explicitly given by
$[\omega]\mapsto\frac{1}{2\pi\sqrt{-1}}\cdot\iint_{X}\omega$ for each global
section $\omega$ of $\mathcal{E}^{(2)}_{X}$ representing an element of
$H^{1}(X,\Omega_{X})$ by the Dolbeault theorem.

Under the identification
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}_{\mathrm{par}}])=H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$
resulting from Propositions 3.6 and 5.2, it defines a nondegenerate bilinear
form

(142)
$\displaystyle(-,-)_{\mathscr{E}^{\spadesuit}}:H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})\times
H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})\rightarrow\mathbb{C}$

on $H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$. Also, we obtain a
sesquilinear pairing

(143)
$\displaystyle\langle-,-\rangle_{\mathscr{E}^{\spadesuit}}:H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})\times
H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})\rightarrow\mathbb{C}$

given by $\langle
v,w\rangle_{\mathscr{E}^{\spadesuit}}:=(-1)\cdot(v,\overline{w})_{\mathscr{E}^{\spadesuit}}$.
Since $(-,-)_{\mathscr{E}^{\spadesuit}}$ is skew-symmetric and the subspace
$H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})$ of
$H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$ is invariant under
$\overline{(-)}$, we have $\langle v,v\rangle_{\mathscr{E}^{\spadesuit}}=0$
for any $v\in H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})$.

Next, we shall describe $\langle-,-\rangle_{\mathscr{E}^{\spadesuit}}$ in
terms of integration of functions. Let $\varpi:\mathbb{H}\twoheadrightarrow
X^{*}$ denote a covering map from
$\mathbb{H}\left(=\\{z\in\mathbb{C}\,|\,\mathrm{Im}(z)>0\\}\right)$ to $X^{*}$
arising from the Fuchsian uniformization. Since $\mathbb{H}$ is contractible,
there exists an identification $\alpha_{\mathfrak{g}}$ of
$\varpi^{*}(\mathbb{L}_{\mu,\mathbb{C}})$ with the trivial
$\mathfrak{g}$-valued local system on $\mathbb{H}$. The exact sequence

(144) $\displaystyle
0\rightarrow\mathbb{L}_{\mu,\mathbb{C}}\xrightarrow{\mathrm{inclusion}}\mathbb{L}_{\mu,\mathbb{C}}\otimes_{\mathbb{C}}\mathcal{E}_{X^{*}}^{(0)}\xrightarrow{d_{0}}\mathbb{L}_{\mu,\mathbb{C}}\otimes_{\mathbb{C}}\mathcal{E}_{X^{*}}^{(1)}\xrightarrow{d_{1}}\mathbb{L}_{\mu,\mathbb{C}}\otimes_{\mathbb{C}}\mathcal{E}_{X^{*}}^{(2)}\rightarrow
0$

defines a fine resolution of $\mathbb{L}_{\mu,\mathbb{C}}$, where $d_{i}$
denotes the $i$-th exterior derivative. Note that
$H_{P}^{1}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$ may be identified with
$H^{1}_{P}(X^{*},\mathbb{L}_{\mu,\mathbb{C}})$ (by Proposition 3.6), which
coincides with the image of the natural morphism
$H^{1}_{c}(X^{*},\mathbb{L}_{\mu,\mathbb{C}})\rightarrow
H^{1}(X^{*},\mathbb{L}_{\mu,\mathbb{C}})$ (cf. [Loo, Lemma 5.3]). Hence, each
element $v$ of $H_{P}^{1}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$ can be
represented, via pull-back along $\varpi$ and using the identification
$\alpha_{\mathfrak{g}}$, by $v_{[z]}dz+v_{[\overline{z}]}d\overline{z}$ for
some $v_{[z]},v_{[\overline{z}]}\in
H^{0}_{c}(\mathbb{H},\mathcal{E}_{\mathbb{H}}^{(0)}\otimes_{\mathbb{R}}\mathfrak{g}_{\mathbb{R}})$.
By using such a representation, we can describe the pairing
$\langle-,-\rangle_{\mathscr{E}^{\spadesuit}}$ as

(145) $\displaystyle\langle v,w\rangle_{\mathscr{E}^{\spadesuit}}$
$\displaystyle=\frac{-1}{2\pi\sqrt{-1}}\cdot\iint_{F}\left(\kappa(v_{[z]},\overline{w_{[\overline{z}]}})-\kappa(v_{[\overline{z}]},\overline{w_{[z]}})\right)dz\wedge
d\overline{z}$
$\displaystyle\left(=\frac{1}{\pi}\cdot\iint_{F}\left(\kappa(v_{[z]},\overline{w_{[\overline{z}]}})-\kappa(v_{[\overline{z}]},\overline{w_{[z]}})\right)dx\wedge
dy\right),$

where $F$ denotes a fundamental domain for
$\Gamma\left(\subseteq\mathrm{PSL}_{2}(\mathbb{R})\right)$ in $\mathbb{H}$,
and $\overline{(-)}$ denotes the complex conjugation on
$H^{0}_{c}(\mathbb{H},\mathcal{E}_{\mathbb{H}}^{(0)}\otimes_{\mathbb{R}}\mathfrak{g}_{\mathbb{R}})$
given by $f\otimes v\mapsto\overline{f}\otimes v$ for
$f\in\mathcal{E}_{\mathbb{H}}^{(0)}$, $v\in\mathfrak{g}_{\mathbb{R}}$.

###  6.2. Hermitian inner product on the maximal torus

We abuse notation by writing $\kappa$ for the bilinear form on
$\mathcal{O}_{\mathbb{H}}\otimes_{\mathbb{C}}\mathfrak{g}$ induced naturally
from the Killing form $\kappa$ on $\mathfrak{g}$. Recall that the pairing
$\mathfrak{t}\times\mathfrak{t}\rightarrow\mathbb{C}$ given by
$(v,w)\mapsto\kappa(v,\overline{w})$ specifies a hermitian inner product on
$\mathfrak{t}$ (cf. [FuHa, § 14.2, (14. 22)]). For each $j\in\mathbb{Z}_{\geq
0}$ with
$\left(\mathfrak{g}_{\mathbb{R},j}\otimes_{\mathbb{R}}\mathbb{C}=:\right)\mathfrak{g}_{j}\neq\\{0\\}$,
the morphism $\mathrm{ad}(q_{-1})^{j}:\mathfrak{g}_{j}\rightarrow\mathfrak{t}$
(where $\mathrm{ad}(q_{-1})^{0}:=\mathrm{id}_{\mathfrak{g}_{0}}$) is
injective. Hence, since $q_{-1}$ has been chosen as an element of
$\mathfrak{g}_{\mathbb{R},-1}$, we obtain a hermitian metric

(146) $\displaystyle\langle-,-\rangle_{\mathbb{H},j}$

on $\mathcal{O}_{\mathbb{H}}\otimes_{\mathbb{C}}\mathfrak{g}_{j}$ defined as
the trivial family over $\mathbb{H}$ of the hermitian inner product on
$\mathfrak{g}_{j}$ given by
$(v,w)\mapsto\kappa(\mathrm{ad}(q_{-1})^{j}(v),\mathrm{ad}(q_{-1})^{j}(\overline{w}))$.

Next, let $j$ (resp., $j^{\prime}$) be an integer and $v_{j}$ (resp.,
$v_{j^{\prime}}$) a local section of
$\mathcal{O}_{\mathbb{H}}\otimes_{\mathbb{C}}\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})}$
(resp.,
$\mathcal{O}_{\mathbb{H}}\otimes_{\mathbb{C}}\mathfrak{g}_{j^{\prime}}^{\mathrm{ad}(q_{1})}$).
Since $\mathrm{ad}(q_{-1})^{l}(\mathfrak{g}_{j})\subseteq\mathfrak{g}_{j-l}$
for any $j$, $l$ and
$\mathrm{ad}(q_{-1})^{l}(\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})})=\\{0\\}$ for
any $l\geq 2j+1$, the associativity of the Killing form $\kappa$ (i.e.,
$\kappa(\mathrm{ad}(a)(b),c)=-\kappa(b,\mathrm{ad}(a)(c))$ for
$a,b,c\in\mathfrak{g}$) implies

(147)
$\displaystyle\kappa(\mathrm{ad}(q_{-1})^{l}(v_{j}),\mathrm{ad}(q_{-1})^{l^{\prime}}(\overline{v_{j^{\prime}}}))=\begin{cases}(-1)^{l-j}\cdot\langle
v_{j},v_{j}\rangle_{\mathbb{H},j}&\text{if $j=j^{\prime}$ and
$j+j^{\prime}=l+l^{\prime}$}\\\ 0&\text{if otherwise.}\end{cases}.$

###  6.3. Discreteness of the intersection

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a permissible
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
$G$-oper on $\mathscr{X}$. Write
$\delta:=\mathrm{dev}_{\mathscr{E}^{\spadesuit}}$, i.e., the developing map of
$\mathscr{E}^{\spadesuit}$ (cf. (105)).

######  Theorem 6.1.

For each $i=1,2$, let $v_{i}:=\sum_{j=1}^{\mathrm{rk}(G)}v_{ij}$ be an element
of
$H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))=\bigoplus_{j=1}^{\mathrm{rk}(G)}H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G,j}(-D))$,
where $v_{ij}\in H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G,j}(-D))$. Note that the
pull-back $\varpi^{*}(v_{ij})$ can be described as $dz\otimes w_{ij}$ for some
$w_{ij}\in
H^{0}(\mathbb{H},\mathcal{O}_{\mathbb{H}}\otimes_{\mathbb{C}}\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})})$.
Then, we have

(148) $\displaystyle\langle v_{1},v_{2}\rangle_{\mathscr{E}^{\spadesuit}}$
$\displaystyle=\frac{-1}{2\pi\sqrt{-1}}\cdot\sum_{j=1}^{\mathrm{rk}(G)}\frac{2^{2j}}{(2j)!}\iint_{F}\frac{\mathrm{Im}(\delta)^{2j}\cdot\langle
w_{1j},w_{2j}\rangle_{\mathbb{H},j}}{|\delta^{\prime}|^{2j}}dz\wedge
d\overline{z}$
$\displaystyle\left(=\frac{1}{\pi}\cdot\sum_{j=1}^{\mathrm{rk}(G)}\frac{2^{2j}}{(2j)!}\iint_{F}\frac{\mathrm{Im}(\delta)^{2j}\cdot\langle
w_{1j},w_{2j}\rangle_{\mathbb{H},j}}{|\delta^{\prime}|^{2j}}dx\wedge
dy\right),$

where $F$ denotes a fundamental domain for
$\Gamma\left(\subseteq\mathrm{PSL}_{2}(\mathbb{R})\right)$ in $\mathbb{H}$. In
particular, the restriction of $\langle-,-\rangle_{\mathscr{E}^{\spadesuit}}$
to $H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))$ forms a hermitian inner
product.

###### Proof.

We shall write
$\mathscr{E}_{\odot}^{\spadesuit}:=(\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
for the $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normal $G^{\odot}$-oper with
$\iota_{G*}(\mathscr{E}_{\odot}^{\spadesuit})\cong\mathscr{E}^{\spadesuit}$.
Since the pair $(\mathbb{H},\frac{d}{dz})$ specifies a globally defined (log)
chart on $\mathbb{H}$, we obtain a composite automorphism

(149) $\displaystyle\mathbb{H}\times
G^{\odot}\xrightarrow{\mathrm{triv}_{G^{\odot},(\mathbb{H},\frac{d}{dz})}}{{}^{\dagger}}\mathcal{E}_{G^{\odot},\mathbb{H}}\left(=\varpi^{*}(\mathcal{E}_{G^{\odot},X^{*}})\right)\xrightarrow{\alpha}\mathbb{H}\times
G^{\odot}$

of the trivial $G^{\odot}$-bundle $\mathbb{H}\times G^{\odot}$, where $\alpha$
denotes the isomorphism defined as in (104). The diagonal embedding
$\mathbb{H}\hookrightarrow\mathbb{H}\times\mathbb{P}^{1}$ is compatible with
$(\mathrm{id}_{\mathbb{H}},\delta):\mathbb{H}\rightarrow\mathbb{H}\times\mathbb{P}^{1}$
via the induced automorphism of $\mathbb{H}\times\mathbb{P}^{1}$. It follows
that (149) coincides with the left-translation by the image $\overline{h}$,
via the quotient $\mathrm{SL}_{2}\twoheadrightarrow G^{\odot}$, of the
morphism $h:\mathbb{H}\rightarrow\mathrm{SL}_{2}$ expressed (on a dense open
subset of $\mathbb{H}$) as

(150) $\displaystyle h:=\begin{pmatrix}1/\sqrt{\delta^{\prime}}&0\\\
\delta/\sqrt{\delta^{\prime}}&\sqrt{\delta^{\prime}}\end{pmatrix}\left(=\begin{pmatrix}1&0\\\
\delta&1\end{pmatrix}\begin{pmatrix}1/\sqrt{\delta^{\prime}}&0\\\
0&\sqrt{\delta^{\prime}}\end{pmatrix}\right),$

where $\sqrt{\delta^{\prime}}$ denotes a square root of the derivative
$\delta^{\prime}$ of $\delta$. We shall set $h_{1}:=\begin{pmatrix}1&0\\\
\delta&1\end{pmatrix}$, $h_{2}:=\begin{pmatrix}1/\sqrt{\delta^{\prime}}&0\\\
0&\sqrt{\delta^{\prime}}\end{pmatrix}$ (hence $h=h_{1}h_{2}$), and write
$\overline{h}_{1}$ and $\overline{h}_{2}$ for the images, via the quotient
$\mathrm{SL}_{2}\twoheadrightarrow G^{\odot}$, of $h_{1}$ and $h_{2}$,
respectively. When we identify
$\varpi^{*}(\mathcal{E}_{G,X^{*}})\left(={{}^{\dagger}}\mathcal{E}_{G,\mathbb{H}}\right)$
with $\mathbb{H}\times G$ via $\mathrm{triv}_{G,(\mathbb{H},\frac{d}{dz})}$,
the connection $\varpi^{*}(\nabla|_{X^{*}})$ may be trivialized after applying
the gauge transformation by the left-translation by
$\iota_{G}(\overline{h})\left(=\iota_{G}(\overline{h}_{1})\iota_{G}(\overline{h}_{2})\right)$.
The automorphism $\mathrm{Ad}_{G}(\iota_{G}(\overline{h}))$ of
$\mathcal{O}_{\mathbb{H}}\otimes_{\mathbb{C}}\mathfrak{g}$ maps
$\varpi^{*}(v_{i})$ (considered as an element of
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}])$)
to the element

(151) $\displaystyle
dz\otimes\mathrm{Ad}_{G}(\overline{h})\left(\sum_{j=1}^{\mathrm{rk}(G)}w_{ij}\right)$
$\displaystyle=dz\otimes\mathrm{Ad}_{G}(\overline{h}_{1})\left(\sum_{j=1}^{\mathrm{rk}(G)}\frac{w_{ij}}{\delta^{\prime
j}}\right)$
$\displaystyle=dz\otimes\sum_{s=0}^{\infty}\frac{1}{s!}\cdot\mathrm{ad}(\mathrm{Log}(\overline{h}_{1}))^{s}\left(\sum_{j=1}^{\mathrm{rk}(G)}\frac{w_{ij}}{\delta^{\prime
j}}\right)$
$\displaystyle=dz\otimes\sum_{s=0}^{\infty}\frac{1}{s!}\cdot\mathrm{ad}(\delta
q_{-1})^{s}\left(\sum_{j=1}^{\mathrm{rk}(G)}\frac{w_{ij}}{\delta^{\prime
j}}\right),$

where $\mathrm{Log}(-)$ denotes the logarithm map (cf. [Wak1, § 1.4.2,
(114)]). It follows that the pairing $\langle
v_{1},v_{2}\rangle_{\mathscr{E}^{\spadesuit}}$ can be calculated by

(152) $\displaystyle\ \ \ \,\langle
v_{1},v_{2}\rangle_{\mathscr{E}^{\spadesuit}}$
$\displaystyle=\frac{-1}{2\pi\sqrt{-1}}\cdot\iint_{F}\kappa\Biggl{(}\sum_{s=0}^{\infty}\frac{1}{s!}\cdot\mathrm{ad}(\delta
q_{-1})^{s}\left(\sum_{j=1}^{\mathrm{rk}(G)}\frac{w_{1j}}{\delta^{\prime
j}}\right),$ $\displaystyle\hskip
113.81102pt\sum_{s=0}^{\infty}\frac{1}{s!}\cdot\mathrm{ad}(\overline{\delta}q_{-1})^{s}\left(\sum_{j=1}^{\mathrm{rk}(G)}\frac{\overline{w_{2j}}}{\overline{\delta^{\prime}}^{j}}\right)\Biggl{)}dz\wedge
d\overline{z}$
$\displaystyle=\frac{-1}{2\pi\sqrt{-1}}\cdot\sum_{j=1}^{\mathrm{rk}(G)}\iint_{F}\sum_{s=0}^{\infty}\kappa\Bigg{(}\frac{1}{s!}\cdot\mathrm{ad}(\delta
q_{-1})^{s}\left(\frac{w_{1j}}{\delta^{\prime j}}\right),$
$\displaystyle\hskip
156.49014pt\frac{1}{(2j-s)!}\cdot\mathrm{ad}(\overline{\delta}q_{-1})^{2j-s}\left(\frac{\overline{w_{2j}}}{\overline{\delta^{\prime}}^{j}}\right)\Bigg{)}dz\wedge
d\overline{z}$
$\displaystyle=\frac{-1}{2\pi\sqrt{-1}}\cdot\sum_{j=1}^{\mathrm{rk}(G)}(-1)^{j}\cdot\iint_{F}\left(\sum_{s=0}^{\infty}\frac{1}{s!\cdot(2j-s)!}\cdot\delta^{s}\cdot(-\overline{\delta})^{2j-s}\right)\cdot$
$\displaystyle\hskip
156.49014pt\kappa\left(\mathrm{ad}(q_{-1})^{j}\left(\frac{w_{1j}}{{\delta^{\prime}}^{j}}\right),\mathrm{ad}(q_{-1})^{j}\left(\frac{\overline{w_{2j}}}{\overline{\delta^{\prime}}^{j}}\right)\right)dz\wedge
d\overline{z}$
$\displaystyle=\frac{-1}{2\pi\sqrt{-1}}\cdot\sum_{j=1}^{\mathrm{rk}(G)}(-1)^{j}\cdot\iint_{F}\frac{1}{(2j)!}\cdot(2\sqrt{-1}\cdot\mathrm{Im}(\delta))^{2j}\cdot\frac{1}{|\delta^{\prime}|^{2j}}\cdot$
$\displaystyle\hskip
156.49014pt\kappa\left(\mathrm{ad}(q_{-1})^{j}\left(w_{1j}\right),\mathrm{ad}(q_{-1})^{j}\left(\overline{w_{2j}}\right)\right)dz\wedge
d\overline{z}$
$\displaystyle=\frac{-1}{2\pi\sqrt{-1}}\cdot\sum_{j=1}^{\mathrm{rk}(G)}\frac{2^{2j}}{(2j)!}\iint_{F}\frac{\mathrm{Im}(\delta)^{2j}\cdot\langle
w_{1j},w_{2j}\rangle_{\mathbb{H},j}}{|\delta^{\prime}|^{2j}}dz\wedge
d\overline{z},$

where the first equality follows from (145), and both the third and last
equalities follow from (147). This completes the proof of the assertion. ∎

######  Corollary 6.2.

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a permissible
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
$G$-oper on $\mathscr{X}$ having real monodromy. Note that the image of the
monodromy map $\mu:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}}$ can be assumed to
be contained in $G(\mathbb{R})$. (In particular, we obtain the
$\mathbb{R}[\Gamma]$-module $\mathfrak{g}(\mathbb{R})_{\mu}$.) Then, the
equality

(153) $\displaystyle H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})\cap
H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))=\\{0\\}$

between subspaces of $H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$ holds.

###### Proof.

Since the subspace $H_{P}^{1}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})$ of
$H_{P}^{1}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$ is isotropic with respect
to the pairing $\langle-,-\rangle_{\mathscr{E}^{\spadesuit}}$, it suffices to
show that $\langle q,q\rangle_{\mathscr{E}^{\spadesuit}}\neq 0$ for any $q\in
H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))\setminus\\{0\\}$. Thus, the
assertion follows from Theorem 6.1. ∎

By applying the above assertion, we obtain the following generalization of a
result by Faltings (cf. [Fal, Corollary]), which is the first main result of
the present paper.

######  Theorem 6.3 (cf. Theorem A).

The two real 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. In particular, the space of
permissible $G$-opers with real monodromy forms a discrete set.

###### Proof.

The assertion follows from Corollary 6.2 together with the commutativity of
the diagrams (66) and (138). ∎

When varying the underlying pointed Riemann surface, we may encounter special
situations where the intersection
$\mathrm{Op}_{P}(\mathscr{X},G(\mathbb{R}))\left(=\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))\cap\mathrm{Op}_{P}(\mathscr{X},G)\right)$
has, outside the locus of permissible $G$-opers, singular points or higher
dimensional components instead of forming a discrete set as a whole. However,
on a general pointed Riemann surface, we can expect the transversality
condition to be still satisfied at every point, as in the case of
$G=G^{\odot}$ already shown by Faltings. Regarding that manner, we shall
describe the following conjecture.

######  Conjecture 6.4.

Suppose that $\mathscr{X}$ is sufficiently general (in some sense). Then, the
two real submanifold $\mathrm{Op}_{P}(\mathscr{X},G)$,
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))$ of
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{C}))$ transversally intersect at every
point. That is to say, their intersection
$\mathrm{Op}_{P}(\mathscr{X},G(\mathbb{R}))$ forms a discrete set.

###  6.4. Eichler-Shimura decompositions for $G^{\odot}$-opers

Next, we establish the Eichler-Shimura isomorphism associated to a
$G^{\odot}$-oper. For each $\mathbb{C}$-vector space $V$, we denote by
$\overline{V}$ its complex conjugation.

Let $\mathscr{E}^{\spadesuit}_{\odot}$ be a $G^{\odot}$-oper classified by
$\mathrm{Op}_{P}(\mathscr{X},G^{\odot}(\mathbb{R}))$. One can suppose that the
image of the monodromy map
$\mu:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}_{\odot}}$ is contained in
$G(\mathbb{R})$. Note that the induced $G$-oper
$\mathscr{E}^{\spadesuit}:=\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})$
belongs to
$\mathrm{Rep}_{P}^{0}(\Gamma,G(\mathbb{R}))\cap\mathrm{Op}_{P}(\mathscr{X},G)$,
and that its monodromy map coincides (up to conjugation) with the composite
$\mu_{G}:\Gamma\xrightarrow{\mu}G^{\odot}(\mathbb{R})\hookrightarrow
G(\mathbb{R})$. To slightly simplify the notation, we write
$\mathfrak{g}(K)_{\mu}$ (where $K$ denotes either $\mathbb{R}$ or
$\mathbb{C}$) instead of $\mathfrak{g}(K)_{\mu_{G}}$.

Consider the $\mathbb{C}$-linear injection

(154)
$\displaystyle\overline{\varepsilon}:\overline{H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}\hookrightarrow\overline{H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu}),$

where the first arrow is the complex conjugation of $\varepsilon$ (cf. (133)),
and the second arrow is the $\mathbb{C}$-linear isomorphism given by $v\otimes
a\mapsto v\otimes\overline{a}$ for $v\in
H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})$ and $a\in\mathbb{C}$ under
the natural identification
$H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})=H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})\otimes_{\mathbb{R}}\mathbb{C}$.
This injection allows us to consider
$\overline{H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}$ as a
$\mathbb{C}$-vector subspace of
$H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$. By an argument similar to
the proof of Theorem 6.1, we see that the restriction of
$(-1)\cdot\langle-,-\rangle_{\mathscr{E}^{\spadesuit}}$ to
$\overline{H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}$ forms a hermitian
inner product.

Since $H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})$ may be identified
with the subspace of $H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$
invariant under the complex conjugation, it follows from Corollary 6.2 that
the morphism

(155)
$\displaystyle\varepsilon\oplus\overline{\varepsilon}:H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))\oplus\overline{H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}\rightarrow
H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$

is an isomorphism. (In fact, if we take an element $v$ of
$H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))\cap\overline{H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}$,
then $v+\overline{v}$ is invariant under the complex conjugation, which means
$v+\overline{v}\in H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})\cap
H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))=\\{0\\}$. Hence, the equality
$v+\overline{v}=0$ holds, so we have $\overline{\sqrt{-1}\cdot
v}=-\sqrt{-1}\cdot\overline{v}=\sqrt{-1}\cdot v$. It follows that
$\sqrt{-1}\cdot v\in H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{R})_{\mu})\cap
H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))=\\{0\\}$. This implies $v=0$,
and thus, we obtain
$H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))\cap\overline{H^{0}_{P}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}=\\{0\\}$.)
That is to say, the following assertion holds.

######  Theorem 6.5.

Let $\mathscr{E}^{\spadesuit}_{\odot}$ be a $G^{\odot}$-oper classified by
$\mathrm{Op}_{P}(\mathscr{X},G^{\odot}(\mathbb{R}))$ with monodromy map
$\mu\left(:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}_{\odot}}\right):\Gamma\rightarrow
G^{\odot}(\mathbb{R})$. Then, it induces a canonical decomposition

(156) $\displaystyle H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$
$\displaystyle=H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))\oplus\overline{H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}.$

of the $1$-st parabolic cohomology group
$H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu})$ of the
$\mathbb{C}[\Gamma]$-module $\mathfrak{g}(\mathbb{C})_{\mu}$.

Next, suppose further that $G=\mathrm{PSL}_{N}$ ($N\geq 2$). In particular,
since $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{N}$, we have
$\mathrm{dim}_{\mathbb{C}}(\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})})=1$ if $1\leq
j\leq N-1$, and
$\mathrm{dim}_{\mathbb{C}}(\mathfrak{g}_{j}^{\mathrm{ad}(q_{1})})=0$ if
otherwise.

Let $K$ denote either $\mathbb{R}$ or $\mathbb{C}$. Given each integer
$l\in\mathbb{Z}_{>0}$, we obtain the $K$-vector space

(157) $\displaystyle
V_{l,K}:=\mathrm{Sym}^{l}(Kx+Ky)\left(=\bigoplus_{a=0}^{l}Kx^{l-a}y^{a}\right)$

of homogenous degree $l$ polynomials in $K[x,y]$; it has an
$\mathrm{SL}_{2}(K)$-action given by $h(P)(x,y)=P(ax+cy,bx+dy)$ for any $P\in
V_{l,K}$ and $h:=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(K)$.
This action induces a $\Gamma$-action on $V_{l,K}$ by using a lifting
$\widetilde{\mu}:\Gamma\rightarrow\mathrm{SL}_{2}(K)$ of $\mu$. (According to
[Fal, Theorem 1], such a lifting always exists.) If $l$ is even, then the
$\Gamma$-action factors through the quotient
$\mathrm{SL}_{2}\twoheadrightarrow G^{\odot}$ and does not depend on the
choice of $\widetilde{\mu}$; so it makes sense to write

(158) $\displaystyle V_{l,K,\mu}$

for the resulting $K[\Gamma]$-module. It is clear that
$H^{1}_{P}(\Gamma,V_{l,\mathbb{R},\mu})\otimes_{\mathbb{R}}\mathbb{C}=H^{1}_{P}(\Gamma,V_{l,\mathbb{C},\mu})$.

For each $j=1,\cdots,N-1$, the $\mathbb{C}$-linear injection
$\varsigma_{j}:\left(V_{2j,\mathbb{C}}\cong\right)V_{2j,\mathbb{C}}\otimes_{\mathbb{R}}\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad(q_{1})}}\hookrightarrow\mathfrak{g}$
given by assigning $x^{2j-s}y^{s}\otimes
v\mapsto\frac{(2j-s)!}{(2j)!}\cdot\mathrm{ad}(q_{-1})^{s}(v)$ (for
$v\in\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad}(q_{1})}$ and $s=0,\cdots,2j$)
preserves the $\Gamma$-action. The morphism of $\mathbb{C}[\Gamma]$-modules

(159)
$\displaystyle\bigoplus_{j=1}^{N-1}\varsigma_{j}:\left(\bigoplus_{j=1}^{N-1}V_{2j,\mathbb{C},\mu}\cong\right)\bigoplus_{j=1}^{N-1}V_{2j,\mathbb{C},\mu}\otimes_{\mathbb{R}}\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad}(q_{1})}\rightarrow\mathfrak{g}(\mathbb{C})_{\mu}$

is an isomorphism, and hence it induces an isomorphism

(160)
$\displaystyle\left(\bigoplus_{j=1}^{N-1}H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})\cong\right)\bigoplus_{j=1}^{N-1}H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})\otimes_{\mathbb{R}}\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad}(q_{1})}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{P}(\Gamma,\mathfrak{g}(\mathbb{C})_{\mu}).$

On the other hand, the isomorphism (85) gives rises to a composite isomorphism

(161) $\displaystyle
H^{1}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigoplus_{j=1}^{N-1}H^{1}(X,{{}^{\dagger}}\mathcal{V}_{G,j}(-D))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigoplus_{j=1}^{N-1}H^{1}(X,\Omega^{\otimes(j+1)}(-D))\otimes_{\mathbb{R}}\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad}(q_{1})}.$

Its complex conjugation determines

(162)
$\displaystyle\overline{H^{1}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigoplus_{j=1}^{N-1}\overline{H^{1}(X,\Omega^{\otimes(j+1)}(-D))}\otimes_{\mathbb{R}}\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad}(q_{1})}.$

It follows from the various definitions involved that the decomposition (156)
preserves the direct sum decompositions (160), (161), and (162). In
particular, by considering the $j$-th factor of this decomposition (which is
independent of the choice of $N$), we obtain the following assertion,
generalizing the classical Eichler-Shimura isomorphism for even weight.

######  Theorem 6.6 (cf. Theorem B).

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

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

of the $1$-st parabolic cohomology group
$H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})$ of the $\mathbb{C}[\Gamma]$-module
$V_{2j,\mathbb{C},\mu}$.

######  Remark 6.7 (Comparison with the classical Eichler-Shimura
isomorphism).

For comparison with the classical situation, we here give an explicit
description of the decomposition (163). Let us take an arbitrary element $v$
of $H^{0}(X,\Omega^{\otimes(j+1)}(-D))\subseteq
H^{0}(X,{{}^{\dagger}}\mathcal{V}_{G}(-D))$. By fixing a generator $u$ of the
$1$-dimensional $\mathbb{R}$-vector space
$\mathfrak{g}_{\mathbb{R},j}^{\mathrm{ad}(q_{1})}$, we have
$\varpi^{*}(v)=wdz\otimes u$ for some $w\in
H^{0}(\mathbb{H},\mathcal{O}_{\mathbb{H}})$. A trivialization of the flat
$G$-bundle
$({{}^{\dagger}}\mathcal{E}_{G,\mathbb{H}},\varpi^{*}(\nabla|_{X^{*}}))$
constructed as in the proof of Theorem 6.1 allows us to express
$\varpi^{*}(v)$ as the element of
$H^{0}(\mathcal{O}_{\mathbb{H}},\Omega_{\mathbb{H}}\otimes_{\mathbb{C}}\mathfrak{g})$
defined by

(164) $\displaystyle\sum_{s=0}^{2j}\frac{1}{s!}\cdot\mathrm{ad}(\delta
q_{-1})^{s}\left(\frac{w_{j}u}{\delta^{\prime
j}}\right)=\sum_{s=0}^{2j}\frac{\delta^{s}\cdot w_{j}}{s!\cdot\delta^{\prime
j}}\cdot\mathrm{ad}(q_{-1})^{s}(u)$

(cf. (151)). The corresponding global section of
$\Omega_{\mathbb{H}}\otimes_{\mathbb{C}}V_{2j,\mathbb{C}}$
($=\Omega_{\mathbb{H}}\otimes_{\mathbb{C}}(V_{2j,\mathbb{C}}\otimes_{\mathbb{C}}\mathfrak{g}_{2j}^{\mathrm{ad}(q_{1})})$
under the identification given by $f\leftrightarrow f\otimes u$ for
$f\in\Omega_{\mathbb{H}}\otimes_{\mathbb{C}}V_{2j,\mathbb{C}}$) via
$\varsigma_{j}$ coincides with
$\sum_{s=0}^{2j}\binom{2j}{s}\cdot\frac{\delta^{s}\cdot w_{j}}{\delta^{\prime
j}}\cdot x^{2j-s}y^{s}=\frac{w_{j}}{\delta^{\prime j}}\cdot(x+\delta y)^{2j}$.
It follows that the $1$-cocycle $I_{v}$ in
$H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})$ representing $\varepsilon(v)$ is
given by

(165) $\displaystyle\gamma\mapsto
I_{v}(\gamma):=\int_{\widetilde{x}_{0}}^{\gamma(\widetilde{x}_{0})}\frac{w_{j}}{\delta^{\prime
j}}\cdot(x+\delta y)^{2j}dz\in V_{2j,\mathbb{C}}$

for each $\gamma\in\Gamma$, where $\widetilde{x}_{0}\in\mathbb{H}$ denotes a
fixed lifting of $x_{0}$. The $1$-cocycle associated to
$\overline{\varepsilon}(\overline{v})$ is defined in a similar vein.

If, moreover, $\mathscr{E}^{\spadesuit}_{\odot}$ is the canonical
$G^{\odot}$-oper arising from the Fuchsian uniformization, then the developing
map $\delta$ coincides with the inclusion
$\mathbb{H}\hookrightarrow\mathbb{P}^{1}$ (hence $\delta^{\prime}=1$).
Therefore, it follows from (165) that the decomposition (163) is, up to
multiplication by a constant factor, the same as the classical Eichler-Shimura
isomorphism (cf. [Eic], [Shi1], [Shi2]). (Moreover, the pairing
$\langle-,-\rangle_{\mathscr{E}^{\spadesuit}}$ restricted to
$H^{0}(X,\Omega^{\otimes(j+1)}(-D))$ becomes the Petersson inner product.)

###  6.5. Hodge structures associated to $G^{\odot}$-opers

In the case of $r=0$, D. Gallo, M. Kapovich, and A. Marden proved (cf. [GKM,
Theorem 1.1.1]) that a $G^{\odot}(\mathbb{C})$-representation
$\mu:\Gamma\rightarrow G^{\odot}(\mathbb{C})$ is conjugate to the monodromy
map of some $G^{\odot}$-oper (= projective structure) on a genus-$g$ compact
Riemann surface if and only if $\mu$ lifts to a representation
$\Gamma\rightarrow\mathrm{SL}_{2}(\mathbb{C})$ and the image of $\mu$ is non-
elementary (in the sense of [GKM, § 1.2]). For a representation $\mu$
satisfying these properties, there are infinitely many distinct pairs
$(X,\mathscr{E}^{\spadesuit})$ of a genus-$g$ compact Riemann surface $X$ and
a $G^{\odot}$-oper $\mathscr{E}^{\spadesuit}$ on $X$ with
$\mu=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}}$ up to conjugation. Also,
according to [BaGu, Theorem 1.1], such $X$’s (for a fixed $\mu$) form a dense
subset of the moduli space $\mathcal{M}_{g}$ of genus-$g$ compact Riemann
surfaces. The study of characterizing $G^{\odot}$-opers with prescribed
monodromy was investigated in, e.g., [Bab], [BaGu], [Dum], [Hub], and [Kap].
In what follows, we propose a framework for doing this kind of research from
the viewpoint of global Torelli problem by restating Theorem 6.6 in terms of
Hodge structures.

Let $V$ be a finite dimensional $\mathbb{R}$-vector space, so we have a
complex conjugation $\overline{(-)}$ on its complexification
$V_{\mathbb{C}}:=V\otimes_{\mathbb{R}}\mathbb{C}$. Recall that a real Hodge
structure of weight $k$ ($\in\mathbb{Z}_{>0}$) on $V$ is a direct sum
decomposition

(166) $\displaystyle V_{\mathbb{C}}=\bigoplus_{\begin{subarray}{c}p+q=k,\\\
(p,q)\in\mathbb{Z}_{\geq 0}^{2}\end{subarray}}V^{p,q}\ \ \text{with}\ \
V^{p,q}=\overline{V^{q,p}}.$

Also, a polarization of a given Hodge structure
$V_{\mathbb{C}}=\bigoplus_{p,q}V^{p,q}$ is a nondegenerate bilinear form
$(-,-)$ on $V_{\mathbb{C}}$, defined over $\mathbb{R}$, satisfying the
following conditions:

  * •

$(x,y)=(-1)^{k}\cdot(y,x)$ for any $x,y\in V_{\mathbb{C}}$ (i.e., $(-,-)$ is
symmetric or skew-symmetric depending on whether $k$ is even or odd);

  * •

$\langle x,y\rangle:=\sqrt{-1}^{p-q}\cdot(x,\overline{y})$ is a positive
definite hermitian inner product on $V^{p,q}$ for every $(p,q)$;

  * •

the decomposition $V_{\mathbb{C}}=\bigoplus_{p+q=k}V^{p,q}$ is orthogonal, in
the sense that $(x,y)=0$ if $x\in V^{p,q}$, $y\in V^{p^{\prime},q^{\prime}}$
with $(p,q)\neq(q^{\prime},p^{\prime})$).

By a polarized real Hodge structure of weight $k$ on $V$, we mean a real Hodge
structure of weight $k$ on $V$ together with its polarization. P. Bayer and J.
Neukirch (cf. [BaNe, Theorem 5.2]) gave, in the framework of Hodge theory
using such notions, a direct proof of the Eichler-Shimura isomorphisms for
Fuchsian groups of the first kind. The following assertion may be regarded as
a generalization of that result.

######  Theorem 6.8 (cf. Theorem B).

Let $\mathscr{E}^{\spadesuit}_{\odot}$ be a $G^{\odot}$-oper classified by
$\mathrm{Op}_{P}(\mathscr{X},G^{\odot}(\mathbb{R}))$ with monodromy map
$\mu\left(:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}_{\odot}}\right):\Gamma\rightarrow
G^{\odot}(\mathbb{R})$. Then, for each positive integer $j$, the decomposition
(163) together with the bilinear form
$\frac{1}{\sqrt{-1}}\cdot(-,-)_{\mathscr{E}^{\spadesuit}_{\odot}}$ (cf. (142))
restricted to $H^{1}_{P}(\Gamma,V_{2j,\mathbb{C},\mu})$ (with respect to the
decomposition (160)) defines a polarized real Hodge structure of weight $1$ on
the $\mathbb{R}$-vector space $H^{1}_{P}(\Gamma,V_{2j,\mathbb{R},\mu})$.

###### Proof.

The assertion follows from Theorems 6.1 and 6.5. ∎

######  Remark 6.9 (Weight of the Hodge structure).

Since we only deal with the case of even weight $2j$, the Hodge structure
asserted in the above theorem may be considered, after multiplying
$(-,-)_{\mathscr{E}^{\spadesuit}_{\odot}}$ by a nonzero constant factor, as
being of weight $2j+1$ (where only two graded pieces are nonzero) in
accordance with the formulation of [BaNe].

By the above theorem, we have extended the construction of Hodge structures to
all $G^{\odot}$-opers with real monodromy and of radii $([0],\cdots,[0])$. It
follows that the question of whether or not a Torelli-type uniqueness theorem
for such objects holds now makes sense. That is to say, it is expected that
such $G^{\odot}$-opers can be reconstructed from the polarized Hodge
structures associated to them, as formulated as follows.

######  Conjecture 6.10.

For each $i=1,2$, let $\mathscr{E}_{\odot,i}^{\spadesuit}$ be a
$G^{\odot}$-oper classified by
$\mathrm{Op}_{P}(\mathscr{X},G^{\odot}(\mathbb{R}))$ with monodromy map
$\mu_{i}\left(:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}_{\odot,i}}\right):\Gamma\rightarrow
G^{\odot}(\mathbb{R})$. Suppose that $\mu:=\mu_{1}=\mu_{2}$. Then, the
following conditions are equivalent to each other:

  * •

$X_{1}$ is biholomorphic to $X_{2}$;

  * •

The polarized real Hodge structures on
$H^{1}_{P}(\Gamma,V_{2j,\mathbb{R},\mu})$ associated to
$\mathscr{E}_{\odot,1}^{\spadesuit}$ and $\mathscr{E}_{\odot,2}^{\spadesuit}$
are identical (up to a constant factor) for every $j\in\mathbb{Z}_{>0}$.

(It follows from the injectivity of (131) that these conditions moreover
implies
$\mathscr{E}^{\spadesuit}_{\odot,1}\cong\mathscr{E}^{\spadesuit}_{\odot,2}$.)
In particular, each pair $(\mathscr{X},\mathscr{E}^{\spadesuit}_{\odot})$ with
$\mathscr{E}^{\spadesuit}_{\odot}\in\mathrm{ob}(\mathrm{Op}_{P}(\mathscr{X},G(\mathbb{R})))$
can be reconstructed from the associated data of the monodromy map
$\mu:=\mathrm{Mon}_{\mathscr{E}^{\spadesuit}_{\odot}}$ and the polarized real
Hodge structure on $H^{1}_{P}(\Gamma,V_{2j,\mathbb{R},\mu})$.

###  6.6. Acknowledgements

We are grateful for the many constructive conversations we had with $G$-opers
on Riemann surfaces, who live in the world of mathematics! Our work was
partially supported by Grant-in-Aid for Scientific Research (KAKENHI No.
21K13770).

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