What do Hecke eigensheaves actually look like?

Question

Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands correspondence constructs a geometrically irreducible Hecke eigensheaf on $Bun_2(C)$ associated to $\rho$, which is a perverse sheaf.

What is this sheaf's generic rank (as a function of $g$, presumably)? What is its singular locus?

It seems to me that the singular locus should be more-or-less independent of $\rho$ because in the complex-analytic picture it is supposed to depend smoothly on $\rho$, but the coordinates of $\rho$ are $\ell$-adic and the coordinates of the singular locus are in $\overline{\mathbb F}_q$ so there is no natural way for the second to depend naturally on the first other than being locally constant. If the singular locus is locally constant, because the complex-analytic space of local systems is connected, it should be globally constant as well. By similar logic it seems like the generic rank should be independent of the local system.

I could come up with some plausible guess for what the singular locus should be (maybe the unstable locus?), but I have no idea what the generic rank should be (some polynomial in $g$?).

Answer 1

The characteristic cycle of Hecke eigensheaves (for irreducible local systems) is the zero fiber of the Hitchin fibration, counting with multiplicity. This is almost obvious for eigen-D-modules constructed from opers in characteristic zero (by Beilinson and Drinfeld), but for $GL_2$ it should not be hard to check this from Drinfeld's formula and verify that the answer still holds, especially if you only care about their smoothness locus and rank. Anyway, the answer for $GL_2$ is as follows:

• Generic rank is $2^{\dim H^0(C,\Omega^{\otimes 2})}=2^{3g-3}$
• The singularity locus consists of bundles $E$ that admit a nilpotent Higgs field; explicitly, this means that $E$ has a line-subbundle $L\subset E$ such that $H^0(C,L^{\otimes 2}\otimes\det(E)^{-1}\otimes\Omega)\ne 0$, which is equivalent to $H^1(C,\det(E)\otimes L^{\otimes -2})\ne 0$.