2.1 Notations related to reductive groups

Let $G$ be a reductive algebraic group over $k$ of rank $l$ . We denote by $\breve{G}$ its Langlands dual group over $k$ . We denote by $\mathfrak{g}$ (respectively by $\breve{\mathfrak{g}}$ ) the Lie algebra of $G$ (respectively $\breve{G}$ ). Let $T$ denote the abstract Cartan of $G$ with its Lie algebra $\mathfrak{t}$ . The counterparts on the Langlands dual side are denoted by $\breve{T},\breve{\mathfrak{t}}$ . We denote by $\text{W}$ the abstract Weyl group of $G$ , which acts on $T$ and $\breve{T}$ . We denote by $\mathbb{X}^{\bullet }(T)$ or simply by $\mathbb{X}^{\bullet }$ (respectively by $\mathbb{X}_{\bullet }(T)$ or simply by $\mathbb{X}_{\bullet }$ ) the character (respectively the cocharacter) group of $T$ . Let $\unicode[STIX]{x1D6F7}\subset \mathbb{X}^{\bullet }(T)$ be the set of roots. Sometimes, we also fix a set of simple roots $\{\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{l}\}$ and an embedding $\mathfrak{t}\subset \mathfrak{g}$ . Then for $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}$ , let $\mathfrak{g}_{\unicode[STIX]{x1D6FC}}\subset \mathfrak{g}$ denote the corresponding root subspace.

From now on, we assume that the $\operatorname{char}~k=p$ is zero or $p\nmid |\text{W}|$ . We fix a $\text{W}$ -invariant non-degenerate bilinear form $(\,,\,):\mathfrak{t}\times \mathfrak{t}\rightarrow k$ and identify $\mathfrak{t}$ with $\breve{\mathfrak{t}}$ using $(\,,\,)$ . This invariant form also determines a unique $G$ -invariant non-degenerate bilinear form $\mathfrak{g}\times \mathfrak{g}\rightarrow k$ , still denoted by $(\,,\,)$ . Let $\mathfrak{g}\simeq \mathfrak{g}^{\ast }$ be the resulting $G$ -equivariant isomorphism.

2.2 Hitchin map

Let $k[\mathfrak{g}]$ and $k[\mathfrak{t}]$ be the algebras of polynomial functions on $\mathfrak{g}$ and on $\mathfrak{t}$ respectively. By Chevalley’s theorem, we have an isomorphism $k[\mathfrak{g}]^{G}\simeq k[\mathfrak{t}]^{\text{W}}$ . Moreover, $k[\mathfrak{t}]^{\text{W}}$ is isomorphic to a polynomial ring of $l$ variables $u_{1},\ldots ,u_{l}$ and each $u_{i}$ is homogeneous in degree $e_{i}$ . Let $\mathfrak{c}=\operatorname{Spec}(k[\mathfrak{t}]^{\text{W}})$ . The natural $\mathbb{G}_{m}$ action on $\mathfrak{g}$ induces a $\mathbb{G}_{m}$ -action on $\mathfrak{c}$ and under the isomorphism $\mathfrak{c}\simeq \operatorname{Spec}(k[u_{1},\ldots ,u_{l}])\simeq \mathbb{A}^{l}$ the action is given by

Let $\unicode[STIX]{x1D712}:\mathfrak{g}\rightarrow \mathfrak{c}$ be the map induced by $k[\mathfrak{c}]\simeq k[\mathfrak{g}]^{G}{\hookrightarrow}k[\mathfrak{g}]$ . It is a $G\times \mathbb{G}_{m}$ -equivariant map where $G$ acts trivially on $\mathfrak{c}$ . Similarly, let $\boldsymbol{\unicode[STIX]{x1D70B}}:\mathfrak{t}\rightarrow \mathfrak{c}$ be the map induced by $k[\mathfrak{c}]{\hookrightarrow}k[\mathfrak{t}]$ , which is also $\mathbb{G}_{m}$ -equivariant. Let ${\mathcal{L}}$ be an invertible sheaf on $C$ and ${\mathcal{L}}^{\times }$ be the corresponding $\mathbb{G}_{m}$ -torsor. We denote by $\mathfrak{g}_{{\mathcal{L}}}=\mathfrak{g}\times ^{\mathbb{G}_{m}}{\mathcal{L}}^{\times }$ , $\mathfrak{t}_{{\mathcal{L}}}=\mathfrak{t}\times ^{\mathbb{G}_{m}}{\mathcal{L}}^{\times }$ , and $\mathfrak{c}_{{\mathcal{L}}}=\mathfrak{c}\times ^{\mathbb{G}_{m}}{\mathcal{L}}^{\times }$ the $\mathbb{G}_{m}$ -twist of $\mathfrak{g}$ , $\mathfrak{t}$ , and $\mathfrak{c}$ with respect to the natural $\mathbb{G}_{m}$ -action.

Let $\operatorname{Higgs}_{G,{\mathcal{L}}}=\operatorname{Sect}(C,[\mathfrak{g}_{{\mathcal{L}}}/G])$ be the stack of sections of $[\mathfrak{g}_{{\mathcal{L}}}/G]$ over $C$ . That is, for each $k$ -scheme $S$ the groupoid $\operatorname{Higgs}_{G,{\mathcal{L}}}(S)$ consist of maps

or, equivalently, those maps

such that the composition of $h_{E,\unicode[STIX]{x1D719}}$ with the projection $[\mathfrak{g}/G\times \mathbb{G}_{m}]\rightarrow B\mathbb{G}_{m}$ is given by the $\mathbb{G}_{m}$ -torsor ${\mathcal{L}}^{\times }$ . Explicitly, $\operatorname{Higgs}_{G,{\mathcal{L}}}(S)$ consist of pairs $(E,\unicode[STIX]{x1D719})$ (called Higgs bundles), where $E$ is an $G$ -torsor over $C\times S$ and $\unicode[STIX]{x1D719}$ is an element in $\unicode[STIX]{x1D6E4}(C\times S,\operatorname{ad}(E)\otimes {\mathcal{L}})$ known as the Higgs field. If the group $G$ is clear from the context, we simply write $\operatorname{Higgs}_{{\mathcal{L}}}$ for $\operatorname{Higgs}_{G,{\mathcal{L}}}$ .

Let $B_{{\mathcal{L}}}=\operatorname{Sect}_{\operatorname{Spec}k}(C,\mathfrak{c}_{{\mathcal{L}}})$ be the scheme of sections of $\mathfrak{c}_{{\mathcal{L}}}$ over $C$ . That is, for each $k$ -scheme $S$ , $B_{{\mathcal{L}}}(S)$ is the set of sections

or, equivalently, those maps

such that the composition of $b$ with the projection $[\mathfrak{c}/\mathbb{G}_{m}]\rightarrow B\mathbb{G}_{m}$ is given by ${\mathcal{L}}^{\times }$ . It is called the Hitchin base of $G$ .

The natural $G$ -invariant projection $\unicode[STIX]{x1D712}:\mathfrak{g}\rightarrow \mathfrak{c}$ induces a map

or more generally

The map $[\unicode[STIX]{x1D712}_{{\mathcal{L}}}]$ induces a natural map

For any $b\in B_{{\mathcal{L}}}(S)$ we denote by $\operatorname{Higgs}_{{\mathcal{L}},b}$ the fiber product $S\times _{B_{{\mathcal{L}}}}\operatorname{Higgs}_{{\mathcal{L}}}$ .

Observe that the invariant bilinear form $\mathfrak{t}\times \mathfrak{t}\rightarrow k$ induces a canonical isomorphism $\mathfrak{t}\simeq \mathfrak{t}^{\ast }=:\breve{\mathfrak{t}}$ , compatible with the $\text{W}$ -action. Therefore, there is a canonical isomorphism $\mathfrak{c}\simeq \breve{\mathfrak{c}}$ and $B_{{\mathcal{L}}}\simeq \breve{B}_{{\mathcal{L}}}$ . In what follows, we will identify them.

Let $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{C}$ be the canonical line bundle of $C$ . We are mostly interested in the case ${\mathcal{L}}=\unicode[STIX]{x1D714}$ . For simplicity, from now on we denote $B=B_{\unicode[STIX]{x1D714}}$ , $\operatorname{Higgs}=\operatorname{Higgs}_{\unicode[STIX]{x1D714}}$ , $h=h_{\unicode[STIX]{x1D714}}:\operatorname{Higgs}\rightarrow B$ , and $\operatorname{Higgs}_{b}=\operatorname{Higgs}_{\unicode[STIX]{x1D714}_{C},b}$ . We sometimes also write $\operatorname{Higgs}_{G}$ for $\operatorname{Higgs}$ to emphasize the group $G$ . Observe that the bilinear form as in §2.1 together with the Serre duality induces an isomorphism $\operatorname{Higgs}\simeq T^{\ast }\operatorname{Bun}_{G}$ (cf. [Reference HitchinHit87]).

2.3 The Kostant section

In this section, we recall the construction of the Kostant section of the Hitchin map $h_{{\mathcal{L}}}$ . For each simple root $\unicode[STIX]{x1D6FC}_{i}$ we choose a non-zero vector $f_{i}\in \mathfrak{g}_{-\unicode[STIX]{x1D6FC}_{i}}$ . Let $f=\bigoplus _{i=1}^{l}f_{i}\in \mathfrak{g}$ . We complete $f$ into an $\mathfrak{s}\mathfrak{l}_{2}$ triple $\{f,h,e\}$ and denote by $\mathfrak{g}^{e}$ the centralizer of $e$ in $\mathfrak{g}$ . A theorem of Kostant says that $f+\mathfrak{g}^{e}$ consist of regular elements in $\mathfrak{g}$ and the restriction of $\unicode[STIX]{x1D712}:\mathfrak{g}\rightarrow \mathfrak{c}$ to $f+\mathfrak{g}^{e}$ is an isomorphism onto $\mathfrak{c}$ . We denote by

the inverse of $\unicode[STIX]{x1D712}|_{f+\mathfrak{g}^{e}}$ . Let $\unicode[STIX]{x1D70C}(\mathbb{G}_{m})$ denote the following $\mathbb{G}_{m}$ -action on $\mathfrak{g}$ : it acts trivially on $\mathfrak{t}$ , and on $\mathfrak{g}_{\unicode[STIX]{x1D6FC}}$ by $\unicode[STIX]{x1D70C}(t)x=t^{\operatorname{ht}(\unicode[STIX]{x1D6FC})}x$ where $\operatorname{ht}(\unicode[STIX]{x1D6FC})=\sum n_{i}$ if $\unicode[STIX]{x1D6FC}=\sum n_{i}\unicode[STIX]{x1D6FC}_{i}$ . We have $\unicode[STIX]{x1D70C}(t)f=t^{-1}f$ and $\unicode[STIX]{x1D70C}(t)e=te$ , in particular $\mathfrak{g}^{e}$ is invariant under $\unicode[STIX]{x1D70C}(\mathbb{G}_{m})$ . We define a new $\mathbb{G}_{m}$ -action on $\mathfrak{g}$ by $\unicode[STIX]{x1D70C}^{+}(t)=t\unicode[STIX]{x1D70C}(t)$ . Then $\unicode[STIX]{x1D70C}^{+}(t)f=f$ and $\unicode[STIX]{x1D70C}^{+}(\mathbb{G}_{m})$ preserves $f+\mathfrak{g}^{e}$ . With respect to this action, the isomorphism $\text{kos}:\mathfrak{c}\simeq f+\mathfrak{g}^{e}$ is $\mathbb{G}_{m}$ -equivariant.

The diagonal map $\mathbb{G}_{m}\rightarrow \mathbb{G}_{m}\times \mathbb{G}_{m}$ induces a map

By precomposing with the map $[\mathfrak{c}/\mathbb{G}_{m}]\stackrel{\text{kos}}{\simeq }[f+\mathfrak{g}^{e}/\unicode[STIX]{x1D70C}^{+}(\mathbb{G}_{m})]\rightarrow [\mathfrak{g}/\unicode[STIX]{x1D70C}^{+}(\mathbb{G}_{m})]$ we obtain

If the action of $\unicode[STIX]{x1D70C}(\mathbb{G}_{m})$ on $\mathfrak{g}$ factors through the adjoint action of $G$ , for example when $G$ is adjoint, then there is a map $[\mathfrak{g}/\mathbb{G}_{m}\times \unicode[STIX]{x1D70C}(\mathbb{G}_{m})]\rightarrow [\mathfrak{g}/\mathbb{G}_{m}\times G]$ which defines a section

of (2.2.1), and in particular, we get a section of $h_{{\mathcal{L}}}$ . In general, the action $\unicode[STIX]{x1D70C}(\mathbb{G}_{m})$ does not necessarily factor through $G$ , but its square does since it is given by the cocharacter $2\unicode[STIX]{x1D70C}:\mathbb{G}_{m}\rightarrow G$ where $2\unicode[STIX]{x1D70C}$ is the sum of positive coroots. So if we denote $\mathbb{G}_{m}^{[2]}\rightarrow \mathbb{G}_{m}$ the square map (so $\mathbb{G}_{m}^{[2]}$ is isomorphic to $\mathbb{G}_{m}$ , but regarded as its the double cover), we get a map

Let ${\mathcal{L}}^{1/2}$ be a square root of ${\mathcal{L}}$ . Then every $b:S\times C\rightarrow [\mathfrak{c}/\mathbb{G}_{m}]$ in $B_{{\mathcal{L}}}(S)$ factors through a unique map $b^{1/2}:S\times C\rightarrow [\mathfrak{c}/\mathbb{G}_{m}^{[2]}]$ . Therefore, by composing with $\unicode[STIX]{x1D702}^{1/2}$ , we get a lift of $b$ :

The assignment $b\rightarrow \unicode[STIX]{x1D702}^{1/2}(b)$ defines a section

of the Hitchin map $h_{{\mathcal{L}}}$ .

We fix a square root $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D714}^{1/2}$ (called a theta characteristic) of $\unicode[STIX]{x1D714}$ and write $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D705}}:B\rightarrow \operatorname{Higgs}$ .

2.4 Cameral curve

For any $b\in B_{{\mathcal{L}}}(S)$ , the cameral curve $\widetilde{C}_{b}$ is defined as the fiber product, as follows.

When $b=\operatorname{id}:B_{{\mathcal{L}}}\rightarrow B_{{\mathcal{L}}}$ , the corresponding cameral curve $\widetilde{C}_{{\mathcal{L}}}:=\widetilde{C}_{b}$ is called the universal cameral curve. For simplicity, we will write $\widetilde{C}=\widetilde{C}_{\unicode[STIX]{x1D714}}$ , $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{b}:\widetilde{C}\rightarrow C\times B$ .

2.5 The universal centralizer group schemes

Consider the group scheme $I$ over $\mathfrak{g}$ consisting of pairs

We define $J=\text{kos}^{\ast }I$ , where $\text{kos}:\mathfrak{c}\rightarrow \mathfrak{g}$ is the Kostant section. This is called the universal centralizer group scheme of $\mathfrak{g}$ (see Proposition 2.5.1). To study it, it is convenient to introduce two auxiliary group schemes. We define $J^{1}=\operatorname{Res}_{\mathfrak{t}/\mathfrak{c}}(T)^{\text{W}}$ and let $J^{0}$ to be the neutral component of $J^{1}$ . All the group schemes $J$ , $J^{0}$ and $J^{1}$ are smooth commutative group schemes over $\mathfrak{c}$ . The following proposition is proved in [Reference NgôNgô06] (see also [Reference Donagi and GaitsgoryDG02]).

All the above constructions can be twisted. Namely, there are $\mathbb{G}_{m}$ -actions on $I$ , $J$ , $J^{1}$ and $J^{0}$ . Moreover, the $\mathbb{G}_{m}$ -action on $I$ can be extended to a $G\times \mathbb{G}_{m}$ -action given by $(h,t)\cdot (x,g)=(t\cdot hxh^{-1},hgh^{-1})$ . The natural morphisms $J\rightarrow \mathfrak{c}$ and $I\rightarrow \mathfrak{g}$ are $\mathbb{G}_{m}$ -equivariant, and therefore we can twist everything by the $\mathbb{G}_{m}$ -torsor ${\mathcal{L}}^{\times }$ to get $J_{{\mathcal{L}}}\rightarrow \mathfrak{c}_{{\mathcal{L}}}$ , $I_{{\mathcal{L}}}\rightarrow \mathfrak{g}_{{\mathcal{L}}}$ where $J_{{\mathcal{L}}}=J\times ^{\mathbb{G}_{m}}{\mathcal{L}}^{\times }$ and $I_{{\mathcal{L}}}=I\times ^{\mathbb{G}_{m}}{\mathcal{L}}^{\times }$ . Similarly, we have $J_{{\mathcal{L}}}^{0}\rightarrow \mathfrak{c}_{{\mathcal{L}}}$ and $J_{{\mathcal{L}}}^{1}\rightarrow \mathfrak{c}_{{\mathcal{L}}}$ , and there are natural inclusions $J_{{\mathcal{L}}}^{0}\subset J_{{\mathcal{L}}}\subset J_{{\mathcal{L}}}^{1}$ . The group scheme $I_{{\mathcal{L}}}$ over $\mathfrak{g}_{{\mathcal{L}}}$ is equivariant under the $G$ -action, hence it descends to a group scheme $[I_{{\mathcal{L}}}]$ over $[\mathfrak{g}_{{\mathcal{L}}}/G]$ .

2.6 Symmetries of Hitchin fibration

Let $b:S\rightarrow B_{{\mathcal{L}}}$ be an $S$ -point of $B_{{\mathcal{L}}}$ , corresponding to a map $b:C\times S\rightarrow \mathfrak{c}_{{\mathcal{L}}}$ . Pulling back $J_{{\mathcal{L}}}\rightarrow \mathfrak{c}_{{\mathcal{L}}}$ along this map, we obtain a smooth group scheme $J_{b}=b^{\ast }J$ over $C\times S$ .

Let $\mathscr{P}_{b}$ be the Picard category of $J_{b}$ -torsors over $C\times S$ . The assignment $b\rightarrow \mathscr{P}_{b}$ defines a Picard stack over $B$ , denoted by $\mathscr{P}_{{\mathcal{L}}}$ . Let us fix $b\in B_{{\mathcal{L}}}(S)$ , and let $(E,\unicode[STIX]{x1D719})\in \operatorname{Higgs}_{{\mathcal{L}},b}$ corresponding to the map $h_{E,\unicode[STIX]{x1D719}}:C\times S\rightarrow [\mathfrak{g}_{{\mathcal{L}}}/G]$ . Observe that the morphism $\unicode[STIX]{x1D712}^{\ast }J\rightarrow I$ in Proposition 2.5.1 induces $[\unicode[STIX]{x1D712}_{{\mathcal{L}}}]^{\ast }J_{{\mathcal{L}}}\rightarrow [I_{{\mathcal{L}}}]$ of group schemes over $[\mathfrak{g}_{{\mathcal{L}}}/G]$ . Pulling back to $C\times S$ using $h_{E,\unicode[STIX]{x1D719}}$ , we get a map

which allows us to twist $(E,\unicode[STIX]{x1D719})\in \operatorname{Higgs}_{{\mathcal{L}},b}$ by a $J_{b}$ -torsor. This construction defines an action of $\mathscr{P}_{{\mathcal{L}}}$ on $\operatorname{Higgs}_{{\mathcal{L}}}$ over $B_{{\mathcal{L}}}$ .

Let $\operatorname{Higgs}_{{\mathcal{L}}}^{\text{reg}}$ be the open stack of $\operatorname{Higgs}_{{\mathcal{L}}}$ consisting of $(E,\unicode[STIX]{x1D719}):C\rightarrow [\mathfrak{g}_{{\mathcal{L}}}/G]$ that factors through $C\rightarrow [(\mathfrak{g}^{\text{reg}})_{{\mathcal{L}}}/G]$ . If $(E,\unicode[STIX]{x1D719})\in \operatorname{Higgs}_{{\mathcal{L}}}^{\text{reg}}$ , then $a_{E,\unicode[STIX]{x1D719}}$ above is an isomorphism. The Kostant section $\unicode[STIX]{x1D702}_{{\mathcal{L}}^{1/2}}:B_{{\mathcal{L}}}\rightarrow \operatorname{Higgs}_{{\mathcal{L}}}$ factors through $\unicode[STIX]{x1D702}_{{\mathcal{L}}^{1/2}}:B_{{\mathcal{L}}}\rightarrow \operatorname{Higgs}_{{\mathcal{L}}}^{\text{reg}}$ . Following [Reference NgôNgô06, §4], we define $B_{{\mathcal{L}}}^{0}$ as the open sub-scheme of $B_{{\mathcal{L}}}$ consisting of $b\in B_{{\mathcal{L}}}(k)$ such that the image of the map $b:C\rightarrow \mathfrak{c}_{{\mathcal{L}}}$ intersects the discriminant divisor transversally. The following proposition can be extracted from [Reference Donagi and GaitsgoryDG02, Reference Donagi and PantevDP12, Reference NgôNgô06].

2.7 The tautological section $\unicode[STIX]{x1D70F}:\mathfrak{c}\rightarrow \operatorname{Lie}\,J$

Recall that by Proposition 2.5.1, there is a canonical isomorphism $\unicode[STIX]{x1D712}^{\ast }J|_{\mathfrak{g}^{\text{reg}}}\simeq I|_{\mathfrak{g}^{\text{reg}}}$ . The sheaf of Lie algebras $\operatorname{Lie}\,(I|_{\mathfrak{g}^{\text{reg}}})\subset \mathfrak{g}^{\text{reg}}\times \mathfrak{g}$ admits a tautological section $\tilde{\unicode[STIX]{x1D70F}}:\mathfrak{g}^{\text{reg}}\rightarrow \operatorname{Lie}\,(I|_{\mathfrak{g}^{\text{reg}}})$ given by $x\mapsto x\in \operatorname{Lie}\,I_{x}$ for $x\in \mathfrak{g}^{\text{reg}}$ . This section descends to a tautological section $\unicode[STIX]{x1D70F}:\mathfrak{c}\rightarrow \operatorname{Lie}\,J$ .Footnote ² Recall the following property of $\unicode[STIX]{x1D70F}$ [Reference Chen and ZhuCZ15, Lemma 2.2].

Let us regard $\operatorname{Lie}\,J$ as a scheme over $\mathfrak{c}$ . Besides the section $\unicode[STIX]{x1D70F}$ , there is a canonical map $c:\operatorname{Lie}\,J\rightarrow \mathfrak{c}$ such that $c\unicode[STIX]{x1D70F}=\operatorname{id}$ . Namely, if we regard $\operatorname{Lie}\,(I|_{\mathfrak{g}^{\text{reg}}})$ as a scheme, then there is a natural map $\operatorname{Lie}\,(I|_{\mathfrak{g}^{\text{reg}}})\rightarrow \mathfrak{c}$ given by

which also descends to a morphism $c:\operatorname{Lie}\,J\rightarrow \mathfrak{c}$ .

The morphisms $\unicode[STIX]{x1D70F}$ and $c$ have global counterparts (see also [Reference Chen and ZhuCZ15, §2.3]). Observe that $\mathbb{G}_{m}$ acts on $\mathfrak{g}\times \mathfrak{g}^{\text{reg}}$ via natural homotheties on both factors, and therefore on $\unicode[STIX]{x1D712}^{\ast }\operatorname{Lie}\,J|_{\mathfrak{g}^{\text{reg}}}\simeq \operatorname{Lie}\,(I|_{\mathfrak{g}^{\text{reg}}})\subset \mathfrak{g}\times \mathfrak{g}^{\text{reg}}$ . This $\mathbb{G}_{m}$ -action on $\unicode[STIX]{x1D712}^{\ast }\operatorname{Lie}\,J|_{\mathfrak{g}^{\text{reg}}}$ descends to a $\mathbb{G}_{m}$ -action on $\operatorname{Lie}\,J$ , and for any line bundle ${\mathcal{L}}$ on $C$ the ${\mathcal{L}}^{\times }$ -twist $(\operatorname{Lie}\,J)\times ^{\mathbb{G}_{m}}{\mathcal{L}}^{\times }$ is $\operatorname{Lie}\,J_{{\mathcal{L}}}\otimes {\mathcal{L}}$ , where $J_{{\mathcal{L}}}$ is introduced in §2.5. In addition, both maps $\unicode[STIX]{x1D70F}$ and $c$ are $\mathbb{G}_{m}$ -equivariant with respect to this $\mathbb{G}_{m}$ -action on $\operatorname{Lie}\,J$ and the natural $\mathbb{G}_{m}$ -action on $\mathfrak{c}$ . Therefore, if we define a vector bundle $B_{J,{\mathcal{L}}}$ over $B_{{\mathcal{L}}}$ , whose fiber over $b\in B_{{\mathcal{L}}}$ is $\unicode[STIX]{x1D6E4}(C,\operatorname{Lie}\,J_{b}\otimes {\mathcal{L}})$ , then by twisting $\unicode[STIX]{x1D70F}$ and $c$ by ${\mathcal{L}}$ , we obtain

which is a canonical section of the projection $\operatorname{pr}:B_{J,{\mathcal{L}}}\rightarrow B_{{\mathcal{L}}}$ , and a canonical map

such that $c_{{\mathcal{L}}}\unicode[STIX]{x1D70F}_{{\mathcal{L}}}=\operatorname{id}$ . As before, we omit the subscript $_{{\mathcal{L}}}$ if ${\mathcal{L}}=\unicode[STIX]{x1D714}$ for brevity.

Likewise, we introduce the vector bundle $B_{J,{\mathcal{L}}}^{\ast }$ over $B_{{\mathcal{L}}}$ whose fiber over $b$ is $\unicode[STIX]{x1D6E4}(C,(\operatorname{Lie}\,J_{b})^{\ast }\otimes \,{\mathcal{L}})$ . Observe that $B_{J,{\mathcal{L}}}^{\ast }$ is not the dual of $B_{J,{\mathcal{L}}}$ . Rather, when ${\mathcal{L}}=\unicode[STIX]{x1D714}$ , it is the pullback $e^{\ast }T^{\ast }(\mathscr{P}_{{\mathcal{L}}}/B_{{\mathcal{L}}})$ of the cotangent bundle of $\mathscr{P}_{{\mathcal{L}}}\rightarrow B_{{\mathcal{L}}}$ along the unit section $e:B_{{\mathcal{L}}}\rightarrow \mathscr{P}_{{\mathcal{L}}}$ and will also be denoted by $\mathbb{T}_{e}^{\ast }(\mathscr{P}_{{\mathcal{L}}})$ interchangeably later on. We construct a section

as follows. The non-degenerate bilinear form $(\,,\,)$ we fixed in 2.1 induces $\mathfrak{g}\simeq \mathfrak{g}^{\ast }$ , which restricts to a map $\operatorname{Lie}\,I_{x}\rightarrow (\operatorname{Lie}\,I_{x})^{\ast }$ for every $x\in \mathfrak{g}^{\text{reg}}$ . This map descends to give

which is $\mathbb{G}_{m}$ -equivariant. We define $\unicode[STIX]{x1D70F}_{{\mathcal{L}}}^{\ast }$ as the twist of $\mathfrak{c}\stackrel{\unicode[STIX]{x1D70F}}{\rightarrow }\operatorname{Lie}\,J\stackrel{\unicode[STIX]{x1D704}}{\rightarrow }(\operatorname{Lie}\,J)^{\ast }$ . As before, we omit the subscript $_{{\mathcal{L}}}$ if ${\mathcal{L}}=\unicode[STIX]{x1D714}$ .

We give another interpretation of this map. Observe that the Kostant section $\unicode[STIX]{x1D705}$ induces the map

over $B$ , and therefore we have the following.

By the similar argument, we have the following lemma, which will be used in §4. Let $\text{j}^{1}:\boldsymbol{\unicode[STIX]{x1D70B}}^{\ast }J\rightarrow T\times \mathfrak{t}$ be the map in Proposition 2.5.1, and let

denote its differential. Consider the pullback $\boldsymbol{\unicode[STIX]{x1D70B}}^{\ast }\unicode[STIX]{x1D70F}:\mathfrak{t}\rightarrow \unicode[STIX]{x1D70B}^{\ast }\operatorname{Lie}\,J$ of $\unicode[STIX]{x1D70F}:\mathfrak{c}\rightarrow \operatorname{Lie}\,J$ along $\boldsymbol{\unicode[STIX]{x1D70B}}:\mathfrak{t}\rightarrow \mathfrak{c}$ .

3.1 Galois description of $\mathscr{P}$

We first introduce several auxiliary Picard stacks.

Let $\widetilde{C}\rightarrow B$ be the universal cameral curve. There is a natural action of $\text{W}$ on $\widetilde{C}$ . For a $T$ -torsor $E_{T}$ on $\widetilde{C}$ , and an element $w\in \text{W}$ , there are two ways to produce a new $T$ -torsor. Namely, the first is via the pullback $w^{\ast }E_{T}=\widetilde{C}\times _{w,\widetilde{C}}E_{T}$ , and the second is via the induction $E_{T}\times ^{T,w}T$ . We denote

Clearly, the assignment $E_{T}\mapsto w(E_{T})$ defines an action of $W$ on $\operatorname{Bun}_{T}(\widetilde{C}/B)$ , i.e., for every $w,w^{\prime }\in \text{W}$ , there is a canonical isomorphism $w(w^{\prime }(E_{T}))\simeq (ww^{\prime })(E_{T})$ satisfying the usual cocycle conditions.

Let $\operatorname{Bun}_{T}^{\text{W}}(\widetilde{C}/B)$ (or $\operatorname{Bun}_{T}^{\text{W}}$ for simplicity) denote the Picard stack of strongly $\text{W}$ -equivariant $T$ -torsors on $\widetilde{C}/B$ . By definition, for a $B$ -scheme $S$ , $\operatorname{Bun}_{T}^{\text{W}}(\widetilde{C}/B)(S)$ is the groupoid of $(E_{T},\{\unicode[STIX]{x1D6FE}_{w},w\in W\})$ , where $E_{T}$ is a $T$ -torsor on $\widetilde{C}_{S}$ , and $\unicode[STIX]{x1D6FE}_{w}:w(E_{T})\simeq E_{T}$ is an isomorphism, satisfying the natural compatibility conditions. Another way to formulate these compatibility conditions is provided in [Reference Donagi and GaitsgoryDG02]. Namely, for a $T$ -torsor $E_{T}$ , let $\operatorname{Aut}_{\text{W}}(E_{T})$ be the group consisting of $(w,\unicode[STIX]{x1D6FE}_{w})$ , where $w\in \text{W}$ and $\unicode[STIX]{x1D6FE}_{w}:w(E_{T})\simeq E_{T}$ is an isomorphism. Then there is a natural projection $\operatorname{Aut}_{\text{W}}(E_{T})\rightarrow \text{W}$ . Then an object of $\operatorname{Bun}_{T}^{\text{W}}(\widetilde{C}/B)(S)$ is a pair $(E_{T},\unicode[STIX]{x1D6FE})$ , where $\unicode[STIX]{x1D6FE}:\text{W}\rightarrow \operatorname{Aut}_{\text{W}}(E_{T})$ is a splitting of the projection.

For later purpose, it is worthwhile to give another description of $\operatorname{Bun}_{T}^{\text{W}}$ . Namely, there is a non-constant group scheme ${\mathcal{T}}=\widetilde{C}\times ^{\text{W}}T$ on the stack $[\widetilde{C}/\text{W}]$ . Then the pullback functor induces an isomorphism from the stack $\operatorname{Bun}_{{\mathcal{T}}}$ of ${\mathcal{T}}$ -torsors on $[\widetilde{C}/W]$ to $\operatorname{Bun}_{T}^{\text{W}}$ .

In [Reference Donagi and GaitsgoryDG02], a Galois description of $\mathscr{P}$ in terms of $\operatorname{Bun}_{T}^{\text{W}}$ is given. We here refine their description.

Let $\mathscr{P}^{1}$ be the Picard stack over $B$ classifying $J^{1}$ -torsors on $C\times B$ . First, we claim that there is a canonical morphism

To construct $j^{1,\mathscr{P}}$ , recall that $J^{1}=(\unicode[STIX]{x1D70B}_{\ast }(T\times \widetilde{C}))^{W}$ , where $\unicode[STIX]{x1D70B}:\widetilde{C}\rightarrow C\times B$ is the projection, and therefore, for any $J^{1}$ -torsor $E_{J^{1}}$ on $C\times S$ (where $b:S\rightarrow B$ is a test scheme), one can form a $T$ -torsor on $\widetilde{C}_{S}$ by

Clearly, $E_{T}$ carries on a strongly $\text{W}$ -equivariant structure $\unicode[STIX]{x1D6FE}$ , and $j^{1}(E_{J^{1}})=(E_{T},\unicode[STIX]{x1D6FE})$ defines the morphism $j^{1,\mathscr{P}}$ .

The morphism $j^{1,\mathscr{P}}$ , in general, is not an isomorphism. Let us describe the image. Let $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}$ be a root and let $i_{\unicode[STIX]{x1D6FC}}:\widetilde{C}_{\unicode[STIX]{x1D6FC}}\rightarrow \widetilde{C}$ be the inclusion of the fixed point subscheme of the reflection $s_{\unicode[STIX]{x1D6FC}}$ . Let $T_{\unicode[STIX]{x1D6FC}}=T/(s_{\unicode[STIX]{x1D6FC}}-1)$ be the torus of coinvariants of the reflection $s_{\unicode[STIX]{x1D6FC}}$ . Then $s_{\unicode[STIX]{x1D6FC}}(E_{T})|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}}\times ^{T}T_{\unicode[STIX]{x1D6FC}}$ is canonically isomorphic to $E_{T}|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}}\times ^{T}T_{\unicode[STIX]{x1D6FC}}$ and therefore $\unicode[STIX]{x1D6FE}_{s_{\unicode[STIX]{x1D6FC}}}|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}}$ induces an automorphism of the $T_{\unicode[STIX]{x1D6FC}}$ -torsor $E_{T}\times ^{T}T_{\unicode[STIX]{x1D6FC}}$ . In other words, there is a natural map

It is easy to see that $r\circ j^{1,\mathscr{P}}$ is trivial, and one can show the following.

Recall that in [Reference Donagi and GaitsgoryDG02, Reference NgôNgô06], an open embedding $J\rightarrow J^{1}$ is constructed. To describe the cokernel, we need some notations. Let $\breve{\unicode[STIX]{x1D6FC}}\in \breve{\unicode[STIX]{x1D6F7}}$ be a coroot. Let

This is either trivial, or $\unicode[STIX]{x1D707}_{2}$ , depending on whether $\breve{\unicode[STIX]{x1D6FC}}$ is primitive or not. Let $\unicode[STIX]{x1D707}_{\breve{\unicode[STIX]{x1D6FC}}}\times \widetilde{C}_{\unicode[STIX]{x1D6FC}}$ be the constant group scheme over $\widetilde{C}_{\unicode[STIX]{x1D6FC}}$ , regarded as a sheaf of groups over $\widetilde{C}_{\unicode[STIX]{x1D6FC}}$ , and let $(i_{\unicode[STIX]{x1D6FC}})_{\ast }(\unicode[STIX]{x1D707}_{\breve{\unicode[STIX]{x1D6FC}}}\times \widetilde{C}_{\unicode[STIX]{x1D6FC}})$ be its push forward to $\widetilde{C}$ . Now, the result of [Reference Donagi and GaitsgoryDG02, §§11 and 12] can be reformulated as follows: there is a natural exact sequence of sheaves of groups on $C$ .

As a result, we obtain a short exact sequence of Picard stacks (see §A.2)

To simplify the notation, we will denote $\operatorname{Res}_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}/B}(\unicode[STIX]{x1D707}_{\breve{\unicode[STIX]{x1D6FC}}}\times \widetilde{C}_{\unicode[STIX]{x1D6FC}})$ by $\unicode[STIX]{x1D707}_{\breve{\unicode[STIX]{x1D6FC}}}(\widetilde{C}_{\unicode[STIX]{x1D6FC}})$ in what follows.

Consider the composition

Combining Lemma 3.1.2 and (3.1.4), we recover a description of $\mathscr{P}$ in terms of $\operatorname{Bun}_{T}^{\text{W}}(\widetilde{C}/B)$ as given in [Reference Donagi and GaitsgoryDG02, §16.3]. Namely, given a strongly $\text{W}$ -equivariant $T$ -torsor $(E_{T},\unicode[STIX]{x1D6FE})$ , one obtains a canonical trivialization

given by $(E_{T}|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}})\times ^{T,\unicode[STIX]{x1D6FC}}\mathbb{G}_{m}\times ^{\mathbb{G}_{m},\breve{\unicode[STIX]{x1D6FC}}}T\simeq E_{T}|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}}\otimes s_{\unicode[STIX]{x1D6FC}}(E_{T}^{-1})|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}}$ . The condition that $r_{\unicode[STIX]{x1D6FC}}(E,\unicode[STIX]{x1D6FE})=1$ is equivalent to the condition that (3.1.5) comes from a trivialization

In addition, the set of all such $c_{\unicode[STIX]{x1D6FC}}$ form a $\unicode[STIX]{x1D707}_{\breve{\unicode[STIX]{x1D6FC}}}$ -torsor. Consider the following Picard stack $\operatorname{Bun}_{T}^{\text{W}}(\widetilde{C}/B)^{+}$ : for any $B$ -scheme $S$ , its $S$ -points form the Picard groupoid of triples

where $(E_{T},\unicode[STIX]{x1D6FE})$ is a strongly $\text{W}$ -equivariant $T$ -torsor on $\widetilde{C}_{S}$ , and $c_{\unicode[STIX]{x1D6FC}}:(E_{T}|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}})\times ^{T,\unicode[STIX]{x1D6FC}}\mathbb{G}_{m}\simeq \mathbb{G}_{m}\times \widetilde{C}_{\unicode[STIX]{x1D6FC}}$ is a trivialization, which induces (3.1.5) and is compatible with the $\text{W}$ -equivariant structure. We call those trivializations $\{c_{\unicode[STIX]{x1D6FC}}\}_{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}}$ a $+$ -structure on $(E_{T},\unicode[STIX]{x1D6FE})$ . Note that, by Lemma 3.1.2, we have the following short exact sequence of Picard stacks:

Here is an application of the above discussion. Observe there is the norm map

We claim that $\operatorname{Nm}$ admits a canonical lifting

To show this, we need to exhibit a canonical trivialization

compatible with the strongly $\text{W}$ -equivariant structure. However, for any $T$ -torsor $E_{T}$ , there is a canonical isomorphism $(E_{T}|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}}\otimes s_{\unicode[STIX]{x1D6FC}}(E_{T})|_{\widetilde{C}_{\unicode[STIX]{x1D6FC}}})\times ^{T,\unicode[STIX]{x1D6FC}}\mathbb{G}_{m}\simeq \mathbb{G}_{m}\times \widetilde{C}_{\unicode[STIX]{x1D6FC}}$ , and therefore, we obtain $c_{\unicode[STIX]{x1D6FC}}$ by writing

where $s_{\unicode[STIX]{x1D6FC}}\setminus W$ denotes the quotient of $W$ by the subgroup generated by $s_{\unicode[STIX]{x1D6FC}}$ . The compatibility of the collection $\{c_{\unicode[STIX]{x1D6FC}}\}$ with the $\text{W}$ -equivariant structure is clear.

3.2 Galois description of $\mathscr{P}$ -torsors

The above description of $\mathscr{P}$ in terms of $\operatorname{Bun}_{T}^{\text{W}}(\widetilde{C}/B)$ can be generalized as follows. Let $\mathscr{D}$ be a $J$ -gerbe on $C\times B$ . Similarly to (3.1.2), we define

as the $T$ -gerbe on $\widetilde{C}$ induced from $\mathscr{D}$ using maps $\unicode[STIX]{x1D70B}:\widetilde{C}\rightarrow C\times B$ and $j^{1}:\unicode[STIX]{x1D70B}^{\ast }J\rightarrow T\times \widetilde{C}$ (see §A.5 and §A.6 for the notion of gerbes and functors between them). Since the map $j^{1}$ is $\text{W}$ -equivariant the gerbe $\mathscr{D}_{T}$ is strongly $\text{W}$ -equivariant. Equivalently, this means that $\mathscr{D}_{T}$ descends to a ${\mathcal{T}}$ -gerbe on $[\widetilde{C}/\text{W}]$ .

Let $\mathscr{T}_{\mathscr{D}}$ be the stack of splittings of $\mathscr{D}$ over $B$ . By definition, for every $S\rightarrow B$ , $\mathscr{T}_{\mathscr{D}}(S)$ is the groupoid of the splittings of the gerbe $\mathscr{D}|_{C\times S}$ . This is a (pseudo) $\mathscr{P}$ -torsor. On the other hand, let $\mathscr{T}_{\mathscr{D}_{T}}^{\text{W}}$ denote the stack of strongly $\text{W}$ -equivariant splittings of $\mathscr{D}_{T}$ , i.e., $\mathscr{T}_{\mathscr{D}_{T}}^{\text{W}}(S)$ is the groupoid of the splittings of $\mathscr{D}_{T}|_{[\widetilde{C}/\text{W}]\times _{B}S}$ . Our goal is to give a description of $\mathscr{T}_{\mathscr{D}}$ in terms of $\mathscr{T}_{\mathscr{D}_{T}}^{\text{W}}$ .

Let $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}$ . Similarly to $E_{T}^{\unicode[STIX]{x1D6FC}}$ and $E_{T}^{\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}}$ as defined in (3.1.5) and (3.1.6), let $\mathscr{D}_{T}^{\unicode[STIX]{x1D6FC}}$ , $\mathscr{D}_{T}^{\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}}$ denote the restrictions to $\widetilde{C}_{\unicode[STIX]{x1D6FC}}$ of the $\mathbb{G}_{m}$ - and $T$ -gerbes on $\widetilde{C}$ induced from $\mathscr{D}_{T}$ using the maps $\unicode[STIX]{x1D6FC}:T\rightarrow \mathbb{G}_{m}$ and $\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}:T\rightarrow T$ respectively. The strongly $\text{W}$ -equivariant structure on $\mathscr{D}_{T}$ implies that the $T$ -gerbe $\mathscr{D}_{T}^{\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}}$ has a canonical splitting $F_{\unicode[STIX]{x1D6FC}}^{0}$ . Moreover, by a similar argument in §3.1, one can show that: (i) there is a canonical splitting $E_{\unicode[STIX]{x1D6FC}}^{0}$ of the $\mathbb{G}_{m}$ -gerbe $\mathscr{D}_{T}^{\unicode[STIX]{x1D6FC}}$ , which induces $F_{\unicode[STIX]{x1D6FC}}^{0}$ via the canonical map $\mathscr{D}_{T}^{\unicode[STIX]{x1D6FC}}\rightarrow \mathscr{D}_{T}^{\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}}$ and: (ii) for any strongly $\text{W}$ -equivariant splitting $(E,\unicode[STIX]{x1D6FE})$ of $\mathscr{D}_{T}$ there is a canonical isomorphism of splittings

where $E^{\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}}$ is the splitting of $\mathscr{D}_{T}^{\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}}$ induces by $E$ via the canonical map $\mathscr{D}_{T}^{\unicode[STIX]{x1D6FC}}\rightarrow \mathscr{D}_{T}^{\breve{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D6FC}}$ . We define $\mathscr{T}_{\mathscr{D}_{T}}^{\text{W},+}$ as the stack over $B$ whose $S$ -points consist of

where $(E,\unicode[STIX]{x1D6FE})$ is a strongly $\text{W}$ -equivariant splittings of $\mathscr{D}_{T}$ and

is an isomorphism of splittings of $\mathscr{D}_{T}^{\unicode[STIX]{x1D6FC}}$ , which induces (3.2.1) and is compatible with the $\text{W}$ -equivariant structure. It is clear that $\mathscr{T}_{\mathscr{D}_{T}}^{\text{W},+}$ is a $\mathscr{P}=\operatorname{Bun}_{T}^{\text{W}}(\widetilde{C}/B)^{+}$ -torsor.

3.3 The Abel–Jacobi map

From now on till the end of this section, we restrict to the open subset $B^{0}$ of the Hitchin base. To simplify the notations, we use $B$ to denote $B^{0}$ unless specified. Recall from Proposition 2.6.1 that the cameral curve $\widetilde{C}$ is smooth over $B^{0}$ .

Let

be the Abel–Jacobi map given by $(x,\breve{\unicode[STIX]{x1D706}})\mapsto {\mathcal{O}}(\breve{\unicode[STIX]{x1D706}}x):={\mathcal{O}}(x)\times ^{\mathbb{G}_{m},\breve{\unicode[STIX]{x1D706}}}T$ . By composition with $\operatorname{Nm}^{\mathscr{P}}$ , we obtain a morphism

It is $\text{W}$ -equivariant, where $\text{W}$ acts on $\widetilde{C}\times \mathbb{X}_{\bullet }(T)$ diagonally and on $\mathscr{P}$ trivially, and is commutative and multiplicative with respect to the group structures on $\mathbb{X}_{\bullet }(T)$ and on $\mathscr{P}$ . Observe that for any $x\in \widetilde{C}_{\unicode[STIX]{x1D6FC}}$ , $\operatorname{AJ}^{\mathscr{P}}(x,\breve{\unicode[STIX]{x1D6FC}})$ is the unit in $\mathscr{P}$ . This follows from

being canonically trivialized, and the trivialization is compatible with the $\text{W}$ -equivariant structure. Here, as before, $W/s_{\unicode[STIX]{x1D6FC}}$ is the quotient of $W$ by the subgroup generated by $s_{\unicode[STIX]{x1D6FC}}$ .

By pulling back the line bundles, we thus obtain

where $\operatorname{Pic}\,^{m}(\widetilde{C}\times \mathbb{X}_{\bullet }(T))^{\text{W}}$ denotes the Picard stack over $B$ of $\text{W}$ -equivariant line bundles on $\widetilde{C}\times \mathbb{X}_{\bullet }(T)$ which are multiplicative with respect to $\mathbb{X}_{\bullet }(T)$ . Observe that there is the canonical isomorphism $\operatorname{Bun}_{\breve{T}}^{\text{W}}(\widetilde{C}/B)\rightarrow \operatorname{Pic}\,^{m}(\widetilde{C}\times \mathbb{X}_{\bullet }(T))^{\text{W}}$ given by $(E_{\breve{T}},\unicode[STIX]{x1D6FE})\mapsto {\mathcal{L}}$ , where ${\mathcal{L}}|_{(x,\breve{\unicode[STIX]{x1D706}})}=E_{\breve{T}}^{\unicode[STIX]{x1D706}}|_{x}$ . Therefore, we can regard $(\operatorname{AJ}^{\mathscr{P}})^{\vee }$ as a morphism

We claim that $(\operatorname{AJ}^{\mathscr{P}})^{\vee }$ canonically lifts to a morphism

Let ${\mathcal{L}}$ be a multiplicative line bundle on $\mathscr{P}$ . We thus need to show that

admits a canonical trivialization, which is compatible with the $\text{W}$ -equivariance structure. However, this follows from $\operatorname{AJ}^{\mathscr{P}}((x,\breve{\unicode[STIX]{x1D6FC}}))$ is the unit of $\mathscr{P}$ and a multiplicative line bundle on $\mathscr{P}$ is canonically trivialized over the unit. To summarize, we have constructed the following commutative diagram.

Now, the classical duality theorem reads as follows.

The proof of this theorem occupies §§3.4–3.6 below.

3.4 First reductions

We first show that $\mathfrak{D}_{\text{cl}}$ induces an isomorphism

For any $S$ -point $b\in B^{0}$ , $\mathscr{P}_{b}$ is a Beilinson $1$ -motive (Appendix A). We have

Observe that

By Corollary A.4.3

Let us also recall the description of $\unicode[STIX]{x1D70B}_{0}(\mathscr{P})$ as given in [Reference NgôNgô10, §§4.10 and 5.5]. As we restrict $\mathscr{P}$ to $B^{0}$ , the answer is very simple. Namely, the Abel–Jacobi map

induces a surjective map

which induces

Therefore, as abstract groups, $\unicode[STIX]{x1D70B}_{0}(\mathscr{P}^{\vee })\simeq \unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}})$ .

Since $\unicode[STIX]{x1D70B}_{0}(\mathscr{P}^{\vee })\simeq \unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}})$ are finitely generated abelian groups and are isomorphic abstractly, to show that $\unicode[STIX]{x1D70B}_{0}(\mathfrak{D}_{\text{cl}})$ is an isomorphism, it is enough to show the following.

Next, we see that

is an isomorphism. Indeed, we can construct $\operatorname{AJ}^{\breve{\mathscr{P}}}\,:\,\widetilde{C}\times \mathbb{X}^{\bullet }(T)\,\rightarrow \,\breve{\mathscr{P}}$ , and therefore $\breve{\mathfrak{D}}_{\text{cl}}\,:\,\breve{\mathscr{P}}^{\vee }\,\rightarrow \,\mathscr{P}$ . By the same argument, it induces an isomorphism $\unicode[STIX]{x1D70B}_{0}(\breve{\mathfrak{D}}_{\text{cl}}):\unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}}^{\vee })\rightarrow \unicode[STIX]{x1D70B}_{0}(\mathscr{P})$ . It is easy to check that $\breve{\mathfrak{D}}_{\text{cl}}=\mathfrak{D}_{\text{cl}}^{\vee }$ , and therefore $W_{0}(\mathfrak{D}_{\text{cl}})$ is also an isomorphism.

Therefore, it is enough to show that $D_{\text{cl}}\,:\,P^{\vee }\,\rightarrow \,\breve{P}$ is an isomorphism, where $P$ (respectively  $\breve{P}$ ) is the neutral connected component of the coarse moduli space of $\mathscr{P}$ (respectively $\breve{\mathscr{P}}$ ), and $D_{\text{cl}}$ is the map induced by $\mathfrak{D}_{\text{cl}}$ . We can prove this fiberwise, and therefore we fix $b\in B(k)$ . However, to simplify the notation, in the following discussion we write $\widetilde{C},\mathscr{P}$ instead of $\widetilde{C}_{b},\mathscr{P}_{b}$ , etc.

3.5 The calculation of the coarse moduli

We introduce a few more notations. Let $\mathscr{P}^{0}$ be the Picard stack of $J^{0}$ -torsors on $C$ , and let $P^{0}$ (respectively $P^{1}$ ) be the neutral connected components of the coarse moduli space of $\mathscr{P}^{0}$ (respectively $\mathscr{P}^{1}$ ).

We first understand $P^{1}$ . Let $\operatorname{Jac}$ denote the Jacobi variety of $\widetilde{C}$ . Then $\operatorname{Jac}\otimes \mathbb{X}_{\bullet }$ is the neutral connected component of the coarse moduli space of $\operatorname{Bun}_{T}$ .

As a result, for any prime $\ell \neq p$ ,

In addition, observe that from the definition of $D_{\text{cl}}$ , the map $P^{1}\subset \operatorname{Jac}\otimes \mathbb{X}_{\bullet }\stackrel{\operatorname{Nm}}{\rightarrow }P^{1}$ factors as

Therefore $D_{\text{cl}}$ is a prime-to- $p$ isogeny. In addition, the map

can be identified with

On the other hand, as $J^{0}$ is connected, the norm map $\operatorname{Nm}:\unicode[STIX]{x1D70B}_{\ast }T\rightarrow J^{1}=(\unicode[STIX]{x1D70B}_{\ast }T)^{\text{W}}$ factors as $\unicode[STIX]{x1D70B}_{\ast }T\rightarrow J^{0}\rightarrow J^{1}$ . Therefore, $\operatorname{Nm}:\operatorname{Jac}\otimes \mathbb{X}_{\bullet }\rightarrow P^{1}$ also factors as

It follows that $P^{0}\rightarrow P^{1}$ is also a prime-to- $p$ isogeny, and for $\ell \neq p$ there is a factorization

We need the following key result.

Now, let $A^{\prime }=\ker (P^{0}\rightarrow P)$ , and $A=\ker (P\rightarrow P^{1})$ . Then by the above proposition,

As both $A^{\prime }$ and $\breve{A}$ are finite étale groups, it is enough to show that $|A^{\prime }|=|\breve{A}|$ , where for a finite group $\unicode[STIX]{x1D6E4}$ , $|\unicode[STIX]{x1D6E4}|$ denotes the number of its elements. This is the subject of the next subsection.

3.6 Calculation of finite groups

Let us understand $A$ . In fact, it is better to pick up $\infty \in C$ away from the ramification loci. Let ${\mathcal{O}}_{\infty }$ denote the completed local ring of $C$ at $\infty$ . Let $J_{\infty }$ be the dilatation of $J$ along the unit of the fiber of $J$ at $\infty$ . By definition (see [Reference Bosch, Lutkebohmert and RaynaudBLR90, §2] for details), $J_{\infty }$ is the unique smooth group scheme over $C$ equipped with a natural map $J_{\infty }\rightarrow J$ , which is an isomorphism away from $\infty$ and induces an isomorphism from $J_{\infty }({\mathcal{O}}_{\infty })$ to the first congruence subgroup of $J({\mathcal{O}}_{\infty })$ . Let $\mathscr{P}_{\infty }$ be the Picard stack of $J_{\infty }$ -torsors on $C$ . One can also interpret $\mathscr{P}_{\infty }$ as the Picard stack of $J$ -torsors on $C$ together with a trivialization at $\infty$ . Observe that $\mathscr{P}_{\infty }$ is in fact a scheme. Let $P_{\infty }$ denote the neutral connected component of $\mathscr{P}_{\infty }$ . Similarly, one can define $J_{\infty }^{0},J_{\infty }^{1},P_{\infty }^{0},P_{\infty }^{1}$ etc. Let $A_{\infty }=\ker (P_{\infty }\rightarrow P_{\infty }^{1})$ and $A_{\infty }^{\prime }=\ker (P_{\infty }^{0}\rightarrow P_{\infty })$ .

As a corollary, we can write

and

Therefore to show that $|\breve{A}|=|A^{\prime }|$ , it is enough to show that:

1. (1) $|\unicode[STIX]{x1D6E4}(C,\breve{J}^{1}/\breve{J})|=|\unicode[STIX]{x1D6E4}(C,J/J^{0})|$ ;

2. (2) $|\!\operatorname{coker}(\operatorname{Aut}_{\breve{\mathscr{P}}}(e)\rightarrow \operatorname{Aut}_{\breve{\mathscr{P}}^{1}}(e))|=|\!\operatorname{coker}(\unicode[STIX]{x1D70B}_{0}(\mathscr{P})^{\ast }\rightarrow \unicode[STIX]{x1D70B}_{0}(\mathscr{P}^{0})^{\ast })|$ ;

3. (3) $|\!\ker (\unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}})\rightarrow \unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}}^{1}))|=|\!\ker (\operatorname{Aut}_{\mathscr{P}}(e)^{\ast }\rightarrow \operatorname{Aut}_{\mathscr{P}^{0}}(e)^{\ast })|$ .

We first prove (1). By (3.1.3),

Observe that $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}\neq 0$ if and only if $\unicode[STIX]{x1D6FC}$ is not a primitive root, i.e., $\unicode[STIX]{x1D6FC}/2\in \mathbb{X}^{\bullet }$ . On the other hand, from

one can see that the character group of $\unicode[STIX]{x1D6E4}(C,J/J_{0})$ is $(\bigoplus _{x\in \sqcup \widetilde{C}_{\unicode[STIX]{x1D6FC}}}(\mathbb{Q}\unicode[STIX]{x1D6FC}\cap \mathbb{X}^{\bullet })/\mathbb{Z}\unicode[STIX]{x1D6FC})^{\text{W}}$ . Then (1) follows.

Next, we prove (2). In fact, it follows from §3.4 and [Reference NgôNgô10, §4.10] that both maps can be identified with the natural inclusion $Z(\breve{G})\rightarrow \breve{T}^{\text{W}}$ .

Finally, we show (3). Recall that $\operatorname{Aut}_{\mathscr{P}}(e)=\{t\in T\mid \unicode[STIX]{x1D6FC}(t)=1,~\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}\}$ . On the other hand, from the above description of $\unicode[STIX]{x1D6E4}(C,J/J_{0})^{\ast }$ ,

Therefore,

To calculate $\ker (\unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}})\rightarrow \unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}}^{1}))$ , we choose ${\tilde{y}}\in \widetilde{C}_{\unicode[STIX]{x1D6FC}_{{\tilde{y}}}}$ above $y\in C-U$ for every point in the ramification loci. The restriction of $1\rightarrow \breve{J}\rightarrow \breve{J}^{1}\rightarrow \breve{J}^{1}/\breve{J}\rightarrow 0$ at $y$ then can be identified with

It follows that the coboundary map $\unicode[STIX]{x1D6E4}(C,\breve{J}^{1}/\breve{J})\rightarrow \text{H}^{1}(C,\breve{J})$ can be identified with

where $\operatorname{AJ}^{\breve{\mathscr{P}}}$ is the Abel–Jacobi map introduced before. Of course, this map does not really depend on the choice of liftings of $y\in C-U$ since $\operatorname{AJ}^{\breve{\mathscr{P}}}({\tilde{y}},\unicode[STIX]{x1D706}_{{\tilde{y}}})=\operatorname{AJ}^{\breve{\mathscr{P}}}(w{\tilde{y}},w\unicode[STIX]{x1D706}_{{\tilde{y}}})$ .

Now, as in the proof of Lemma 3.6.1, we have a right exact sequence

Since the Abel–Jacobi map induces $\mathbb{X}^{\bullet }/\mathbb{Z}\unicode[STIX]{x1D6F7}\simeq \unicode[STIX]{x1D70B}_{0}(\breve{\mathscr{P}})$ , we deduce that

Therefore, (3) follows and the proof of Theorem 3.3.1 is complete.◻

3.7 A property of $\mathfrak{D}_{\text{cl}}$

In this subsection, we assume that $G$ is semi-simple. We show that the classical duality $\mathfrak{D}_{\text{cl}}$ intertwines certain homomorphisms of Picard stacks over the Hitchin base $B^{0}$ . As before, we omit the subscript $^{0}$ .

Let $Z(\breve{G})\text{-}\text{tors}(C)$ denote the Picard stack of $Z(\breve{G})$ -torsors on $C$ . We start with the construction of two homomorphisms

The definition of $\breve{\mathfrak{l}}_{J}$ is easy. It is induced by the natural map of group schemes

for every $b:S\rightarrow B$ . For $K\in Z(\breve{G})\text{-}\text{tors}(C)$ , let

Next we define $\mathfrak{l}_{J}$ . For the purpose, we need to generalize a construction of [Reference Beilinson and DrinfeldBD91, §4.1]. Let $\unicode[STIX]{x1D70B}:{\mathcal{C}}\rightarrow B$ be a smooth proper relative curve over an affine base $B$ (later on ${\mathcal{C}}=C\times B$ ). Let

be an extension of smooth affine group schemes on ${\mathcal{C}}$ with $\unicode[STIX]{x1D6F1}$ commutative finite étale. Let $\unicode[STIX]{x1D6F1}^{\vee }=\operatorname{Hom}(\unicode[STIX]{x1D6F1},\mathbb{G}_{m})$ be its Cartier dual, which is assumed to be étale as well (in particular, the order of $\unicode[STIX]{x1D6F1}$ is prime to $\operatorname{char}~k$ ), and let $\unicode[STIX]{x1D6F1}^{\vee }\text{-}\text{tors}({\mathcal{C}}/B)$ denote the Picard stack (over $B$ ) of $\unicode[STIX]{x1D6F1}^{\vee }$ -torsors on ${\mathcal{C}}$ relative to $B$ . We construct a Picard functor

of Picard stacks over $B$ as follows. First, let $\unicode[STIX]{x1D6F1}\text{-}\text{gerbes}({\mathcal{C}}/B)$ denote the Picard 2-stack of $\unicode[STIX]{x1D6F1}$ -gerbes on ${\mathcal{C}}$ relative to $B$ , regarded as a Picard stack. Then there is the generalized (or categorical) Chern class map

that assigns every $B$ -scheme $S$ and a ${\mathcal{G}}$ -torsor $E$ on ${\mathcal{C}}_{S}$ , the Picard groupoid of the lifting of $E$ to a $\tilde{{\mathcal{G}}}$ -torsor. We have the following.

We follow [Reference Beilinson and DrinfeldBD91, §4.1.5] for a ‘scientific interpretation’ of this lemma and refer to [Reference Beilinson and DrinfeldBD91, §§4.1.2–4.1.4] for the precise construction. As explained in §A.1, the Picard stack $\unicode[STIX]{x1D6F1}\text{-}\text{gerbes}({\mathcal{C}}/B)$ is incarnated by the complex $\unicode[STIX]{x1D70F}_{{\geqslant}-1}R\unicode[STIX]{x1D70B}_{\ast }\unicode[STIX]{x1D6F1}[2](1)$ , and $\unicode[STIX]{x1D6F1}^{\vee }\text{-}\text{tors}({\mathcal{C}}/B)$ is incarnated by the complex $\unicode[STIX]{x1D70F}_{{\leqslant}0}R\unicode[STIX]{x1D70B}_{\ast }\unicode[STIX]{x1D6F1}^{\vee }[1]$ . Let $\unicode[STIX]{x1D707}_{\infty }^{\prime }$ denote the group of prime-to- $p$ roots of unit. Note that $\unicode[STIX]{x1D70B}^{!}\unicode[STIX]{x1D707}_{\infty }^{\prime }\simeq \unicode[STIX]{x1D707}_{\infty }^{\prime }[2](1)$ . Then by the Verdier duality,

By shifting by $[1]$ and truncating $\unicode[STIX]{x1D70F}_{{\leqslant}0}$ , one obtains the lemma. As explained in [Reference Beilinson and DrinfeldBD91, §4.1.5], working in the framework of derived categories is not enough to turn the above heuristics into a proof. One can either give a concrete construction as in [Reference Beilinson and DrinfeldBD91, §§4.1.2–4.1.4] or understand the above argument in the framework of stable $\infty$ -categories.

Therefore, each $K\in \unicode[STIX]{x1D6F1}^{\vee }\text{-}\text{tors}({\mathcal{C}}/B)$ defines a morphism

or equivalently a line bundle ${\mathcal{L}}_{{\mathcal{G}},K}$ on $\operatorname{Bun}_{{\mathcal{G}}}({\mathcal{C}}/B)$ and the assignment $K\rightarrow {\mathcal{L}}_{{\mathcal{G}},K}$ defines a homomorphism of Picard stacks

which factors through the $n$ -torsion of $\operatorname{Pic}(\operatorname{Bun}_{{\mathcal{G}}})({\mathcal{C}}/B)$ where $n$ is the order of $\unicode[STIX]{x1D6F1}^{\vee }$ .

Note that in the above discussion we do not assume that ${\mathcal{G}}$ is commutative. But if ${\mathcal{G}}$ is commutative, $\operatorname{Bun}_{{\mathcal{G}}}({\mathcal{C}}/B)$ has a natural structure of Picard stack over $B$ and one can check that $\mathfrak{l}_{{\mathcal{G}}}$ factors through a homomorphism $\mathfrak{l}_{{\mathcal{G}}}:\unicode[STIX]{x1D6F1}^{\vee }\text{-}\text{tors}({\mathcal{C}}/B)\rightarrow (\operatorname{Bun}_{{\mathcal{G}}}({\mathcal{C}}/B))^{\vee }$ .

Now let ${\mathcal{C}}=C\times B$ , where $B$ is the Hitchin base as before. Let ${\mathcal{G}}=J_{b}$ and $\tilde{{\mathcal{G}}}=(J_{\text{sc}})_{b}$ , where $b:B\rightarrow B$ is the identity map, and $J_{\text{sc}}$ is the universal regular centralizer for $G_{\text{sc}}$ , the simply connected cover of $G$ . Then $\unicode[STIX]{x1D6F1}(1)=\unicode[STIX]{x1D6F1}_{G}(1)$ is the fundamental group and $\unicode[STIX]{x1D6F1}_{G}^{\vee }$ is canonical isomorphic to the center $Z(\breve{G})$ of $\breve{G}$ . Therefore, the above construction gives $\mathfrak{l}_{J}$ as promised in (3.7.1).

Note that similarly we can set ${\mathcal{G}}=G\times {\mathcal{C}}$ and ${\mathcal{G}}=T\times {\mathcal{C}}$ in the above construction so we obtain

For $K\in Z(\breve{G})\text{-}\text{tors}(C)$ , let ${\mathcal{L}}_{G,K}:=\mathfrak{l}_{G}(K)\in \operatorname{Pic}(\operatorname{Bun}_{G})$ , ${\mathcal{L}}_{J,K}:=\mathfrak{l}_{J}(\{K\}\times B)\in (\mathscr{P})^{\vee }(B)$ . The following lemma will be used in §5.6.

We write $l_{G},l_{T},l_{J}$ for the induced maps between the corresponding coarse moduli spaces. The following lemma is a specialization of our construction of the duality given in Lemma 3.7.1.

Now we are ready to state the result in this subsection.