ag.algebraic geometry - A good example of a curve for geometric Langlands

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 D-modules (these NOT all Hecke-eigensheves).

In papers with Dmitry Talalaev we described classical Hitchin system on such curves.
http://arxiv.org/abs/hep-th/0303069
Hitchin system on singular curves I

http://arxiv.org/abs/hep-th/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/hep-th/0404106

The breakthrough that Dmitry Talalaev found
http://arxiv.org/abs/hep-th/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 GL-oper explicitly (moreover it gives "universal" GL-oper
meaning that its coefficents are quantum Hitchin (Gaudin) hamiltonians, but not complex numbers). Fixing values of Hitchin's hamiltonians we get complex-valued GL-oper, which corresponds by Langlands to these Hitchin's hamiltonians. So the Langlands correspondence: Hitchin D-module -> GL-oper is made very explicit.

3) His formula makes explicit the idea that "GL-oper is quantization of the spectral curve"

To some extent this solves the questions about the Laglands for GL-Hitchin's system.
We have not write down the proof of "Hecke-eigenvaluedness" of Hitchin's D-modules.
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 D-modules 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 finite-dimensional integrable systems: a geometric approach
V. Kuznetsov, P. Vanhaecke

http://arxiv.org/abs/nlin/0110045 Hitchin Systems - Symplectic Hecke Correspondence and Two-dimensional Version
A.M. Levin, M.A. Olshanetsky, A. Zotov

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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, Calogero-Moser space, and deformed Harish-Chandra homomorphis

It would be very nice (and should be simple) to explicitly describe
the Hecke-Bacclund transformations and their action on CalogeroMoser system and so on...