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 $\bigoplus H^0(C, \omega_C^{\otimes i})$ (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 $A$)?

  • $\begingroup$ you didn't define $X$. Is $X=C$? $\endgroup$ – IMeasy Feb 4 '13 at 7:23
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    $\begingroup$ Just to illustrate the necessity of cameral covers for general groups. If you have a regular Higgs field $\phi\in H^0(X, \mathrm{ad} P\otimes K)$, its centralizer $c\subset \mathrm{ad} P$ will be a subbundle of abelian Lie algebras (its rank is the rank of the group). When $G=GL_n$, the multiplication on $\mathrm{ad} P$ descends to a commutative algebra structure on $c$, so you can take the relative Spec of $c$ to obtain the spectral cover... $\endgroup$ – Pavel Safronov Feb 4 '13 at 16:47
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    $\begingroup$ ... In general, however, $c$ does not carry an algebra structure and there is no analogous $\mathrm{rk} G:1$ cover, only a (ramified) $|W|:1$ cover parametrizing Borels containing $c$. $\endgroup$ – Pavel Safronov Feb 4 '13 at 16:49

Here is a very brief explanation of what is going on. You can find more details in the papers that Ana lists and in arxiv.org/abs/math/0604617 as you say.

If $G$ is a reductive group, the Hitchin base $\text{Hitch}_{G}$ is defined to be the moduli of cameral covers of $X$. Concretely

$$ \tag{1} \text{Hitch}_{G} = H^{0}(X, tot(\mathfrak{t}\otimes \omega_{X})/W)$$

where $\mathfrak{t}$ is a Cartan subalgera in $\mathfrak{g}$. This space is non-canonically isomorphic to

$$ \oplus_{i = 1}^{r} H^{0}(X, \omega_{X}^{\otimes d_{i}})$$

where $d_{1}, \ldots, d_{r}$ are the degrees of the members of a basis of homogeneous invariant polynomials on $\mathfrak{g}$.

It is better to work with the description (1) since it does not involve choices. Given a section $b \in H^{0}(X, tot(\mathfrak{t}\otimes \omega_{X})/W)$ we can construct a cameral cover $\widetilde{X}_{b} \to X$ as the pull back of the natural $W$-cover

$$tot(\mathfrak{t}\otimes \omega_{X}) \to tot(\mathfrak{t}\otimes \omega_{X})/W$$

by the map $b : X \to tot(\mathfrak{t}\otimes \omega_{X})/W$.

The universal cameral cover then is a subvariety

$$ \widetilde{X} \subset tot(\mathfrak{t}\otimes \omega_{X}) \times \text{Hitch}_{G}$$

(rather than a subvariety in $tot(\omega_{X})\times \text{Hitch}_{G}$).

Furthermore the fibers of the Hitchin map are not the relative Picard varieties along the fibers of the universal cameral cover. They are generalized Prym varieties associated with this relative Picard varieties and the $W$-action. Roughly you will have

$$T^{*}\text{Bun}^{0}_{G} = \text{Hom}(\Lambda,Pic(\widetilde{X}^{0}/\text{Hitch}^{0}_{G}))^{W} $$

where $\Lambda$ is the character lattice of a maximal torus of $G$.

However this description is only correct up to isogeny. The precise description is given in the Donagi-Gaitsgory paper and is also recalled in our paper arxiv.org/abs/math/0604617.



You should have a look at Hitchin's stable bundles and integrable systems, where he constructs the spectral curve for types B, C and D.

For more general groups one needs to use cameral covers. References for that are Scognamillo's papers, Donagi's The decomposition of spectral covers, Donagi and Gaitsgory's The gerbe of Higgs bundles or Ngô's Fibration de Hitchin et endoscopie and Le lemme fondamental pour les algèbres de Lie.

Hope this helps!

  • $\begingroup$ Thanks! I was having look at those references. I was having trouble finding the definition of cameral covers in the general case there though (but arxiv.org/abs/math/0604617 seemed relevant). Do you know if an analogue of the result $T^* \text{Bun}_n^0=\text{Pic}(\tilde{X}^0/\text{Hitch}_n^0)$ would hold in the general case, if we use cameral covers instead? Also, how precisely would one define $\tilde{X}^0$ in the general case? $\endgroup$ – Vinoth Feb 4 '13 at 23:49

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