I'm currently working through Frenkel's beautiful paper: http://arxiv.org/PS_cache/hepth/pdf/0512/0512172v1.pdf. I'm looking for a good example of a projective curve to get my hands dirty, and go through the general constructions that Frenkel shows there and try to do them manually for this example of a curve. Are there any good instructive examples for doing this? (Or does it always get out of hand very quickly?)
3 Answers
Unfortunately I don't think geometric Langlands is very easy on any curve. The only curve where the objects are readily accessible is $P^1$, but even there the general statement is kind of tricky (see Lafforgue's note here). I would look at Frenkel's writings on the Gaudin model, which is a concrete illustration of the BeilinsonDrinfeldFeiginFrenkel approach to geometric Langlands for $P^1$ with several punctures. Also Arinkin and Lysenko worked out explicitly a case of geometric Langlands (in a stronger sense) on $P^1$ minus 4 points  see the first four papers on a mathscinet search for Arinkin. So the answer is try $P^1$ with some punctures, but don't be surprised if things are rather tricky already there.
(I also think geometric Langlands on an elliptic curve should be accessible, but as far as I know it hasn't been worked out very explicitly.)

$\begingroup$ Thank you very much for taking the time to answer my question. I'd love to try and see what can be said for geometric Langlands for elliptic curves. By saying it hasn't been worked out ''very explicitly", do you mean that no work has been done at all in this direction, or that there is some paper with partial results in this direction? $\endgroup$ Dec 19, 2009 at 6:23

3$\begingroup$ In genus one there's a very explicit description of an open locus in the moduli stack of bundles (namely the Atiyah description of moduli of semistable vector bundles and its variants). This can be seen easily eg from the FourierMukai transform. One can then try to write down Dmodules explicitly  this is closely related to elliptic CalogeroMoser systems. One paper I like about this is by Nevins, 0804.4170. There are also many related arXiv papers by B. Enriquez (with obvious titles), in particular writing down quantum Hitchin hamiltonians explicitly. But there's still a long way to go.. $\endgroup$ Dec 19, 2009 at 21:53

$\begingroup$ Thank you, that was extremely helpful. I'll keep reading Frenkel's paper, and those you have suggested, and try to digest what you have said. $\endgroup$ Dec 20, 2009 at 13:15
This is a kind of question we asked ourselves about 10 years ago :)
Our answer  $P^1$ with nodes and cusps (and more general singularities) are very good examples for doing this. The answer is actually motivated by Serre's "Algebraic groups and algebraic class fields ..." where he works with generalized Jacobions and abelian Langlands (i.e. class field theory).
We concerned the part of the Langlands which treats the Hitchin's Dmodules (these NOT all Heckeeigensheves).
In papers with Dmitry Talalaev we described classical Hitchin system on such curves. http://arxiv.org/abs/hepth/0303069 Hitchin system on singular curves I
http://arxiv.org/abs/hepth/0309059  here are more general singularities
The second step was to quantize Hitchin's hamiltonians. Actually it is the same as to quantize Gaudin's hamiltonian's. Naive recipe  works only for sl(2), sl(3)  http://arxiv.org/abs/hepth/0404106
The breakthrough that Dmitry Talalaev found http://arxiv.org/abs/hepth/0404153 Quantization of the Gaudin System
how to do TWO things simultaneously, he proposed a beautiful formula for:
1) All quantum Hitchin (Gaudin) Hamiltonionas (later generalized to whole center of universal enveloping for loop algebra)
2) At the same time it gives the GLoper explicitly (moreover it gives "universal" GLoper meaning that its coefficents are quantum Hitchin (Gaudin) hamiltonians, but not complex numbers). Fixing values of Hitchin's hamiltonians we get complexvalued GLoper, which corresponds by Langlands to these Hitchin's hamiltonians. So the Langlands correspondence: Hitchin Dmodule > GLoper is made very explicit.
3) His formula makes explicit the idea that "GLoper is quantization of the spectral curve"
To some extent this solves the questions about the Laglands for GLHitchin's system. We have not write down the proof of "Heckeeigenvaluedness" of Hitchin's Dmodules. But it seems that is rather clear(may be not the ritht word), if you take appropriate point of view on Hecke's transformations  as in the paper by A. Braverman, R. Bezrukavnikov http://arxiv.org/abs/math/0602255 Geometric Langlands correspondence for Dmodules in prime characteristic: the GL(n) case
One of key ideas  that you can do everything in "classical limit" and than quantize. They worked for finite fields  so they can use some trick to go from classical to quantum, over complex numbers we have explicit formulas by Talalaev so they should do the same.
Let me also mention that Hecke transformations are also known as Backlund transformations in integrability and relevant papers are:
http://arxiv.org/abs/nlin/0004003 Backlund transformations for finitedimensional integrable systems: a geometric approach V. Kuznetsov, P. Vanhaecke
http://arxiv.org/abs/nlin/0110045 Hitchin Systems  Symplectic Hecke Correspondence and Twodimensional Version A.M. Levin, M.A. Olshanetsky, A. Zotov
It would be very nice project to consider from this point of view $P^1$ with cusp, the cotangent to moduli space of vector bundles is $[X,Y]=0/GL(n)$, the same thing which is considered in Etingof's Ginzburg's paper
http://arxiv.org/abs/math/0011114 Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphis
It would be very nice (and should be simple) to explicitly describe the HeckeBacclund transformations and their action on CalogeroMoser system and so on...
Mosuli space of SL_2 bundles on any genus 2 curve  is P^3 by Narashimhan and Ramanan. So working with P^3 should be accessible, but may be not so easy as might seems...
Classical Hitchin system has been described by van Geemen, Previato, Gawedzki, et. al.
http://arxiv.org/abs/alggeom/9410015
http://arxiv.org/abs/solvint/9710025
Quantum hamiltonians should be contained in Geemen de Jong paper: http://arxiv.org/abs/alggeom/9701007
So one needs to 1) Quantize (?may be done) it 2) Make Hecke transform and see that result is as predicted by Beilison and Drinfeld  product of initial Hitchin Dmodule on Dmodule give by SL_2oper.
If I would think on this I would try to do the following:
1) find Lax matrix L(z)  it might be done by Gawedzki, et.al.
2) try to guess what is the "qauntum spectral curve" "det"(d/dz  L(z))  it gives both quantum Hitchin hamiltonians and SL_2oper (this done by Talalaev for GaudinHitchin for P^1 http://arxiv.org/abs/hepth/0404153 )
3) try to prove that Hecke transformation agrees with "qauntum spectal curve"
As in previous post I would strongly suggest to use downtoearth point of view on Hecke transformation.
So in such a way one may prove that the Hitchin's Dmodule is Hecke eigensheave with "eigenvalue" given by the SL_2oper="quantum spectral curve" (Dmodule on basic curve).
Considerations of other Hecke eigensheaves (not Hitchin's) is another story...
In our paper http://arxiv.org/abs/hepth/0303069 we described classical Hitchin system for the degenerate genus 2 curve y^2 = (xa)^3 (xb)^3. However we did not check that our "analogs" of NarasimhanRamanan coordinates are indeed limits of true NarashimhanRamanan coordinates, so it might not be very helpful. Also Talalaev's formula cannot be applied directly for the Lax matrix in these coordinates. It is coordinate dependent, it is solvable problem but requires some work.