as a generalization of that result.) Also, see [Ogu2, Theorem 3.1, Remark 3.2]
for a version of elliptic $F$-$T$-crystals.

######  Theorem 5.2.

Let
$\mathscr{F}^{\heartsuit}_{\odot}:=(\mathcal{F}_{\odot},\nabla_{\odot},\\{\mathcal{F}^{j}_{\odot}\\}_{j=0}^{2})$
be an $\mathrm{SL}_{2}$-oper on $\mathscr{X}$. Write $\varTheta$ for the theta
characteristic associated to $\mathscr{F}^{\heartsuit}$ (cf. § 3.4). Also,
denote by $\mathrm{Sym}^{n-1}\mathscr{F}_{\odot}$ the $(n-1)$-st symmetric
product of the underlying flat $\mathrm{SL}_{2}$-bundle $\mathscr{F}_{\odot}$
of $\mathscr{F}^{\heartsuit}_{\odot}$. Then, the following assertions hold:

  * (i)

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

(205) $\displaystyle 0\rightarrow
H^{0}(X,{{}^{\Box}}(\varTheta^{\otimes(n+1)}))\rightarrow
H^{1}_{\mathrm{dR}}(X,{{}^{\Box}}(\mathrm{Sym}^{n-1}\mathscr{F}_{\odot}))\rightarrow
H^{1}(X,{{}^{\Box}}(\varTheta^{\otimes(-n+1)}))\rightarrow 0.$

Moreover, there exists a canonical isomorphism

(206) $\displaystyle
H^{1}_{\mathrm{dR}}(X,{{}^{c}}(\mathrm{Sym}^{n-1}\mathscr{F}_{\odot}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{n-1}\mathscr{F}_{\odot})^{\vee}$

via which the short exact sequences (205) for $\Box$’s in both cases are
compatible with each other.

  * (ii)

Suppose further that $\mathscr{F}^{\heartsuit}_{\odot}$ is of radii
$[0]^{\times r}\in\mathfrak{c}(k)^{\times r}$ (where $[0]^{\times
r}:=\emptyset$ if $r=0$). Then, there exists a canonical short exact sequence

(207) $\displaystyle 0\rightarrow
H^{0}(X,\varTheta^{\otimes(n+1)}(-D))\rightarrow
H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{n-1}\mathscr{F}_{\odot})\rightarrow
H^{1}(X,\varTheta^{\otimes(-n+1)})\rightarrow 0.$

Moreover, there exists a canonical isomorphism

(208) $\displaystyle
H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{n-1}\mathscr{F}_{\odot})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{n-1}\mathscr{F}_{\odot})^{\vee}$

via which the short exact sequence (207) is compatible with its dual.

###### Proof.

We only prove assertion (ii) because the proof of assertion (i) is similar.
For simplicity, we shall write
$(\mathcal{F},\nabla):=(\mathrm{Sym}^{n-1}\mathcal{F}_{\odot},\mathrm{Sym}^{n-1}\nabla_{\odot})$.

First, we shall consider the former assertion. Under the identifications
$\mathcal{F}^{n-1}=\varTheta^{\otimes(n-1)}$ and
$\mathcal{F}/\mathcal{F}^{1}=\varTheta^{\otimes(-n+1)}$ given by (52), the
sequence (188) defined for the $\mathrm{SL}_{n}$-oper
$\mathrm{Sym}^{n-1}\mathscr{F}_{\odot}^{\heartsuit}$ becomes a short exact
sequence

(209) $\displaystyle 0\rightarrow
H^{0}(X,\Omega\otimes_{\mathcal{O}_{X}}\varTheta^{\otimes(n-1)}(-D))\rightarrow
H_{\mathrm{dR},\mathrm{par}}^{1}(X,\mathrm{Sym}^{n-1}\mathscr{F}_{\odot})\rightarrow
H^{1}(X,\varTheta^{\otimes(-n+1)})\rightarrow 0.$

Moreover, since (51) gives an identification $\Omega=\varTheta^{\otimes 2}$,
we have
$H^{0}(X,\Omega\otimes_{\mathcal{O}_{X}}\varTheta^{\otimes(n-1)}(-D))\cong
H^{0}(X,\varTheta^{\otimes(n+1)}(-D))$. Hence, the assertion follows
immediately from Proposition 5.1, (ii).

Next, we shall prove the latter assertion of (ii). To do this, we may assume
that the $\mathrm{SL}_{2}$-oper $\mathscr{F}_{\odot}^{\heartsuit}$ comes, via
the isomorphism “$\Lambda_{\vartheta}^{\clubsuit\Rightarrow\diamondsuit}$”
obtained in [Wak4, § 4.6.6, (553)], from a $(2,\vartheta)$-projective
connection (cf. [Wak4, Definition 4.37, (ii)]), where $\vartheta$ denotes a
$2$-theta characteristic of $X^{\mathrm{log}}$ (cf. [Wak4, Definition 4.31,
(i)]) determined by $\varTheta$ in the manner of [Wak4, Example 4.34]. Then,
by considering the matrix form of the log connection $\nabla_{\odot}$ (cf.
[Wak4, Remarks 4.30, 4.39]), the dual $\mathscr{F}_{\odot}^{\vee}$ of
$\mathscr{F}_{\odot}$ is isomorphic to $\mathscr{F}_{\odot}$ itself. We fix an
isomorphism
$\mathscr{F}_{\odot}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{F}_{\odot}^{\vee}$,
which induces an isomorphism
$\varpi:(\mathcal{F},\nabla)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathcal{F}^{\vee},\nabla^{\vee})$
(cf. [Can, § 3]).

Now, let $U$ and $\iota$ be as in § 4.3, and consider the
$\mathcal{O}_{X}$-bilinear pairing

(210)
$\displaystyle\widetilde{\varpi}^{\blacktriangleright}:\iota_{*}(\iota^{*}(\mathcal{F}))\times\iota_{*}(\iota^{*}(\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F}))\rightarrow\iota_{*}(\iota^{*}(\Omega))$

arising from the isomorphism $\varpi$. In the following discussion, we shall
prove (by an argument entirely similar to the proof of Proposition 4.3) the
claim that $\widetilde{\varpi}^{\blacktriangleright}$ restricts to a
nondegenerate $\mathcal{O}_{X}$-bilinear pairing

(211)
$\displaystyle\varpi^{\blacktriangleright}:\Omega_{\mathrm{par}}^{0,c}(\mathcal{F})\times\Omega_{\mathrm{par}}^{1}(\mathcal{F})\rightarrow{{}^{c}}\Omega.$

Since the restriction of $\widetilde{\varpi}^{\blacktriangleright}$ to $U$ is
nondegenerate, the problem is reduced to examining the pairing
$\widetilde{\varpi}^{\blacktriangleright}$ around the marked points. Let us
choose $i\in\\{1,\cdots,r\\}$ and choose a local function
$t\in\mathcal{O}_{X}$ defining the closed subscheme
$\mathrm{Im}(\sigma_{i})\subseteq X$. Then, the formal neighborhood $Q$ of
$\sigma_{i}$ is naturally isomorphic to $\mathrm{Spec}(k[\\![t]\\!])$, and we
have ${{}^{c}}\Omega|_{Q}=k[\\![t]\\!]dt$. After fixing a trivialization
$\varTheta|_{Q}=k[\\![t]\\!]$, we can associate, to the local function $t$, a
canonical identification $\mathcal{F}|_{Q}=k[\\![t]\\!]^{\oplus n}$ (cf.
[Wak4, Remark 4.30]); it restricts to
$\mathcal{F}^{j}|_{Q}=k[\\![t]\\!]^{\oplus n-j}\oplus\\{0\\}^{\oplus
j}\left(\subseteq k[\\![t]\\!]^{\oplus n}\right)$. This identification gives

(212)
$\displaystyle\iota_{*}(\iota^{*}(\mathcal{F}))|_{Q}=k(\\!(t)\\!)^{\oplus n}\
\left(\text{resp.,}\
\iota_{*}(\iota^{*}(\Omega\otimes_{\mathcal{O}_{X}}\mathcal{F}))|_{Q}=(k(\\!(t)\\!)dt)^{\oplus
n}\right).$

Moreover, by the equality $\mu_{i}^{\mathrm{Sym}^{n-1}\nabla_{\odot}}=q_{-1}$,
(212) restricts to an identification

(213)
$\displaystyle\Omega_{\mathrm{par}}^{0,c}(\mathcal{F})|_{Q}=(k[\\![t]\\!]t)^{\oplus(n-1)}\oplus
k[\\![t]\\!]\ \left(\text{resp.,}\
\Omega_{\mathrm{par}}^{1}(\mathcal{F})|_{Q}=k[\\![t]\\!]dt\oplus\left(k[\\![t]\\!]\frac{dt}{t}\right)^{\oplus(n-1)}\right).$

According to this description, each section $v$ (resp., $u$) of
$\Omega_{\mathrm{par}}^{0,c}(\mathcal{F})|_{Q}$ (resp.,
$\Omega_{\mathrm{par}}^{1}(\mathcal{F})|_{Q}$) may be described as the sum
$v=t\cdot\sum_{i=1}^{n-1}v_{i}+v_{n}$ (resp.,
$u=u_{1}+\frac{1}{t}\cdot\sum_{i=2}^{n}u_{i}$) for some $v_{1},\cdots,v_{n}\in
k[\\![t]\\!]$ (resp., $u_{1},\cdots,u_{n}\in k[\\![t]\\!]dt$). Then, up to a
multiplicative constant factor, the following equalities hold:

(214) $\displaystyle\widetilde{\varpi}^{\blacktriangleright}(v,u)$
$\displaystyle=\widetilde{\varpi}^{\blacktriangleright}\left(t\cdot\sum_{i=1}^{n-1}v_{i},\frac{1}{t}\cdot\sum_{i=2}^{n}u_{i}\right)+\widetilde{\varpi}^{\blacktriangleright}(v_{n},u_{1})$
$\displaystyle=\widetilde{\varpi}^{\blacktriangleright}\left(\sum_{i=1}^{n-1}v_{i},\sum_{i=2}^{n}u_{i}\right)+\widetilde{\varpi}^{\blacktriangleright}(v_{n},u_{1})$
$\displaystyle=\sum_{i=1}^{n}\widetilde{\varpi}^{\blacktriangleright}(v_{i},u_{n+1-i}).$

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

Just as in the case of (104), the pairing $\varpi^{\blacktriangleright}$
together with the natural nondegenerate pairing
$\varpi^{\triangleright}:\Omega_{\mathrm{par}}^{1,c}(\mathcal{F})\times\Omega_{\mathrm{par}}^{1}(\mathcal{F})\rightarrow{{}^{c}}\Omega$
arising from $\varpi$ yields a $k$-bilinear pairing

(215)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla_{\mathrm{par}}])\times\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])\rightarrow
k.$

The induced morphism
$\varpi_{H}:\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla_{\mathrm{par}}])\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])^{\vee}$
fits into the morphism of exact sequences

(220)

(cf. (110) for a similar diagram), where the horizontal sequences arise from
the Hodge to de Rham spectral sequences involved, and the vertical arrows
except for the middle one arise from the pairings
$\varpi^{\blacktriangleright}$ and $\varpi^{\triangleright}$. By applying the
five lemma to this diagram, we see that the middle vertical arrow $\varpi_{H}$
is an isomorphism. Under the equality
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla_{\mathrm{par}}])=H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{n-1}\mathcal{F}_{\odot})\left(=\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}])\right)$
resulting from Proposition 4.1, $\varpi_{H}$ defines an isomorphism

(221) $\displaystyle
H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{n-1}\mathscr{F}_{\odot})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{n-1}\mathscr{F}_{\odot})^{\vee}.$

Moreover, by the various definitions involved and the local description of
$\widetilde{\varpi}^{\blacktriangleright}$ described above, we see that that
the short exact sequence (207) is compatible with its dual via (221). This
completes the proof of this proposition. ∎

Next, if $n=2l+1$ for some positive integer $l$, then the following assertion
holds. (We leave to the reader the formulation of relevant duality as asserted
in the above theorem.)

######  Theorem 5.3.

Let $\mathscr{E}^{\spadesuit}$ be a $G^{\odot}$-oper on $\mathscr{X}$. Recall
from the discussion in § 3.5 that the $2l$-th symmetric product
$\mathrm{Sym}^{2l}\mathscr{E}_{\odot}^{\spadesuit}$ of
$\mathscr{E}^{\spadesuit}$ can be obtained (cf. (65)). Then, the following
assertions hold:

  * (i)

There exists a canonical short exact sequence of $k$-vector spaces

(222) $\displaystyle 0\longrightarrow
H^{0}(X,\Omega^{\otimes(l+1)})\longrightarrow
H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{2l}\mathscr{E})\longrightarrow
H^{1}(X,\Omega^{\otimes(-l)})\longrightarrow 0,$

where $\mathrm{Sym}^{2l}\mathscr{E}_{\cdot}$ denotes the underlying flat
$\mathrm{SL}_{2l}$-bundle of
$\mathrm{Sym}^{2l}\mathscr{E}^{\spadesuit}_{\cdot}$.

  * (ii)

Suppose further that $\mathscr{E}^{\spadesuit}_{\odot}$ is of radii
$[0]^{\times r}\in\mathfrak{c}(k)^{\times r}$ (where $[0]^{\times
r}:=\emptyset$ if $r=0$). Then, there exists a canonical short exact sequence

(223) $\displaystyle 0\longrightarrow
H^{0}(X,\Omega^{\otimes(l+1)}(-D))\longrightarrow
H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{2l}\mathscr{E})\longrightarrow
H^{1}(X,\Omega^{\otimes(-l)})\longrightarrow 0.$

Finally, when $\mathscr{E}^{\spadesuit}$ comes from an $\mathrm{SL}_{2}$-oper
via projectivization, these short exact sequences coincide with the
corresponding ones in the statement of Theorem 5.2.

###### Proof.

Assertion (i) (resp., (ii)) follows from Proposition 5.1, (i) (resp., (ii))
and an argument similar to the proof of Theorem 5.2, (i) (resp., (ii)). ∎

###  5.3. The irreducible decomposition of the adjoint bundle

Let
$\mathscr{E}^{\spadesuit}_{\odot}:=(\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
be a $\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normal $\mathrm{PGL}_{2}$-oper. We shall write
$\mathscr{E}^{\spadesuit}\left(=(\mathcal{E}_{B},\nabla)\right):=\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})$,
which is a $G$-oper. Also, write
$\mathcal{E}_{\odot}:=\mathcal{E}_{B^{\odot}}\times^{B^{\odot}}G^{\odot}$ and
$\mathcal{E}:=\mathcal{E}_{B}\times^{B}G$. In what follows, we consider a
decomposition of the flat adjoint bundle
$(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$ according to the
irreducible decomposition of $\mathfrak{g}$, regarded as an
$\mathfrak{s}\mathfrak{l}_{2}$-module via $\iota_{\mathfrak{g}}$.

Let us fix an integer $l$ with $-\mathrm{rk}(\mathfrak{g})\leq
l\leq\mathrm{rk}(\mathfrak{g})$. Under the natural identification
$\Omega^{\otimes
l}\otimes_{k}\mathfrak{g}_{l}^{\mathrm{ad}(q_{1})}=\mathcal{V}_{G,l}\left(\subseteq\mathfrak{g}_{\mathcal{E}}\right)$,
each nonzero element $a\in\mathfrak{g}_{l}^{\mathrm{ad}(q_{1})}$ determines an
inclusion $\gamma_{a}:\Omega^{\otimes
l}\hookrightarrow\mathfrak{g}_{\mathcal{E}}$. Note that, for each
$j=0,\cdots,2l+1$, the $k$-vector subspace

(224) $\displaystyle
H_{a}^{j}:=\sum_{s=0}^{2l-j}k\cdot\mathrm{ad}(q_{-1})^{s}(a)\left(=\bigoplus_{s=0}^{2l-j}k\cdot\mathrm{ad}(q_{-1})^{s}(a)\right)$

of $\mathfrak{g}$ is closed under the adjoint $B^{\odot}$-action via
$\iota_{B}$, and that $H_{a}^{0}$ defines an
$\mathfrak{s}\mathfrak{l}_{2}$-submodule of $\mathfrak{g}$. Hence, $H_{a}^{0}$
determines a rank $(2l+1)$ flat subbundle $(\mathcal{H}_{a},\nabla_{a})$ of
$(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$, and the subspaces
$\\{H_{a}^{j}\\}_{j=0}^{2l+1}$ of $H_{a}^{0}$ determine a decreasing
filtration $\\{\mathcal{H}_{a}^{j}\\}_{j=0}^{2l+1}$ on $\mathcal{H}_{a}$. One
may verify that the resulting collection

(225)
$\displaystyle\mathscr{H}_{a}^{\heartsuit}:=(\mathcal{H}_{a},\nabla_{a},\\{\mathcal{H}_{a}^{j}\\}_{j=0}^{2l+1})$

form an $\mathrm{SL}_{2l+1}$-oper on $\mathscr{X}$.

Next, let $\mathcal{D}$ (resp., $\mathcal{D}_{<m}$ for each nonnegative
integer $m$) denote the sheaf of logarithmic differential operators (resp.,
logarithmic differential operators of order $<m$) of $X^{\mathrm{log}}$ (cf.
[Wak4, § 4.2.1]). Then, it follows from the definition of an
$\mathrm{SL}_{2l+1}$-oper that the composite

(226)
$\displaystyle\widetilde{\gamma}_{a}:\mathcal{D}_{<2l+1}\otimes_{\mathcal{O}_{X}}\Omega^{\otimes
l}\longrightarrow\mathcal{D}\otimes_{\mathcal{O}_{X}}\mathcal{H}_{a}^{0}\longrightarrow\mathcal{H}_{a}\hskip
56.9055pt$ $\displaystyle\left(\text{resp.,}\
\widetilde{\gamma}_{\mathrm{sym}}:\mathcal{D}_{<2l+1}\otimes_{\mathcal{O}_{X}}\Omega^{\otimes
l}\longrightarrow\mathcal{D}\otimes_{\mathcal{O}_{X}}\mathrm{Sym}^{2l}\mathcal{E}\longrightarrow\mathrm{Sym}^{2l}\mathcal{E}_{\odot}\right)$

is an isomorphism, where the first arrow denotes the injection arising from
both the natural inclusion $\mathcal{D}_{<2l+1}\hookrightarrow\mathcal{D}$ and
$\gamma_{a}$ (resp., the injection $\Omega^{\otimes
l}\hookrightarrow\mathrm{Sym}^{2l}\mathcal{E}_{\odot}$ deduced from (63)), and
the second arrow arises from the $\mathcal{D}$-module structure on
$\mathcal{H}_{a}^{0}$ (resp., $\mathrm{Sym}^{2l}\mathcal{E}_{\odot}$)
determined by $\nabla_{a}$ (resp., $\mathrm{Sym}^{2l}\nabla_{\odot}$). The
composite

(227)
$\displaystyle\mathrm{Sym}^{2l}\mathcal{E}_{\odot}\xrightarrow{\widetilde{\gamma}_{\mathrm{sym}}^{-1}}\mathcal{D}_{<2l+1}\otimes_{\mathcal{O}_{X}}\Omega^{\otimes
l}\xrightarrow{\widetilde{\gamma}_{a}}\mathcal{H}_{a}\xrightarrow{\mathrm{inclusion}}\mathfrak{g}_{\mathcal{E}}$

is compatible with the respective connections, i.e.,
$\mathrm{Sym}^{2l}\nabla_{\odot}$ and $\nabla^{\mathrm{ad}}$. That is, we have
obtained an injective morphism of flat bundles
$\xi_{a}:\mathrm{Sym}^{2l}\mathscr{E}_{\odot}\hookrightarrow(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$.
The morphism of flat bundles

(228)
$\displaystyle\xi_{l}:\mathrm{Sym}^{2l}\mathscr{E}_{\odot}\otimes_{k}\mathfrak{g}_{l}^{\mathrm{ad}(q_{1})}\hookrightarrow(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$

given by $v\otimes a\mapsto\xi_{a}(v)$ for
$a\in\mathfrak{g}_{l}^{\mathrm{ad}(q_{1})}$ and
$v\in\mathrm{Sym}^{2l}\mathcal{E}_{\odot}$ is well-defined, and the various
$\xi_{l}$’s define an isomorphism

(229)
$\displaystyle\bigoplus_{l=1}^{\mathrm{rk}(\mathfrak{g})}\xi_{l}:\bigoplus_{l=1}^{\mathrm{rk}(\mathfrak{g})}\mathrm{Sym}^{2l}\mathscr{E}_{\odot}\otimes_{k}\mathfrak{g}^{\mathrm{ad}(q_{1})}_{l}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}}).$

By applying the $1$-st hypercohomology functor to this isomorphism, we obtain
an isomorphism of $k$-vector spaces

(230)
$\displaystyle\bigoplus_{l=1}^{\mathrm{rk}(\mathfrak{g})}H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{2l}\mathscr{E}_{\odot})\otimes_{k}\mathfrak{g}^{\mathrm{ad}(q_{1})}_{l}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}]).$

This isomorphism fits into the isomorphism of short exact sequences

(235)

(cf. (222) for the upper horizontal sequence and (120) for the lower
horizontal sequence), where the left-hand and right-hand vertical arrows are
the isomorphism induced from
$\mathcal{V}_{G}\cong\bigoplus_{l=1}^{\mathrm{rk}(\mathfrak{g})}\Omega\otimes_{k}\mathfrak{g}^{\mathrm{ad}(q_{1})}$
(cf. (47)).

Moreover, suppose that $\mathscr{E}^{\spadesuit}_{\odot}$ is of radii
$[0]^{\times r}$ (where $[0]^{\times r}:=\emptyset$ if $r=0$). Then, (229)
yields an isomorphism

(236)
$\displaystyle\bigoplus_{l=1}^{\mathrm{rk}(\mathfrak{g})}H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{2l}\mathscr{E}_{\odot})\otimes_{k}\mathfrak{g}^{\mathrm{ad}(q_{1})}_{l}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla_{\mathrm{par}}^{\mathrm{ad}}]),$

which fits into the isomorphism of short exact sequences

(241)

(cf. (207) for the upper horizontal sequence and (168) for the lower
horizontal sequence). Once we have constructed a canonical self-duality for
$H^{1}_{\mathrm{dR},\mathrm{par}}(X,\mathrm{Sym}^{2l}\mathscr{E}_{\odot})$ as
asserted in Theorem 5.2, (ii), this duality will be verified to be compatible
with (112) via (236).

##  6\. Dormant opers and their deformations

This section discusses opers in positive characteristic, especially, dormant
opers (= opers with vanishing $p$-curvature). The study of the moduli space of
dormant opers was substantially developed in [Jos], [JoPa], [Wak1], [Wak2],
and [Wak4].

One of the important features of this moduli space is the generic étaleness
(cf. [Wak4, Theorem G]). In other words, if the underlying curve is
sufficiently general (and $G$ satisfies a certain additional assumption), then
the moduli spaces of opers and $p$-flat bundles intersect transversally. This
fact will be essential in the proof of our main theorem, as discussed in the
next section.

In the rest of the present paper, we assume that $\mathrm{char}(k)=p>0$ for a
prime $p$, and that either “$p>2h_{G}$” or “$G=\mathrm{PGL}_{n}$ with $1<n<p$”
is fulfilled.

###  6.1. Dormant opers on pointed stable curves

Let $(\mathcal{E},\nabla)$ be a flat $G$-bundle on $\mathscr{X}$. Recall (cf.
[Wak4, Definition 3.8]) that the $p$-curvature of $\nabla$ is the
$\mathcal{O}_{X}$-linear morphism ${{}^{p}}\psi^{\nabla}:\mathcal{T}^{\otimes
p}\rightarrow\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}}$ uniquely
determined by the condition that any local section in $\mathcal{T}^{\otimes
p}$ of the form $\partial^{\otimes p}$ for some $\partial\in\mathcal{T}$ is
mapped to $\nabla(\partial)^{[p]}-\nabla(\partial^{[\partial]})$, where
$(-)^{[p]}$ denotes the result of applying the $p$-power operations (cf.
[Wak4, § 3.2]). In the case where $G$ is a matrix group, this morphism is
equivalent to the classical definition of $p$-curvature described in terms of
vector bundles (cf. e.g., [Katz, § 5]).

By a $p$-flat $G$-bundle, we mean a flat $G$-bundle $(\mathcal{E},\nabla)$
such that $\nabla$ has vanishing $p$-curvature. We shall denote by

(242) $\displaystyle\mathcal{C}onn_{G}^{\psi=0}\ \left(\text{resp.,}\
\mathcal{C}onn_{G,\mu}^{\psi=0}\ \text{for each $\mu\in\mathfrak{g}^{\times
r}$}\right)$

the closed substack of $\mathcal{C}onn_{G}$ (resp., $\mathcal{C}onn_{G,\mu}$)
classifying $p$-flat $G$-bundles.

Next, a $G$-oper $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ is
called dormant if $\nabla$ has vanishing $p$-curvature (cf. [Wak4, Definition
3.15]). Then, we obtain a closed subscheme

(243) $\displaystyle\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G}\
\left(\text{resp.},\ \mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G,\rho}\right)$

of $\mathcal{O}p_{G}$ (resp., $\mathcal{O}p_{G,\rho}$ for each
$\rho\in\mathfrak{c}(k)^{\times r}$) classifying dormant $G$-opers. The
following equality between substacks of $\mathcal{C}onn_{G}$ (resp.,
$\mathcal{C}onn_{G,\widetilde{\rho}}$) holds:

(244)
$\displaystyle\mathcal{O}p_{G}^{{}^{\mathrm{Zzz...}}}=\mathcal{O}p_{G}\cap\mathcal{C}onn_{G}^{\psi=0}\
\left(\text{resp.,}\
\mathcal{O}p_{G,\rho}^{{}^{\mathrm{Zzz...}}}=\mathcal{O}p_{G,\rho}\cap\mathcal{C}onn_{G,\widetilde{\rho}}^{\psi=0}\right).$

Since $G$ has been assumed to be of adjoint type, we can apply [Wak4, Theorem
C] to see that $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G}$ (resp.,
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G,\rho}$) is a nonempty finite
$k$-scheme (resp., a possibly empty finite $k$-scheme).

Now, let
$\mathscr{E}^{\spadesuit}_{\odot}:=(\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
be a $G^{\odot}$-oper on $\mathscr{X}$. The log connection
$\iota_{G*}(\nabla)$ of the $G$-bundle
$\mathcal{E}_{G}:=\mathcal{E}_{B^{\odot}}\times^{B^{\odot},\iota_{B}}G$ (cf.
(43)) satisfies the equality
${{}^{p}}\psi^{\iota_{G*}(\nabla)}=(d\iota_{G})_{\mathcal{E}}\circ{{}^{p}}\psi^{\nabla}$,
where $(d\iota_{G})_{\mathcal{E}}$ denotes the natural injection
$\widetilde{\mathcal{T}}_{\mathcal{E}^{\mathrm{log}}_{G^{\odot}}}\hookrightarrow\widetilde{\mathcal{T}}_{\mathcal{E}_{G}^{\mathrm{log}}}$
(cf. [Wak4, § 3.3.2]). In particular, $\mathscr{E}^{\spadesuit}_{\odot}$ is
dormant if and only if the associated $G$-oper
$\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})$ is dormant. It follows that
(44) (resp., (45)) restricts to a closed immersion

(245)
$\displaystyle\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G^{\odot}}\hookrightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G}\
\left(\text{resp.,}\
\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G^{\odot},\rho_{\odot}}\hookrightarrow\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G,\iota_{\mathfrak{c}}(\rho_{\odot})}\right).$

###  6.2. Example: a dormant $\mathrm{PGL}_{2}$-oper on a Shimura curve

In this subsection, we shall illustrate an example of a dormant
$\mathrm{PGL}_{2}$-oper on a Shimura curve resulting from the discussions by
H. Reimann (cf. [Rei]) and M. Sheng, J. Zhang, K. Zuo (cf. [SZZ]). See [Wak5]
for other examples of dormant $\mathrm{PGL}_{2}$-opers, constructed by using
the Gauss maps for Fermat curves.

Let $F$ be a totally real number field of degree $\geq 2$ in which $p$ is
unramified. Also, let $A$ be a quaternion algebra over $F$ which is split at
one infinite place $\tau:F\hookrightarrow\mathbb{R}$ and ramified at all
remaining infinite places. After fixing an embedding
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$, one obtains
an bijection $\mathrm{Hom}_{\mathbb{Q}}(F,\overline{\mathbb{Q}})$ and
$\coprod_{i=1}^{r}\mathrm{Hom}_{\mathbb{Q}_{p}}(F_{\mathfrak{p}_{i}},\overline{\mathbb{Q}}_{p})$,
where
$\mathfrak{p}_{1}\left(=:\mathfrak{p}\right),\mathfrak{p}_{2},\mathfrak{p}_{3},\cdots,\mathfrak{p}_{r}$
are all primes in $F$ dividing $p$ such that $\tau$ lies in
$\mathrm{Hom}_{\mathbb{Q}_{p}}(F_{\mathfrak{p}},\overline{\mathbb{Q}}_{p})$.
Also, we shall choose a pair $(L,K)$ consisting of a totally imaginary
quadratic extension $L$ of $F$ contained in $A$ such that all
$\mathfrak{p}_{i}$ stay prime in $L$, and an imaginary quadratic extension $K$
of $F$ in which all the primes $\mathfrak{p}_{1},\cdots,\mathfrak{p}_{r}$
split. By regarding both $F^{\times}$ and $K^{\times}$ as $\mathbb{Q}$-groups,
one may obtain a $\mathbb{Q}$-group $G$ which makes the following sequence
exact:

(246) $\displaystyle 1\longrightarrow
F^{\times}\xrightarrow{f\mapsto(f,f^{-1})}A^{\times}\times
K^{\times}\longrightarrow G\longrightarrow 1.$

Suppose $A$ is split at $\mathfrak{p}_{1}$, and write
$\mathcal{O}_{B}:=\mathcal{O}_{A}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{K}$,
where $\mathcal{O}_{A}$ is an order of $A$ containing $\mathcal{O}_{L}$ with
certain additional properties (cf. [Rei, § 2, (2.9)]). Then, for every level
structure $C=C_{p}\times C^{p}\subseteq G(\mathbb{A}_{f})$ (where
$\mathbb{A}_{f}$ denotes the ring of finite adèles of $\mathbb{Q}$) with
$C_{p}=G(\mathbb{Z}_{p})$ and $C^{p}$ sufficiently small (cf. [Rei, § 2,
(2.11)]), there exists a smooth proper $\mathcal{O}_{F_{\mathfrak{p}}}$-scheme
$\mathcal{M}_{C}$ of relative dimension $1$ which is the coarse moduli space
of a certain moduli functor of PEL type with the endomorphism algebra
$\mathcal{O}_{B}$ (cf. [Rei, Definition 2.12, Proposition 2.14, and Corollary
3.14]).

Let us take one of the connected components of the base-change of
$\mathcal{M}_{C}$ to $k$ (via a fixed inclusions
$\mathcal{O}_{F_{\mathfrak{p}}}/p\mathcal{O}_{F_{\mathfrak{p}}}\hookrightarrow\overline{\mathbb{F}}_{p}\hookrightarrow
k$), which we denote by $X$. After possibly replacing $C$ with another, we may
assume that the genus of $X$ is greater than $1$ and that there exists a
universal abelian scheme $f:Y\rightarrow X$. The $1$-st de Rham cohomology
sheaf
$\mathcal{H}:=R^{1}f_{*}(\mathcal{K}^{\bullet}[\mathcal{O}_{Y}\xrightarrow{d}\Omega_{Y/X}])$
on $X$ is equipped with the Gauss-Manin connection $\nabla$, as well as a
$2$-step decreasing filtration $\\{\mathcal{H}^{j}\\}_{j=0}^{2}$ on
$\mathcal{H}$ arising from the Hodge filtration.

If we write
$\mathfrak{p}_{i}\mathcal{O}_{K}=\mathfrak{q}_{i}\overline{\mathfrak{q}}_{i}$
($i=1,\cdots,r$) for distinct primes $\mathfrak{q}_{i}$,
$\overline{\mathfrak{q}}_{i}$ in $K$, then we have
$p\mathcal{O}_{LK}=\prod_{i=1}^{r}\mathfrak{q}_{i}\overline{\mathfrak{q}}_{i}$;
it induces a natural isommorphism
$LK\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\prod_{i=1}^{r}LK_{\mathfrak{q}_{i}}\times
LK_{\overline{\mathfrak{q}}_{i}}$. This isomorphism tensored with
$\mathbb{Q}^{\mathrm{un}}_{p}$ (= the maximal unramified extension of
$\mathbb{Q}_{p}$) restricts to a decomposition

(247)
$\displaystyle\prod_{\phi\in\Phi}w(\phi)\times\overline{w}(\phi):\mathcal{O}_{LK}\otimes_{\mathbb{Z}}W(\overline{\mathbb{F}}_{p})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\prod_{\phi\in\Phi}W(\overline{\mathbb{F}}_{p})\times
W(\overline{\mathbb{F}}_{p}),$

where $\Phi:=\mathrm{Hom}_{\mathbb{Q}}(L,\overline{\mathbb{Q}})$ and for every
$\phi\in\Phi$ we denote by $w(\phi)$ (resp., $\overline{w}(\phi)$) the unique
extension of $\phi$ in $\mathrm{Hom}_{\mathbb{Q}}(LK,\overline{\mathbb{Q}})$
whose restriction to $K$ lies
$\mathrm{Hom}_{\mathbb{Q}_{p}}(K_{\mathfrak{q}_{i}},\overline{\mathbb{Q}}_{p})$
(resp.,
$\mathrm{Hom}_{\mathbb{Q}_{p}}(K_{\overline{\mathfrak{q}}_{i}},\overline{\mathbb{Q}}_{p})$)
for some $i$. By the injection
$\left(\mathcal{O}_{LK}\subseteq\right)\mathcal{O}_{B}\hookrightarrow\mathrm{End}_{X}(Y)$,
$\mathcal{H}$ is equipped with a structure of
$\mathcal{O}_{LK}\otimes_{\mathbb{Z}}W(\overline{\mathbb{F}}_{p})$-module.
Hence the decomposition (247) gives rise to a decomposition of
$(\mathcal{H},\nabla)$ into a direct sum of rank $2$ flat subbundles

(248)
$\displaystyle(\mathcal{H},\nabla)=\bigoplus_{\phi\in\Phi}(\mathcal{H}_{\phi},\nabla_{\phi})\oplus(\mathcal{H}_{\overline{\phi}},\nabla_{\overline{\phi}}),$

(cf. [SZZ, Proposition 4.1]); it restricts to a decomposition
$\mathcal{H}^{1}=\bigoplus_{\phi\in\Phi}\mathcal{H}^{1}_{\phi}\oplus\mathcal{H}_{\overline{\phi}}^{1}$,
where $\mathcal{H}_{\phi}^{1}:=\mathcal{H}^{1}\cap\mathcal{H}_{\phi}$ and
$\mathcal{H}_{\overline{\phi}}^{1}:=\mathcal{H}^{1}\cap\mathcal{H}_{\overline{\phi}}$.
Also, the structure of Dieudonné crystal on $\mathcal{H}$ corresponding to the
abelian scheme $Y/X$ determines, via reduction modulo $p$, an
$\mathcal{O}_{X}$-linear morphism
$\xi:F_{X}^{*}\mathcal{H}\rightarrow\mathcal{H}$ (where $F_{X}$ denotes the
absolute Frobenius endomorphism of $X$). Let us take $\widetilde{\tau}\in\Phi$
with $\widetilde{\tau}|_{F}=\tau$. If $\sigma$ denotes the automorphism
induced by the Frobenius automorphism of $\overline{\mathbb{F}}_{p}$ over
$\mathbb{F}_{p}$, then $\xi$ restricts to an isomorphism

(249)
$\displaystyle\xi_{\widetilde{\tau}}:F_{X}^{*}\mathcal{H}_{\sigma^{-1}\widetilde{\tau}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{H}_{\widetilde{\tau}}$

(cf. [SZZ, Proposition 3.1, (iii)]). Since this isomorphism is compatible with
the connection, we see that $\nabla_{\widetilde{\tau}}$ has vanishing
$p$-curvature. By the latter assertion of [SZZ, Proposition 4.1], both
$\mathcal{H}_{\widetilde{\tau}}^{1}$ and
$\mathcal{H}_{\widetilde{\tau}}/\mathcal{H}_{\widetilde{\tau}}^{1}$ are line
bundles. Moreover, if $p\geq 2g$, then the structure of Higgs field defined as
the Kodaira-Spencer map
$\mathcal{H}_{\widetilde{\tau}}^{1}\rightarrow\Omega_{X}\otimes_{\mathcal{O}_{X}}\left(\mathcal{H}_{\widetilde{\tau}}/\mathcal{H}_{\widetilde{\tau}}^{1}\right)$
is an isomorphism (cf. [SZZ, Proposition 4.4]). It follows that the collection

(250)
$\displaystyle(\mathcal{H}_{\widetilde{\tau}},\nabla_{\widetilde{\tau}},\\{\mathcal{H}^{j}_{\widetilde{\tau}}\\}_{j=0}^{2})$

specifies a dormant $\mathrm{GL}_{2}$-oper. In particular, by taking its
projectivization, one can obtain a dormant $\mathrm{PGL}_{2}$-oper on $X$.

###  6.3. The deformation space of a dormant $G$-oper

Let $(\mathcal{E},\nabla)$ be a flat $G$-bundle on $\mathscr{X}$ with
vanishing $p$-curvature. In the resp’d portion of the following discussion, we
suppose that $(\mathcal{E},\nabla)$ is equipped with a marking, and that
$\nabla$ has residues $\mu\in\mathfrak{g}^{\times r}$ (where $\mu:=\emptyset$
if $r=0$).

Consider the conjugate spectral sequence

(251)
$\displaystyle^{\prime\prime}E_{2}^{a,b}:=H^{a}(X,\mathcal{H}^{b}(\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}]))\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])\hskip
11.38109pt$ (252) $\displaystyle\left(\text{resp.,}\
^{\prime\prime}E_{2}^{a,b}:=H^{a}(X,\mathcal{H}^{b}(\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}]))\Rightarrow\mathbb{H}^{a+b}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])\right)$

associated to $\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}]$ (resp.,
$\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}]$), where
$\mathcal{H}^{b}(\mathcal{K}^{\bullet}[-])$ denotes the $b$-th cohomology
sheaf of the complex $\mathcal{K}^{\bullet}[-]$. This spectral sequence
induces a short exact sequence

(253) $\displaystyle 0\rightarrow
H^{1}(X,\mathrm{Ker}(\nabla^{\mathrm{ad}}))\xrightarrow{{{}^{\prime\prime}e_{\sharp}}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])\xrightarrow{{{}^{\prime\prime}e}_{\flat}}H^{0}(X,\mathrm{Coker}(\nabla^{\mathrm{ad}}))\rightarrow
0\hskip 28.45274pt$ $\displaystyle\left(\text{resp.,}\ 0\rightarrow
H^{1}(X,\mathrm{Ker}({{}^{c}}\nabla^{\mathrm{ad}}))\xrightarrow{{{}^{\prime\prime}e_{\sharp,c}}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])\xrightarrow{{{}^{\prime\prime}e}_{\flat,c}}H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}}))\rightarrow
0\right).$

Recall that the deformation of $p$-curvature may be described as the Cartier
map of the flat bundle $(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$
(cf. [Wak4, Proposition 6.11]), and that this map (resp., this map restricted
to ${{}^{c}}\mathfrak{g}_{\mathcal{E}}$) factors through the quotient
$\Omega\otimes_{\mathcal{O}_{X}}\mathfrak{g}_{\mathcal{E}}\twoheadrightarrow\mathrm{Coker}(\nabla^{\mathrm{ad}})$
(resp., the quotient
$\Omega\otimes_{\mathcal{O}_{X}}{{}^{c}}\mathfrak{g}_{\mathcal{E}}\twoheadrightarrow\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}})$)
(cf. [Ogu1, Proposition 2.2.4]). Hence, the space of first-order deformations
of the flat $G$-bundle $(\mathcal{E},\nabla)$ preserving the condition of
vanishing $p$-curvature (resp., the condition of vanishing $p$-curvature and
fixing residues) may be identified with
$H^{1}(X,\mathrm{Ker}(\nabla^{\mathrm{ad}}))$ (resp.,
$H^{1}(X,\mathrm{Ker}({{}^{c}}\nabla^{\mathrm{ad}}))$). That is, the
isomorphism $\gamma$ (resp., $\gamma_{\mu}$) restricts, via
${{}^{\prime\prime}}e_{\sharp}$ (resp., ${{}^{\prime\prime}}e_{\sharp,c}$), to
an isomorphism of $k$-vector spaces

(254)
$\displaystyle{{}^{p}}\gamma:T_{q}\mathcal{C}onn_{G}^{\psi=0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(X,\mathrm{Ker}(\nabla^{\mathrm{ad}}))\
\left(\text{resp.,}\
{{}^{p}}\gamma_{\mu}:T_{q}\mathcal{C}onn_{G,\mu}^{\psi=0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(X,\mathrm{Ker}({{}^{c}}\nabla^{\mathrm{ad}}))\right).$

Next, suppose further that there exists a structure of $B$-reduction
$\mathcal{E}_{B}$ on $\mathcal{E}$ for which
$\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ specifies a dormant
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
$G$-oper (resp., a dormant
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
$G$-oper of radii $\rho:=\chi(\mu)$, where $\rho:=\emptyset$ if $r=0$). Denote
by $q$ the $k$-rational point of $\mathcal{O}p_{G}^{{}^{\mathrm{Zzz...}}}$
(resp., $\mathcal{O}p_{G,\rho}^{{}^{\mathrm{Zzz...}}}$) classifying
$\mathscr{E}^{\spadesuit}$; we use the same symbol “$q$” to denote the point
of $\mathcal{C}onn_{G}^{\psi=0}$ (resp.,
$\mathcal{C}onn_{G,\widetilde{\rho}}^{\psi=0}$) classifying
$(\mathcal{E},\nabla)$.

We already know the following assertion.

######  Proposition 6.1.

The following equalities hold:

(255) $\displaystyle\mathrm{dim}(T_{q}\mathcal{C}onn_{G}^{\psi=0})$
$\displaystyle=2\cdot\mathrm{dim}(T_{q}\mathcal{O}p_{G})-r\cdot\mathrm{rk}(\mathfrak{g})$
$\displaystyle=2\cdot\mathrm{dim}(T_{q}\mathcal{C}onn_{G}^{\psi=0})+r\cdot\mathrm{rk}(\mathfrak{g})$
$\displaystyle=(2g-2+r)\cdot\mathrm{dim}(\mathfrak{g}).$

In particular, we have

(256)
$\displaystyle\mathrm{dim}(T_{q}\mathcal{O}p_{G})+\mathrm{dim}(T_{q}\mathcal{C}onn_{G}^{\psi=0})=\mathrm{dim}(T_{q}\mathcal{C}onn_{G}^{\psi=0}).$

###### Proof.

The assertion follows from (92), (138), (254), and [Wak4, Proposition 2.23,
6.5, and 6.18]. ∎

We shall set

(257) $\displaystyle
e_{\flat}^{(0)}:H^{0}(X,\mathcal{V}_{G})\xrightarrow{{{}^{\prime}}e_{\sharp}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])\xrightarrow{{{}^{\prime\prime}}e_{\flat}}H^{0}(X,\mathrm{Coker}(\nabla^{\mathrm{ad}}))\hskip
28.45274pt$ $\displaystyle\left(\text{resp.,}\
{{}^{c}}e_{\flat}^{(0)}:H^{0}(X,{{}^{c}}\mathcal{V}_{G})\xrightarrow{{{}^{\prime}}e_{\sharp,c}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])\xrightarrow{{{}^{\prime\prime}}e_{\flat,c}}H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}})).\right)$

The discussion preceding the above proposition (together with the equality
(244)) also shows that the tangent space
$T_{q}\mathcal{O}p_{G}^{{}^{\mathrm{Zzz...}}}$ (resp.,
$T_{q}\mathcal{O}p_{G,\rho}^{{}^{\mathrm{Zzz...}}}$) of
$\mathcal{O}p_{G}^{{}^{\mathrm{Zzz...}}}$ (resp.,
$\mathcal{O}p_{G,\rho}^{{}^{\mathrm{Zzz...}}}$) at $q$ may be identified with
$\mathrm{Ker}(e_{\flat}^{(0)})=\mathrm{Im}({{}^{\prime}}e_{\sharp})\cap\mathrm{Im}({{}^{\prime\prime}}e_{\sharp})$
(resp.,
$\mathrm{Ker}({{}^{c}}e_{\flat}^{(0)})=\mathrm{Im}({{}^{\prime}}e_{\sharp,c})\cap\mathrm{Im}({{}^{\prime\prime}}e_{\sharp,c})$).
That is to say, we obtain an isomorphism of $k$-vector spaces

(258)
$\displaystyle{{}^{p}}\gamma_{\sharp}:=\gamma_{\sharp}\cap{{}^{p}}\gamma:T_{q}\mathcal{O}p_{G}^{{}^{\mathrm{Zzz...}}}\\!\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ker}(e_{\flat}^{(0)})\
\left(\text{resp.,}\
{{}^{p}}\gamma_{\sharp,\rho}:=\gamma_{\sharp,\widetilde{\rho}}\cap{{}^{p}}\gamma_{\rho}:T_{q}\mathcal{O}p_{G,\rho}^{{}^{\mathrm{Zzz...}}}\\!\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ker}({{}^{c}}e_{\flat}^{(0)})\right)$

(cf. [Wak4, Corollary 6.20]).

###  6.4. Ordinariness for dormant $G$-opers

Next, we shall denote by

(259) $\displaystyle\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G,\text{\'{e}t}}$

the étale locus of $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G}$ over $k$. It
follows from [Wak4, Theorem G] that
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G,\text{\'{e}t}}=\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G}$
if $\mathscr{X}$ is a general pointed stable curve and $G$ is either
$\mathrm{PGL}_{n}$ with $2n<p$, $SO_{2l+1}$ with $4l+2<p$, or $PCSp_{2m}$ with
$4m<p$. For example, if $G$ is one of these groups, then this equality holds
for every totally degenerate curve, in the sense of [Wak4, Definition 7.15].

######  Definition 6.2.

We shall say that a dormant $G$-oper $\mathscr{E}^{\spadesuit}$ on
$\mathscr{X}$ is ordinary if it is classified by a point of
$\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{G,\text{\'{e}t}}$, or equivalently, the
morphism $e_{\flat}^{(0)}$ associated to $\mathscr{E}^{\spadesuit}$ is
injective (cf. (258)).

######  Remark 6.3 (Previous studies).

In some of the papers written by the author, the ordinariness for opers and
related notions were discussed in several settings. For example, we refer the
reader to [Wak6, Definitions 2.1.2, 3.4.1] for the ordinariness of
$\mathrm{PGL}_{n}$-opers. On the other hand, the case of dormant indigenous
$(G,H)$-bundles (where $G$ denotes a connected smooth algebraic group and $H$
denotes a suitable closed subgroup of $G$) on a smooth algebraic variety can
be found in [Wak7, Definition 6.7.1].

######  Proposition 6.4.

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a dormant
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
$G$-oper on $\mathscr{X}$. Then, the following three conditions are equivalent
to each other:

  * (a)

$\mathscr{E}^{\spadesuit}$ is ordinary in the sense of Definition 6.2.

  * (b)

The morphism $e_{\flat}^{(0)}$ associated to $\mathscr{E}^{\spadesuit}$ is an
isomorphism.

  * (c)

The composite

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

is an isomorphism.

###### Proof.

The equivalence (b) $\Leftrightarrow$ (c) follows from the exactness of the
sequences (120) and (253). Also, there is nothing to prove for the implication
(b) $\Rightarrow$ (a). Finally, the implication (a) $\Rightarrow$ (b) follows
from the equality

(261)
$\displaystyle\mathrm{dim}H^{0}(X,\mathcal{V}_{G})=\mathrm{dim}H^{0}(X,\mathrm{Coker}(\nabla^{\mathrm{ad}}))\left(=\frac{2g-2+r}{2}\cdot\mathrm{dim}(\mathfrak{g})+\frac{r}{2}\cdot\mathrm{rk}(\mathfrak{g})\right),$

that was already proved in [Wak4, Propositions 2.23 and 6.18] (see also
Proposition 6.1 described above). ∎

######  Corollary 6.5.

Let $\mathscr{E}^{\spadesuit}$ be an ordinary dormant $G$-oper on
$\mathscr{X}$. Then, there exists a canonical decomposition of
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])$ into a direct
sum

(262)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])$
$\displaystyle=H^{0}(X,\mathcal{V}_{G})\oplus
H^{1}(X,\Omega\otimes_{\mathcal{O}_{X}}\mathcal{V}_{G}^{\vee})\left(\cong
H^{0}(X,\mathcal{V}_{G})\oplus
H^{0}(X,{{}^{c}}\mathcal{V}_{G})^{\vee}\right).$

###### Proof.

After possibly replacing $\mathscr{E}^{\spadesuit}$ with its
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normalization (cf. [Wak4, Proposition 2.19]), we may assume that
$\mathscr{E}^{\spadesuit}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal.
Since $e_{\flat}^{(0)}$ is bijective by the equivalence (a) $\Leftrightarrow$
(b) asserted in Proposition 6.4, we obtain the composite surjection

(263)
$\displaystyle\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])\xrightarrow{{{}^{\prime\prime}}e_{\flat}}H^{1}(X,\mathrm{Coker}(\nabla^{\mathrm{ad}}))\xrightarrow{(e_{\flat}^{(0)})^{-1}}H^{0}(X,\mathcal{V}_{G}).$

It specifies a split surjection of the short exact sequence (120) in the case
where $\Box$ denotes the absence of “$c$”. In particular, this split
surjection determines the desired decomposition (262). ∎

Let $\mathscr{E}^{\spadesuit}$ and $q$ be as in § 6.3. Suppose further that
$\mathscr{E}^{\spadesuit}$ is ordinary, which means that the two substacks
$\mathcal{O}p_{G}$, $\mathcal{C}onn_{G}^{\psi=0}$ of $\mathcal{C}onn_{G}$
intersect transversally at $q$. Since $e_{\flat}^{(0)}$ is an isomorphism (cf.
Proposition 6.4), the morphism

(264)
$\displaystyle({{}^{\prime}}e_{\sharp},{{}^{\prime\prime}}e_{\sharp}):H^{0}(X,\mathcal{V}_{G})\oplus
H^{1}(X,\mathrm{Ker}(\nabla^{\mathrm{ad}}))\rightarrow\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])$

turns out to be an isomorphism. Under the identifications between various
$k$-vector spaces given by $\gamma$ (cf. (92)), $\gamma_{\sharp}$ (cf. (138)),
and ${{}^{p}}\gamma$ (cf. (254)), the isomorphism (264) reads a decomposition

(265) $\displaystyle T_{q}\mathcal{O}p_{G}\oplus
T_{q}\mathcal{C}onn_{G}^{\psi=0}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T_{q}\mathcal{C}onn_{G}$

of the tangent space $T_{q}\mathcal{C}onn_{G}$; it can be obtained by
differentiating the immersions
$\mathcal{O}p_{G}\hookrightarrow\mathcal{C}onn_{G}$ and
$\mathcal{C}onn_{G}^{\psi=0}\hookrightarrow\mathcal{C}onn_{G}$ at $q$.

##  7\. Eichler-Shimura-type isomorphisms for dormant opers

This final section is devoted to proving Theorem A, which establishes an
Eichler-Shimura-type decomposition for a dormant $G$-oper on a general pointed
stable curve. This is constructed by relating it to the decomposition of a
tangent space already obtained in (265).

###  7.1. The irreducible decomposition of the deformation space

Let $\mathscr{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla)$ be a dormant
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
$G$-oper on $\mathscr{X}$. Suppose that $\mathscr{E}^{\spadesuit}$ is of radii
$\rho:=(\rho_{i})_{i=1}^{r}\in\mathfrak{c}(k)^{\times r}$, where
$\rho:=\emptyset$ if $r=0$.

Note that the dual
$(\mathfrak{g}_{\mathcal{E}}^{\vee},\nabla^{\mathrm{ad}\vee})$ of the flat
bundle $(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$ is isomorphic to
$(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$ itself because of the
nondegeneracy of the Killing form on $\mathfrak{g}$. Hence, by applying [Wak4,
Corollary 6.16] to the case where “$(\mathcal{V},\nabla)$” is taken to be
$(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$, we obtain an isomorphism
of short exact sequences:

(270)

######  Lemma 7.1.

The isomorphism $e_{\flat}^{(0)}$ associated to $\mathscr{E}^{\spadesuit}$
(cf. Proposition 6.4) restricts to an isomorphism of $k$-vector spaces

(271) $\displaystyle
H^{0}(X,{{}^{c}}\mathcal{V}_{G})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}})).$

###### Proof.

Note that both the log connections $\nabla^{\mathrm{ad}}$ and
${{}^{c}}\nabla^{\mathrm{ad}}$ have vanishing $p$-curvature. Hence, the
morphism
$\delta:\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}})\rightarrow\mathrm{Coker}(\nabla^{\mathrm{ad}})$
induced by the inclusion of complexes
$\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}]\hookrightarrow\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}]$
is injective because of the commutativity of the following square diagram:

(276)

where

  * •

$X^{(1)}$ denotes the Frobenius twist of $X$ over $k$, and $\Omega^{(1)}$
denotes the pull-back of $\Omega$ along the base-change morphism
$X^{(1)}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X$ induced by the
Frobenius automorphism of $k$;

  * •

we regard both $\mathrm{Ker}({{}^{c}}\nabla^{\mathrm{ad}})$ and
$\mathrm{Ker}(\nabla^{\mathrm{ad}})$ as $\mathcal{O}_{X^{(1)}}$-modules via
the underlying homeomorphism of the relative Frobenius morphism $X\rightarrow
X^{(1)}$;

  * •

the right-hand and left-hand vertical arrows denote the isomorphisms induced
by the Cartier operators of
$(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$ and
$({{}^{c}}\mathfrak{g}_{\mathcal{E}},{{}^{c}}\nabla^{\mathrm{ad}})$,
respectively (cf. the discussion following [Ogu1, Proposition 1.2.4]).

In particular, the morphism

(277) $\displaystyle
H^{0}(\delta):H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}}))\rightarrow
H^{0}(X,\mathrm{Coker}(\nabla^{\mathrm{ad}}))$

induced by $\delta$ is injective. It follows that the composite of natural
morphisms
${{}^{c}}\mathcal{V}_{G}[-1]\rightarrow\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}]\rightarrow\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}})[-1]$
gives a restriction

(278) $\displaystyle H^{0}(X,{{}^{c}}\mathcal{V}_{G})\hookrightarrow
H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}}))$

of $e_{\flat}^{(0)}$ via the injection $H^{0}(\delta)$ and the natural
inclusion $H^{0}(X,{{}^{c}}\mathcal{V}_{G})\hookrightarrow
H^{0}(X,\mathcal{V}_{G})$. Since the dual
$(\mathfrak{g}_{\mathcal{E}}^{\vee},\nabla^{\mathrm{ad}\vee})$ of the flat
bundle $(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$ is isomorphic to
$(\mathfrak{g}_{\mathcal{E}},\nabla^{\mathrm{ad}})$ itself because of the
nondegeneracy of the Killing form on $\mathfrak{g}$, it follows from [Wak4,
Corollary 6.16] that $H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}}))$
is isomorphic to $H^{1}(X,\mathrm{Ker}(\nabla^{\mathrm{ad}}))^{\vee}$. Hence,
this fact together with [Wak4, Propositions 2.23 and 6.18] implies the
equality

(279)
$\displaystyle\mathrm{dim}(H^{0}(X,{{}^{c}}\mathcal{V}_{G}))=\mathrm{dim}(H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}})))\left(=\frac{2g-2+r}{2}\cdot\mathrm{dim}(\mathfrak{g})-\frac{r}{2}\cdot\mathrm{rk}(\mathfrak{g})\right).$

Thus, (278) is verified to be an isomorphism, as desired. ∎

Next, suppose that $\mathscr{E}^{\spadesuit}$ is ordinary. The dual of the
composite

(280) $\displaystyle
H^{0}(X,\mathrm{Coker}(\nabla^{\mathrm{ad}}))\xrightarrow{(e_{\flat}^{(0)})^{-1}}H^{0}(X,\mathcal{V}_{G})\xrightarrow{{{}^{\prime}}e_{\sharp}}\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[\nabla^{\mathrm{ad}}])$

determines a split surjection of the lower horizontal sequence in (270).
Hence, it induces a splitting of the upper horizontal one. Under the
identifications between various $k$-vector spaces given by
$\gamma_{\widetilde{\rho}}$ (cf. (93)), ${{}^{p}}\gamma_{\widetilde{\rho}}$
(cf. (254)), and ${{}^{c}}e_{\flat}^{(0)}\circ\gamma_{\sharp,\rho}$ (cf.
(138), (257)), the resulting decomposition of
$\mathbb{H}^{1}(X,\mathcal{K}^{\bullet}[{{}^{c}}\nabla^{\mathrm{ad}}])$
determines a direct sum decomposition

(281) $\displaystyle T_{q}\mathcal{C}onn_{G,\widetilde{\rho}}^{\psi=0}\oplus
T_{q}\mathcal{O}p_{G,\rho}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T_{q}\mathcal{C}onn_{G,\widetilde{\rho}}$

of $T_{q}\mathcal{C}onn_{G,\widetilde{\rho}}$; it coincides with the morphism
induced by differentiating the inclusions
$\mathcal{O}p_{G,\rho}\hookrightarrow\mathcal{C}onn_{G,\widetilde{\rho}}$ and
$\mathcal{C}onn_{G,\widetilde{\rho}}^{\psi=0}\hookrightarrow\mathcal{C}onn_{G,\widetilde{\rho}}$.

Then, the following assertion holds.

######  Theorem 7.2.

Let $\mathscr{E}^{\spadesuit}$ be an ordinary dormant
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal
$G$-oper on $\mathscr{X}$. Let us consider the following two composite
isomorphisms:

(282) $\displaystyle T_{q}\mathcal{C}onn_{G,\widetilde{\rho}}^{\psi=0}$
$\displaystyle\xrightarrow{{{}^{p}}\gamma_{\widetilde{\rho}}}H^{1}(X,\mathrm{Ker}({{}^{c}}\nabla^{\mathrm{ad}}))\xrightarrow{\sim}H^{0}(X,\mathrm{Coker}(\nabla^{\mathrm{ad}}))^{\vee}$
$\displaystyle\xrightarrow{(e_{\flat}^{(0)})^{\vee}}H^{0}(X,\mathcal{V}_{G})^{\vee}\xrightarrow{\gamma_{\sharp}^{\vee}}T^{\vee}_{q}\mathcal{O}p_{G},$
(283) $\displaystyle T_{q}\mathcal{O}p_{G,\rho}$
$\displaystyle\xrightarrow{\gamma_{\sharp,\rho}}H^{0}(X,{{}^{c}}\mathcal{V}_{G})\xrightarrow{{{}^{c}}e_{\flat}^{(0)}}H^{0}(X,\mathrm{Coker}({{}^{c}}\nabla^{\mathrm{ad}}))$
$\displaystyle\xrightarrow{\sim}H^{1}(X,\mathrm{Ker}(\nabla^{\mathrm{ad}}))^{\vee}\xrightarrow{({{}^{p}}\gamma)^{\vee}}T_{q}^{\vee}\mathcal{C}onn_{G}^{\psi=0}$

(cf. (270) for the definitions of the second arrow in (282) and the third
arrow in (283)). Then, these isomorphisms make the following square diagram
commute:

(288)

In particular, if $r=0$, then the subspace $T_{q}\mathcal{C}onn_{G}^{\psi=0}$
of $T_{q}\mathcal{C}onn_{G}$ is Lagrangian with respect to the bilinear form
on $T_{q}\mathcal{C}onn_{G}$ determined by (94).

###### Proof.

The assertion follows from Theorem 4.12, (ii), and the definitions of various
morphisms involved. ∎

######  Remark 7.3 (Case of the parabolic de Rham cohomology).

When $r>0$, we cannot construct a direct sum decomposition of
$T_{q}\mathcal{C}onn_{G,[0]^{\times r}}$ using a dormant $G$-oper in the same
manner as (265) or (281). In fact, as discussed in [Wak4, Remark 3.37], any
dormant $G$-oper cannot have radii $[0]^{\times r}$.

###  7.2. The canonical decomposition derived from ordinariness

Next, let
$\mathscr{E}^{\spadesuit}_{\odot}:=(\mathcal{E}_{B^{\odot}},\nabla_{\odot})$
be a dormant $G^{\odot}$-oper on $\mathscr{X}$, and write
$\mathcal{E}_{\odot}:=\mathcal{E}_{B^{\odot}}\times^{B^{\odot}}G^{\odot}$ and
$\mathscr{E}_{\odot}:=(\mathcal{E}_{\odot},\nabla_{\odot})$.

######  Proposition 7.4.

The following two conditions are equivalent to each other:

  * (a)

The dormant $G$-oper $\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})$ (cf. (43))
is ordinary;

  * (b)

For each integer $l$ with $1\leq l\leq\mathrm{rk}(\mathfrak{g})$, the
composite morphism

(289)
$\displaystyle\kappa_{l}:H^{1}(X,\mathrm{Ker}(\mathrm{Sym}^{2l}\nabla_{\odot}))\xrightarrow{\theta_{l}}H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{2l}\mathscr{E}_{\odot})\longrightarrow
H^{1}(X,\Omega^{\otimes(-l)})$

(cf. (222) for the definition of the second arrow) is an isomorphism, where
the first arrow $\theta_{l}$ arises from the natural inclusion
$\mathrm{Ker}(\mathrm{Sym}^{2l}\nabla_{\odot})[0]\rightarrow\mathcal{K}^{\bullet}[\mathrm{Sym}^{2l}\nabla_{\odot}]$.

###### Proof.

After possibly replacing $\mathscr{E}^{\spadesuit}_{\odot}$ with its
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip
1.42262pt$-normalization, we may assume that
$\mathscr{E}^{\spadesuit}_{\odot}$ is
$\,\,q_{1}\rotatebox[origin={c}]{-10.0}{$\swarrow$}\hskip 1.42262pt$-normal.
We shall write
$\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot}):=(\mathcal{E}_{B},\nabla)$. The
isomorphism (229) restricts to an isomorphism

(290)
$\displaystyle\bigoplus_{l=1}^{\mathrm{rk}(\mathfrak{g})}\mathrm{Ker}(\mathrm{Sym}^{2l}\nabla_{\odot})\otimes_{k}\mathfrak{g}_{l}^{\mathrm{ad}(q_{1})}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ker}(\nabla^{\mathrm{ad}}).$

It induces a commutative square diagram

(295)

By composing (295) and the right-hand square diagram in (235), we obtain a
commutative square

(300)

(cf. (260) for the definition of $e_{\sharp}^{(0)}$). The commutativity of
this diagram implies that $\kappa_{l}$ is an isomorphism for every $l$ if and
only if $e_{\sharp}^{(0)}$ is an isomorphism. Hence, the assertion follows
from the equivalence (a) $\Leftrightarrow$ (c) asserted in Proposition 6.4. ∎

######  Corollary 7.5.

Let $\mathscr{E}_{\odot}^{\spadesuit}$ be a dormant $G^{\odot}$-oper on
$\mathscr{X}$. Also, let $G^{\prime}$ be another algebraic group satisfying
the same conditions imposed on $G$ (cf. § 1.4). Suppose that
$\mathrm{rk}(\mathfrak{g})\geq\mathrm{rk}(\mathfrak{g}^{\prime})$ and that the
$G$-oper $\iota_{G*}(\mathscr{E}^{\spadesuit})$ is ordinary. Then, the dormant
$G^{\prime}$-oper $\iota_{G^{\prime}*}(\mathscr{E}^{\spadesuit})$ is ordinary.

###### Proof.

The assertion can be immediately proved by applying Proposition 7.4 to both
$\iota_{G*}(\mathscr{E}^{\spadesuit})$ and
$\iota_{G^{\prime}*}(\mathscr{E}^{\spadesuit})$. ∎

######  Corollary 7.6.

Let $\mathscr{E}_{\odot}^{\spadesuit}$ be a dormant $G^{\odot}$-oper on
$\mathscr{X}$. Suppose that $\mathscr{X}$ is general in the moduli stack
$\overline{\mathcal{M}}_{g,r}$ of $r$-pointed stable curves of genus $g$ over
$k$. (To be precise, $\mathscr{X}$ specifies a point of
$\overline{\mathcal{M}}_{g,r}$ that lies outside some fixed closed substack.)

  * (i)

Suppose that $\mathrm{rk}(\mathfrak{g})\leq\frac{p-3}{2}$. Then, the dormant
$G$-oper $\iota_{G*}(\mathscr{E}^{\spadesuit}_{\odot})$ associated to any
dormant $G^{\odot}$-oper $\mathscr{E}^{\spadesuit}_{\odot}$ on $\mathscr{X}$
is ordinary.

  * (ii)

For each integer $l$ with $1\leq l\leq\frac{p-3}{2}$, the morphism
$\kappa_{l}$ associated to $\mathscr{E}^{\spadesuit}_{\odot}$ (introduced in
Proposition 7.4) is an isomorphism.

###### Proof.

First, let us prove assertion (i). For simplicity, we shall set
$G^{\circledast}:=\mathrm{PGL}_{\frac{p-1}{2}}$, which has rank
$\frac{p-3}{2}$. By applying Corollary 7.5 to both $G$ and $G^{\circledast}$,
we can reduce the problem to the case of $G=G^{\circledast}$. Denote by
$\mathfrak{O}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{G^{\odot}}$ (resp.,
$\mathfrak{O}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{G^{\circledast}}$) the
Deligne-Mumford stack classifying pairs
$(\mathscr{X},\mathscr{E}^{\spadesuit})$ consisting of
$\mathscr{X}\in\mathrm{Ob}(\overline{\mathcal{M}}_{g,r})$ and a dormant
$G^{\odot}$-oper (resp., a dormant $G^{\circledast}$-oper)
$\mathscr{E}^{\spadesuit}$ on $\mathscr{X}$. According to [Moc2, Chap. II,
Theorem 2.8], the stack
$\mathfrak{O}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{G^{\odot}}$ is irreducible,
and the natural projection
$\mathfrak{O}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{G^{\odot}}\rightarrow\overline{\mathcal{M}}_{g,r}$
is finite and faithfully flat. Hence, one can find a unique irreducible
component $\mathcal{N}$ of
$\mathfrak{O}\mathfrak{p}_{G^{\circledast}}^{{}^{\mathrm{Zzz...}}}$ containing
the image of the $\overline{\mathcal{M}}_{g,r}$-morphism

(301)
$\displaystyle\mathfrak{O}\mathfrak{p}_{G^{\odot}}^{{}^{\mathrm{Zzz...}}}\rightarrow\mathfrak{O}\mathfrak{p}_{G^{\circledast}}^{{}^{\mathrm{Zzz...}}}$

given by assigning
$(\mathscr{X},\mathscr{E}^{\spadesuit})\mapsto(\mathscr{X},\iota_{G*}(\mathscr{E}^{\spadesuit}))$.
If
$\mathfrak{O}\mathfrak{p}_{G^{\circledast},\text{\'{e}t}}^{{}^{\mathrm{Zzz...}}}$
denotes the étale locus of
$\mathfrak{O}\mathfrak{p}_{G^{\circledast}}^{{}^{\mathrm{Zzz...}}}$ over
$\overline{\mathcal{M}}_{g,r}$, then the intersection
$\mathfrak{O}\mathfrak{p}_{G^{\circledast},\text{\'{e}t}}^{{}^{\mathrm{Zzz...}}}\cap\mathcal{N}$
defines a dense open substack of $\mathcal{N}$ (cf. [Wak4, Theorem G]). Since
$\mathfrak{O}\mathfrak{p}_{G^{\circledast}}^{{}^{\mathrm{Zzz...}}}$ is finite
over $\overline{\mathcal{M}}_{g,r}$ (cf. [Wak4, Theorem 3.34]), the
restriction
$\mathfrak{O}\mathfrak{p}_{G^{\odot}}^{{}^{\mathrm{Zzz...}}}\rightarrow\mathcal{N}$
of (301) is dominant. This implies that the inverse image $\mathcal{Q}$ of
$\mathfrak{O}\mathfrak{p}_{G^{\circledast},\text{\'{e}t}}^{{}^{\mathrm{Zzz...}}}\cap\mathcal{N}$
via (301) is a dense open substack of
$\mathfrak{O}\mathfrak{p}_{G^{\odot}}^{{}^{\mathrm{Zzz...}}}$. The complement
in $\overline{\mathcal{M}}_{g,r}$ of the image of
$\mathfrak{O}\mathfrak{p}_{G^{\odot}}^{{}^{\mathrm{Zzz...}}}\setminus\mathcal{Q}$
forms a dense open substack of $\overline{\mathcal{M}}_{g,r}$ and by
definition classifies pointed stable curves $\mathscr{X}$ such that
$\iota_{G^{\circledast}*}(\mathscr{E}^{\spadesuit})$ is ordinary for any
dormant $G^{\odot}$-oper $\mathscr{E}^{\spadesuit}$ on $\mathscr{X}$. This
completes the proof of assertion (i).

Moreover, assertion (ii) follows from assertion (i) together with Proposition
7.4. ∎

We conclude this paper with the following assertions.

######  Theorem 7.7.

Let $\mathscr{E}^{\spadesuit}_{\odot}$ be a dormant $G^{\odot}$-oper on
$\mathscr{X}$, and write $\mathscr{E}_{\odot}$ for the underlying flat
$G^{\odot}$-bundle of $\mathscr{E}^{\spadesuit}_{\odot}$. Also, let $l$ be an
integer with $1\leq l\leq\frac{p-3}{2}$. Suppose that $\mathscr{X}$ is general
in $\overline{\mathcal{M}}_{g,r}$. Then, there exists a canonical splitting of
(222). In particular, we obtain a canonical decomposition

(302) $\displaystyle
H^{1}_{\mathrm{dR}}(X,\mathrm{Sym}^{2l}\mathscr{E}_{\odot})$
$\displaystyle=H^{0}(X,\Omega^{\otimes(l+1)})\oplus
H^{1}(X,\Omega^{\otimes(-l)})$ $\displaystyle\left(\cong
H^{0}(X,\Omega^{\otimes(l+1)})\oplus
H^{0}(X,\Omega^{\otimes(l+1)}(-D))^{\vee}\right)$

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

###### Proof.

It follows from Corollary 7.6, (ii), that $\kappa_{l}$ is an isomorphism for
every $l=1,\cdots,\frac{p-3}{2}$. Then, the composite injection

(303) $\displaystyle
H^{1}(X,\Omega^{\otimes(-l)})\xrightarrow{\kappa_{l}^{-1}}H^{1}(X,\mathrm{Ker}(\mathrm{Sym}^{2l}\nabla_{\odot}))\xrightarrow{\theta_{l}}H_{\mathrm{dR}}^{1}(X,\mathrm{Sym}^{2l}\mathscr{E}_{\odot}).$

specifies the desired splitting. ∎

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

Let $\mathscr{F}^{\heartsuit}_{\odot}$ be a dormant $\mathrm{SL}_{2}$-oper on
$\mathscr{X}$, and $l$ an integer with $1\leq l\leq\frac{p-3}{2}$. Denote by
$\varTheta$ the theta characteristic of $\mathscr{X}$ associated to
$\mathscr{F}^{\heartsuit}_{\odot}$, and by $\mathscr{F}_{\odot}$ the
underlying flat $\mathrm{SL}_{2}$-bundle of
$\mathscr{F}^{\heartsuit}_{\odot}$. Suppose that $\mathscr{X}$ is general in
$\overline{\mathcal{M}}_{g,r}$. Then, there exists a canonical splitting of
(207) for $n=2l+1$. In particular, we obtain a canonical decomposition

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

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

###### Proof.

The assertion follows from Theorem 7.7 and the comment at the end of the
statement in Theorem 5.3. ∎

######  Remark 7.9.

In [Wak3, § 2.4], the author constructed a $p$-adic version of the Eichler-
Shimura isomorphism for the $2$-nd symmetric product of an ordinary nilpotent
indigenous bundle, i.e., a certain $G^{\odot}$-oper with nilpotent
$p$-curvature (cf. [Moc1, Chap. II, Definition 3.1]). This result is an
essential ingredient for proving the main theorem in [Wak3], which compares
canonical symplectic structures on the related moduli spaces. Despite the
similarity in concept, this result seems to have no (at least direct) relation
with the decomposition asserted in the above theorem because the
$G^{\odot}$-opers treated there never have vanishing $p$-curvature. However,
we also expect that the previous $p$-adic version extends to the higher
symmetric products, generalizing Faltings’ construction of the $p$-adic
Eichler-Shimura isomorphisms for modular curves (cf. [Fal, Theorem 6]).

### Acknowledgements

We would like to thank modular curves for their helpful comments on the
Eichler-Shimura isomorphism. Our work was partially supported by Grant-in-Aid
for Scientific Research (KAKENHI No. 21K13770).

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