\varTheta(D)\subseteq{{}^{\ddagger}}\widetilde{\mathcal{V}}$ turns out to be
an isomorphism. That is, the collection
$({{}^{\ddagger}}\widetilde{\mathcal{V}},\widetilde{\nabla},\varTheta(D))$
forms a so-called $\mathrm{SL}_{2}$-oper on $\mathscr{X}$, and its
projectivization specifies a $G^{\odot}$-oper

(100) $\displaystyle\mathscr{E}^{\spadesuit}_{\odot}(\nabla)$

on $\mathscr{X}$. By construction, the radius of
$\mathscr{E}^{\spadesuit}_{\odot}(\nabla)$ at every marked point coincides
with $[0]$.

######  Proposition 4.3.

(Recall that we have assumed that $r$ is even.) The map of sets

(101)
$\displaystyle\mathrm{Per}(\mathscr{X},\varTheta)\rightarrow\mathrm{Op}_{P}(\mathscr{X},G^{\odot})$

given by assigning $\nabla\mapsto\mathscr{E}^{\spadesuit}_{\odot}(\nabla)$ is
bijective.

###### Proof.

Recall from [Fal, § 5] that the set $\mathrm{Per}(\mathscr{X},\varTheta)$ has
a structure of affine space modeled on $H^{0}(X,\Omega^{\otimes 2}(-D))$.
Then, the assertion follows from the fact that the $H^{0}(X,\Omega^{\otimes
2}(-D))$-affine structures on the respective sets
$\mathrm{Per}(\mathscr{X},\varTheta)$,
$\mathrm{Op}_{P}(\mathscr{X},G^{\odot})$ are compatible via (101). ∎

######  Example 4.4 (Canonical $G^{\odot}$-oper).

Denote by $\mathbb{P}^{1}$ the projective line over $\mathbb{C}$ (i.e., the
Riemann sphere) and by $\mathbb{H}$ the upper half plane, i.e.,
$\mathbb{H}:=\left\\{z\in\mathbb{C}\,|\,\mathrm{Im}(z)>0\right\\}$. Since the
automorphism group $\mathrm{Aut}(\mathbb{P}^{1})$ of $\mathbb{P}^{1}$ may be
identified with $G^{\odot}$, $G^{\odot}$-bundles correspond bijectively to
$\mathbb{P}^{1}$-bundles. Moreover, for a fixed $G^{\odot}$-bundle
$\mathcal{E}$ on $X$, we see that giving a $B^{\odot}$-reduction of
$\mathcal{E}$ is equivalent to giving a global section of the projection
$\mathcal{E}\times^{G^{\odot}}\mathbb{P}^{1}\rightarrow X$ of the
corresponding $\mathbb{P}^{1}$-bundle
$\mathcal{E}\times^{G^{\odot}}\mathbb{P}^{1}$. In particular, the diagonal
embedding $\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$
determines a $B^{\odot}$-reduction $\mathcal{E}_{B^{\odot}}^{\mathrm{triv}}$
of the trivial $G^{\odot}$-bundle $\mathbb{P}^{1}\times G^{\odot}$ on
$\mathbb{P}^{1}$, and the resulting pair

(102)
$\displaystyle(\mathcal{E}_{B^{\odot}}^{\mathrm{triv}},\nabla^{\mathrm{triv}})$

forms a $G^{\odot}$-oper on $\mathbb{P}^{1}$.

Next, denote by $\varpi$ a covering map $\mathbb{H}\twoheadrightarrow X^{*}$
arising from the Fuchsian uniformization. The diagonal $\Gamma$-action on
$\mathbb{H}\times G^{\odot}$ preserves the connection $\nabla^{\mathrm{triv}}$
as well as the $B^{\odot}$-reduction
$\mathcal{E}_{B^{\odot}}^{\mathrm{triv}}$. Hence, the $G^{\odot}$-oper
$(\mathcal{E}_{B^{\odot}}^{\mathrm{triv}},\nabla^{\mathrm{triv}})|_{\mathbb{H}}$
on $\mathbb{H}$ descends, via $\varpi$, to a $G^{\odot}$-oper on $X^{*}$; it
extends uniquely to a $G^{\odot}$-oper

(103)
$\displaystyle(\mathcal{E}_{B^{\odot}}^{\mathrm{can}},\nabla^{\mathrm{can}})$

classified by $\mathrm{Op}_{P}(\mathscr{X},G^{\odot})$ and corresponds to the
canonical permissible connection resulting from [Fal, Theorem 4].

###  4.7. Permissible $G$-opers

Proposition 4.3 means that permissible connections are essentially equivalent
to $G^{\odot}$-oper each of whose radii coincides with $[0]$. By taking this
into account, we shall make the following definition for a general case of $G$
(without imposing the assumption that $r$ is even).

######  Definition 4.5.

We shall say that a $G$-oper on $\mathscr{X}$ is permissible if it is
isomorphic to $\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})$ for some
$G^{\odot}$-oper $\mathscr{E}^{\spadesuit}_{\odot}$ classified by
$\mathrm{Op}_{P}(\mathscr{X},G^{\odot})$. (In particular, a $G^{\odot}$-oper
is permissible if and only if it is classified by
$\mathrm{Op}_{P}(\mathscr{X},G^{\odot})$.)

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a permissible
$G$-oper on $\mathscr{X}$. Choose a $G^{\odot}$-oper
$\mathscr{E}_{\odot}^{\spadesuit}:=(\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
classified by $\mathrm{Op}_{P}(\mathscr{X},G^{\odot})$ with
$\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})\cong\mathscr{E}^{\spadesuit}$.
(Such a $G^{\odot}$-oper is uniquely determined up to isomorphism; see [Wak1,
Proposition 2.9, Remark 2.37].) We shall write $\varpi:\mathbb{H}\rightarrow
X^{*}$ for a covering map from the upper half plane $\mathbb{H}$ onto $X^{*}$
arising from the Fuchsian uniformization. Then, the pull-back
$\varpi^{*}(\mathscr{E}^{\spadesuit}_{\odot}):=(\varpi^{*}(\mathcal{E}_{B^{\odot}}),\varpi^{*}(\nabla_{\odot}))$
of $\mathscr{E}^{\spadesuit}_{\odot}$ specifies a $G^{\odot}$-oper on
$\mathbb{H}$. Since $\mathbb{H}$ is contractible, there exists an isomorphism
of flat $G^{\odot}$-bundles

(104)
$\displaystyle\alpha:(\varpi^{*}(\mathcal{E}_{G^{\odot}}),\varpi^{*}(\nabla_{\odot}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathbb{H}\times
G^{\odot},\nabla^{\mathrm{triv}}|_{\mathbb{H}}),$

which induces an isomorphism of $\mathbb{P}^{1}$-bundles
$\alpha_{\mathbb{P}}:\varpi^{*}(\mathcal{E}_{G^{\odot}})\times^{G^{\odot}}\mathbb{P}^{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}\times\mathbb{P}^{1}$.

Denote by $\sigma$ the global section
$\mathbb{H}\rightarrow\varpi^{*}(\mathcal{E}_{G^{\odot}})\times^{G^{\odot}}\mathbb{P}^{1}$
of the natural projection
$\varpi^{*}(\mathcal{E}_{G^{\odot}})\times^{G^{\odot}}\mathbb{P}^{1}\rightarrow\mathbb{H}$
corresponding to the $B^{\odot}$-reduction
$\varpi^{*}(\mathcal{E}_{B^{\odot}})$ of
$\varpi^{*}(\mathcal{E}_{G^{\odot}})$. It follows from the definition of a
$G^{\odot}$-oper that the differential of the composite

(105)
$\displaystyle\mathrm{dev}_{\mathscr{E}^{\spadesuit}}:\mathbb{H}\xrightarrow{\sigma}\varpi^{*}(\mathcal{E}_{G^{\odot}})\times^{G^{\odot}}\mathbb{P}^{1}\xrightarrow{\alpha_{\mathbb{P}}}\mathbb{H}\times\mathbb{P}^{1}\xrightarrow{\mathrm{pr}_{2}}\mathbb{P}^{1}$

is nowhere vanishing. The morphism $\mathrm{dev}_{\mathscr{E}^{\spadesuit}}$
is independent of the choice of $\alpha$ up to post-composition with an
automorphism of $\mathbb{P}^{1}$; we refer to it as the developing map of
$\mathscr{E}^{\spadesuit}$.

Note that the square diagrams

(114) $\displaystyle\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.6028pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 95.27827pt\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
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114.19006pt\raise-40.25pt\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
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46.14224pt\raise-58.2794pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
106.38728pt\raise-51.88776pt\hbox{\hbox{\kern 0.0pt\raise
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14.22636pt\vbox{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.52698pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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are commutative, and the upper horizontal arrow of the left-hand square
preserves both the $G^{\odot}$-action and the log connection. Hence, these
squares together yield an isomorphism of $G^{\odot}$-opers
$\alpha_{\mathscr{E}^{\spadesuit}_{\odot}}:\varpi^{*}(\mathscr{E}^{\spadesuit}_{\odot})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{dev}_{\mathscr{E}^{\spadesuit}}^{*}(\mathcal{E}_{B^{\odot}}^{\mathrm{triv}},\nabla^{\mathrm{triv}})$
on $\mathbb{H}$; it extends, via change of structure group by $\iota_{G}$, to
an isomorphism of $G$-opers

(115)
$\displaystyle\alpha_{\mathscr{E}^{\spadesuit}}:\varpi^{*}(\mathscr{E}^{\spadesuit})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\iota_{G*}(\mathrm{dev}_{\mathscr{E}^{\spadesuit}}^{*}(\mathcal{E}^{\mathrm{triv}}_{B^{\odot}},\nabla^{\mathrm{triv}}))\left(=\mathrm{dev}_{\mathscr{E}^{\spadesuit}}^{*}(\iota_{G*}(\mathcal{E}^{\mathrm{triv}}_{B^{\odot}},\nabla^{\mathrm{triv}}))\right),$

defining an isomorphism between the underlying $G$-bundles
$\varpi^{*}(\mathcal{E}_{G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}\times
G$.

##  5\. Monodromy of $G$-opers

In this section, we consider the parabolic de Rham cohomology associated to a
flat $G$-bundle and a comparison with the parabolic cohomology of the
corresponding local system (cf. Proposition 5.2). The resulting isomorphism
between these cohomology groups will be used to describe the differential of
the morphism
$\mathrm{Op}_{P}(\mathscr{X},G)\rightarrow\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$
(cf. (131)) arising from taking the monodromy maps.

Let $\mathscr{X}$ be as in § 4, and we keep the various notations involved.

###  5.1. Parabolic de Rham cohomology

Each morphism $\nabla^{\prime}:\mathcal{K}^{0}\rightarrow\mathcal{K}^{1}$ of
sheaves may be regarded as a complex concentrated at degrees $0$ and $1$; we
denote this complex by $\mathcal{K}^{\bullet}[\nabla^{\prime}]$. (In
particular, we have $\mathcal{K}^{j}[\nabla^{\prime}]:=\mathcal{K}^{j}$ for
$j=0,1$). For each $l\in\mathbb{Z}_{\geq 0}$, 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 $\mathcal{F}$, 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 holomorphic vector bundle with log connection
$\mathscr{F}:=(\mathcal{F},\nabla:\mathcal{F}\rightarrow\Omega\otimes\mathcal{F})$
on $\mathscr{X}$, we set $\Omega^{0}_{\mathrm{par}}(\mathcal{F}):=\mathcal{F}$
and

(116)
$\displaystyle\Omega^{1}_{\mathrm{par}}(\mathcal{F}):=\Omega\otimes\mathcal{F}(-D)+\mathrm{Im}(\nabla)\left(\subseteq\Omega\otimes\mathcal{F}\right).$

Then, $\nabla$ restricts to a $\mathbb{C}$-linear morphism

(117)
$\displaystyle\nabla_{\mathrm{par}}:\Omega^{0}_{\mathrm{per}}(\mathcal{F})\rightarrow\Omega^{1}_{\mathrm{per}}(\mathcal{F}).$

The hypercohomology group
$\mathbb{H}^{l}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{per}}])$
($l\in\mathbb{Z}_{\geq 0}$) associated to the complex
$\mathcal{K}^{\bullet}[\nabla_{\mathrm{per}}]$ is called the $l$-th parabolic
de Rham cohomology group of $\nabla$.

Let us suppose that the residue $\mathrm{Res}_{\sigma_{i}}(\nabla)$ at
$\sigma_{i}$ is nilpotent for every $i$. Also, let $U_{i}$ be as in § 3.5, and
denote by $\nu_{i}:U_{i}\hookrightarrow X$ the natural inclusion.

######  Proposition 5.1.

The following sequence is exact:

(118) $\displaystyle
0\longrightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])\longrightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla])\xrightarrow{\bigoplus_{i=1}^{r}\nu_{i}^{*}}\bigoplus_{i=1}^{r}\mathbb{H}^{1}(U_{i},\mathcal{K}^{\bullet}[\nabla|_{U_{i}}]),$

where the second arrow arises from the natural inclusion
$\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}]\hookrightarrow\mathcal{K}[\nabla]$.
(By using the second arrow, we occasionally regard
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])$ as a subspace
of $\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla])$.)

###### Proof.

The sheaf $\mathcal{F}$ has a decreasing filtration
$\\{\mathcal{F}(-nD)\\}_{n\in\mathbb{Z}_{\geq 0}}$ (where
$\mathcal{F}(-nD):=\mathcal{F}\otimes\mathcal{O}_{X}(-nD)$) with
$\bigcap_{n\in\mathbb{Z}_{\geq 0}}\mathcal{F}(-nD)=\\{0\\}$. By the nilpotency
assumption on $\mathrm{Res}_{\sigma_{i}}(\nabla)$’s, the morphism between
graded pieces

(119)
$\displaystyle\overline{\nabla}_{n}:\mathcal{F}(-nD)/\mathcal{F}(-(n+1)D)\rightarrow\Omega\otimes(\mathcal{F}(-nD)/\mathcal{F}(-(n+1)D))$

induced by $\nabla$ is an isomorphism for every $n\in\mathbb{Z}_{>0}$. Hence,
since
$\overline{\nabla}_{0}=\bigoplus_{i=1}^{r}\sigma_{i*}(\mathrm{Res}_{\sigma_{i}}(\nabla))$,
the natural morphism

(120)
$\displaystyle\bigoplus_{i=1}^{r}\mathbb{H}^{1}(U_{i},\mathcal{K}^{\bullet}[\nabla|_{U_{i}}])\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\left(\bigoplus_{i=1}^{r}\mathbb{H}^{1}(U_{i},\mathcal{K}^{\bullet}[\sigma_{i*}(\mathrm{Res}_{\sigma_{i}}(\nabla))])=\right)\bigoplus_{i=1}^{r}\mathrm{Coker}(\mathrm{Res}_{\sigma_{i}}(\nabla))$

is an isomorphism. On the other hand, let us consider the natural short exact
sequence of complexes

(121) $\displaystyle
0\longrightarrow\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}]\xrightarrow{\mathrm{inclusion}}\mathcal{K}^{\bullet}[\nabla]\longrightarrow\bigoplus_{i=1}^{r}\sigma_{i*}(\mathrm{Coker}(\mathrm{Res}_{\sigma_{i}}(\nabla)))[-1]\longrightarrow
0.$

It induces the exact sequence of $\mathbb{C}$-vector spaces

(122) $\displaystyle
0\longrightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{per}}])\longrightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla])\longrightarrow\bigoplus_{i=1}^{r}\mathrm{Coker}(\mathrm{Res}_{\sigma_{i}}(\nabla)).$

By composing the last arrow in this sequence with the inverse of (120), we
obtain (118). In particular, (118) turns out to be exact, as desired. ∎

Denote by $\mathbb{L}$ the local system on $X^{*}$ corresponding to
$(\mathcal{F},\nabla)|_{X^{*}}$ via the Riemann-Hilbert correspondence. That
is, $\mathbb{L}$ is obtained as the sheaf of horizontal sections of
$\nabla|_{X^{*}}$. It is well-known that the inclusion of complexes
$\mathbb{L}[0]\rightarrow\mathcal{K}^{\bullet}[\nabla|_{X^{*}}]$ induces an
isomorphism
$H^{1}(X^{*},\mathbb{L})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(X^{*},\mathcal{K}^{\bullet}[\nabla|_{X^{*}}])$.
Thus, we obtain a composite isomorphism

(123) $\displaystyle
H^{1}(X^{*},\mathbb{L})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(X^{*},\mathcal{K}^{\bullet}[\nabla|_{X^{*}}])\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla]),$

where the second arrow follows from [Del, Chap. II, Corollaire 3.15].

######  Proposition 5.2.

(Recall that $\mathrm{Res}_{\sigma_{i}}(\nabla)$ is assumed to be nilpotent
for every $i$.) The isomorphism (123) restricts to an isomorphism of
$\mathbb{C}$-vector spaces

(124) $\displaystyle
H^{1}_{P}(X,\mathbb{L})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}]).$

###### Proof.

Note that both (123) and the natural isomorphisms
$H^{1}(U_{i}^{*},\mathbb{L}|_{U_{i}^{*}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(U_{i},\mathcal{K}^{\bullet}[\nabla|_{U_{i}}])$
(cf. [Con, Theorem 4.2]) make the following diagram commute:

(129)

where the upper and lower horizontal sequences are (45) and (118),
respectively. Hence, the desired isomorphism can be obtained by restricting
(123) to the kernels of the rightmost arrows in the upper and lower sequences
in this diagram. ∎

###  5.2. Monodromy of a $G$-oper

Let us prove the following assertion.

######  Proposition 5.3.

The assignment from each $G$-oper
$\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ to the associated flat
$G$-bundle $(\mathcal{E}_{G},\nabla)$ (where
$\mathcal{E}_{G}:=\mathcal{E}_{B}\times^{B}G$) determines a morphism of
$\mathbb{C}$-stacks

(130)
$\displaystyle\mathrm{Op}_{P}(\mathscr{X},G)\rightarrow\mathcal{C}onn_{P}^{0}(\mathscr{X},G).$

Moreover, this morphism is schematic and an immersion.

###### Proof.

To begin with, we shall prove the former assertion. Let
$\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a $G$-oper classified
by $\mathrm{Op}_{P}(\mathscr{X},G)$. It follows from [Wak1, Proposition 2.9]
that this $G$-oper does not have non-trivial automorphisms, i.e., satisfies
the condition (b) described in § 3.8. For the condition (c), we refer to the
discussion in Remark 4.2, (ii). Thus, the remaining portion of the former
assertion is to prove the condition (a), i.e., that $\mathscr{E}^{\spadesuit}$
is irreducible. Suppose, on the contrary, that there exist a parabolic
subgroup $P$ of $G$ and a $P$-reduction $\mathcal{E}_{P}$ of
$\mathcal{E}_{G}|_{X^{*}}$ preserving the connection $\nabla|_{X^{*}}$. After
possibly replacing $P$ with its conjugate by some element of $B$, we may
assume that $P$ contains $B$. Then, for a local chart $(V,\partial)$, we have
$\mathrm{triv}_{B,(V,\partial)}^{*}(\nabla)^{\mathfrak{g}}\in
H^{0}(V,\mathcal{O}_{X}\otimes\mathfrak{p})$, where $\mathfrak{p}$ denotes the
Lie algebra of $P$. By a well-known fact on Lie algebras, there exists a
proper subset $I\subseteq\Phi$ such that $\mathfrak{p}$ is generated by the
subspaces $\mathfrak{g}^{\alpha}$ for $\alpha\in\Phi$ and
$\mathfrak{g}^{-\alpha}$ for $\alpha\in I$ (cf. [Hum, § 16, Exercise 6]). In
particular, since $I\neq\Phi$, we have $q_{-1}\notin\mathfrak{p}$. This
contradicts the fact that
$\mathrm{triv}_{B,(V,\partial)}^{*}(\nabla)^{\mathfrak{g}}$ belongs to
$H^{0}(V,\mathcal{O}_{X}\otimes\mathfrak{p})$. This proves the irreducibility
of $\mathscr{E}^{\spadesuit}$, thus completing the proof of the former
assertion.

Moreover, the latter assertion follows from [Wak2, Theorem 3.1.1]. ∎

We shall regard $\mathrm{Op}_{P}(\mathscr{X},G)$ as a complex submanifold of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ (of half-dimension) via the
composite injection

(131)
$\displaystyle\mathrm{Op}_{P}(\mathscr{X},G)\xrightarrow{(\ref{EEE55})}\mathcal{C}onn_{P}^{0}(\mathscr{X},G)\xrightarrow{(\ref{ER3})}\mathcal{R}ep_{P}^{0}(\Gamma,G(\mathbb{C}))\rightarrow\mathrm{Rep}^{0}_{P}(\Gamma,G(\mathbb{C})),$

where the last arrow denotes the functor given by sending each object in the
domain to its isomorphism class.

Now, let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a $G$-oper
classified by $\mathrm{Op}_{P}(\mathscr{X},G)$. The monodromy map of the flat
$G$-bundle $(\mathcal{E}_{G},\nabla)$ restricted to $X^{*}$ specifies a
homomorphism

(132) $\displaystyle\mathrm{Mon}_{\mathscr{E}^{\spadesuit}}:\Gamma\rightarrow
G(\mathbb{C}),$

which is determined up to post-composition with an inner automorphism of
$G(\mathbb{C})$. In particular, the point of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ classifying
$\mathrm{Mon}_{\mathscr{E}^{\spadesuit}}$ coincides with the image via (131)
of the point of $\mathrm{Op}_{P}(\mathscr{X},G)$ classifying
$\mathscr{E}^{\spadesuit}$.

Next, suppose further that $\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal.
Write $\mu:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}}$, and write
$\nabla^{\mathrm{ad}}$ for the log connection on $\mathfrak{g}_{\mathcal{E}}$
induced from $\nabla$ via change of structure group by $\mathrm{Ad}_{G}$. In
particular, we obtain a flat vector bundle
$(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$, which corresponds to the
$\mathbb{C}[\Gamma]$-module $\mathfrak{g}(\mathbb{C})_{\mu}$, as well as to
the $\mathfrak{g}$-valued local system $\mathbb{L}_{\mu,\mathbb{C}}$ (cf.
(60)). By applying Propositions 3.6 and 5.2, we obtain a composite injection
between $\mathbb{C}$-vector spaces

(133)
$\displaystyle\varepsilon:H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))\hookrightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{P}(X^{*},\mathbb{L}_{\mu,\mathbb{C}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu}),$

where the first arrow is the injection associated to the morphism of complexes
${{}^{\dagger}}\mathcal{V}_{G}(-D)[-1]\hookrightarrow\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}]$
given by the inclusion
${{}^{\dagger}}\mathcal{V}_{G}(-D)\hookrightarrow\Omega_{\mathrm{par}}^{1}(\mathfrak{g}_{{{}^{\dagger}}\mathcal{E}_{G}})$
(cf. [Wak1, Proposition 6.5, (ii)]). By using this injection, we will regard
$H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))$ as a subspace of
$H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$.

If $T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},G)$ (resp.,
$T_{[\mu]}\mathrm{Rep}(\Gamma,G(\mathbb{C}))$) denotes the tangent space of
$\mathrm{Op}_{P}(\mathscr{X},G)$ (resp., $\mathrm{Rep}(\Gamma,G(\mathbb{C}))$)
at the point classifying $\mathscr{E}^{\spadesuit}$ (resp., $[\mu]$), it
follows from the various definitions involved that the following diagram is
commutative:

(138)

where the left-hand vertical arrow arises from the
$H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))$-affine structure on
$\mathrm{Op}_{P}(\mathscr{X},G)$ (cf. § 4.4). In particular, the two subspaces
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))$,
$T_{\mathscr{E}^{\spadesuit}}\mathrm{Op}_{P}(\mathscr{X},G)$ of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ may be identified with the two
subspaces $H^{1}_{P}(X,\mathfrak{g}(\mathbb{R})_{\mu})$,
$H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))$ of
$H^{1}_{P}(X,\mathfrak{g}(\mathbb{C})_{\mu})$ via the vertical arrows in (66)
and (138).

######  Definition 5.4.

We shall say that a $G$-oper has real monodromy if the image of its monodromy
map is contained in $G(\mathbb{R})$ up to $G(\mathbb{C})$-conjugation.

In particular, the intersection

(139)
$\displaystyle\mathrm{Op}_{P}(\mathscr{X},G(\mathbb{R})):=\mathrm{Rep}_{P}^{0}(\gamma,G(\mathbb{R}))\cap\mathrm{Op}_{P}(\mathscr{X},G)$

forms the moduli space parametrizing $G$-opers having real monodromy each of
whose radii coincides with $[0]\in\mathfrak{c}$.

##  6\. Proofs of the main theorems

This final section is devoted to proving the theorems described in
Introduction. As observed previously, we have the equality between the
dimensions of real manifolds

(140)
$\displaystyle\mathrm{dim}(\mathrm{Op}_{P}(\mathscr{X},G))+\mathrm{dim}(\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R})))=\mathrm{dim}(\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C})))\left(=2\cdot
a(G)\right).$

Hence, it is natural to ask whether or not the intersection of the two real
submanifolds $\mathrm{Op}_{P}(\mathscr{X},G)$,
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))$ of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ forms a discrete set, i.e., they
intersect transversally. We first provide a partial answer to this question by
proving Theorem A (cf. Theorem 6.3). After that, we use this result to
establish the Eichler-Shimura isomorphisms for $G^{\odot}$-opers and the
associated Hodge structures; this completes the proof of Theorem B (cf.
Theorems 6.6, 6.8).

Let $\mathscr{X}$ be as in § 4, and we keep the various notations involved.

###  6.1. Bilinear form on the parabolic cohomology

Before proving Theorem A, we shall prepare some notions. For each Riemann
surface $X^{\prime}$, we denote by $\mathcal{E}_{X^{\prime}}^{(0)}$ (resp.,
$\mathcal{E}^{(1)}_{X^{\prime}}$; resp., $\mathcal{E}_{X^{\prime}}^{(2)}$) the
sheaf of $C^{\infty}$ functions (resp., $C^{\infty}$ differential $1$-forms;
resp., $C^{\infty}$ differential $2$-forms) on $X^{\prime}$.

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a $G$-oper
classified by $\mathrm{Op}_{P}(\mathscr{X},G(\mathbb{R}))$. We may assume that
the image of the monodromy map $\mu:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}}$
is contained in $G(\mathbb{R})$. Recall from [Wak2, Corollary 4.6] that the
following bilinear form on
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])$
is skew-symmetric and nondegenerate:

(141)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])\times\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])\rightarrow
H^{1}(X,\Omega_{X})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{C},$

where

  * •

the first arrow is the morphism induced by the cup product on parabolic de
Rham cohomology and the Killing form
$\kappa:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{C}$ on
$\mathfrak{g}$;

  * •

the second arrow is the isomorphism arising from Serre duality, which is