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I've been learning a bit about the Hitchin fibration, and I wanted to ask about how it works outside of type A.

Background: In type A, the Hitchin fibration is reviewed on pg 14 of this paper of Bezrukavnikov and Braverman. Fix a curve $C$; let $\text{Bun}_n$ be the stack of vector bundles of rank $n$. The Hitchin base $\text{Hitch}_n$ is defined as $\bigoplus H^0(XH^0(C, \omega_X^{\otimes omega_C^{\otimes i})$ (where the sum is for $1 \leq i \leq n$). One also has the Hitchin map $h: T^* \text{Bun}_n \rightarrow \text{Hitch}_n$ (defined on section $4.1$), the "total spectral curve" $\tilde{X} \subset T^* X \times \text{Hitch}_n$ (defined on section $4.2$). There is an certain open substack $\text{Hitch}_0^n \subset \text{Hitch}_n$; let $T^* \text{Bun}_n^0 = h^{-1}(\text{Hitch}_n^0)$ and $\tilde{X}^0 = p^{-1}(\tilde{X})$ (where $p: \tilde{X} \rightarrow \text{Hitch}_n$ is the projection). Then one nice result (Corollary $4.5$) is that:

With the projection $T^* \text{Bun}_n^0 \rightarrow \text{Hitch}_n$, we have $T^* \text{Bun}_n^0 = Pic(\tilde{X}^0 / \text{Hitch}_n)$.

Question: In Chapter 3 of Tsao-Hsien's thesis, the Hitchin fibration in the general case is discussed; in particular see Proposition $3.3.1$ on pg $25$ (there, $\text{Higgs}$ denotes $T^* \text{Bun}_G$ and $B$ denotes the Hitchin base). But I'm unsure about how to define the "total spectral curve" $\widetilde{X}$ (in a way that hopefully gives a direct analogue to the result, Corollary $4.5$ above). How does one define the total spectral curve in the general case (or am I just confused and does it only exist in type $A$)?

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# Hitchin fibration outside of type A

I've been learning a bit about the Hitchin fibration, and I wanted to ask about how it works outside of type A.

Background: In type A, the Hitchin fibration is reviewed on pg 14 of this paper of Bezrukavnikov and Braverman. Fix a curve $C$; let $\text{Bun}_n$ be the stack of vector bundles of rank $n$. The Hitchin base $\text{Hitch}_n$ is defined as $\bigoplus H^0(X, \omega_X^{\otimes i})$ (where the sum is for $1 \leq i \leq n$). One also has the Hitchin map $h: T^* \text{Bun}_n \rightarrow \text{Hitch}_n$ (defined on section $4.1$), the "total spectral curve" $\tilde{X} \subset T^* X \times \text{Hitch}_n$ (defined on section $4.2$). There is an certain open substack $\text{Hitch}_0^n \subset \text{Hitch}_n$; let $T^* \text{Bun}_n^0 = h^{-1}(\text{Hitch}_n^0)$ and $\tilde{X}^0 = p^{-1}(\tilde{X})$ (where $p: \tilde{X} \rightarrow \text{Hitch}_n$ is the projection). Then one nice result (Corollary $4.5$) is that:

With the projection $T^* \text{Bun}_n^0 \rightarrow \text{Hitch}_n$, we have $T^* \text{Bun}_n^0 = Pic(\tilde{X}^0 / \text{Hitch}_n)$.

Question: In Chapter 3 of Tsao-Hsien's thesis, the Hitchin fibration in the general case is discussed; in particular see Proposition $3.3.1$ on pg $25$ (there, $\text{Higgs}$ denotes $T^* \text{Bun}_G$ and $B$ denotes the Hitchin base). But I'm unsure about how to define the "total spectral curve" $\widetilde{X}$ (in a way that hopefully gives a direct analogue to the result, Corollary $4.5$ above). How does one define the total spectral curve in the general case (or am I just confused and does it only exist in type $A$)?