[Paper Review] Outline of the proof of the geometric Langlands conjecture for GL(2)

Dennis Gaitsgory|arXiv (Cornell University)|Feb 11, 2013

Advanced Algebra and Geometry20 citations

TL;DR

This paper outlines a strategy to prove the categorical geometric Langlands conjecture for GL(2) by reducing it to two conjectures: one involving the extended Whittaker category and another concerning spectral side ind-coherent sheaves on local systems. The key contribution is a framework that resolves the failure of Eisenstein series functors to preserve compactness by introducing a modified spectral side category, leading to a fully faithful comparison via extended Whittaker coefficients and gluing functors.

ABSTRACT

We outline a proof of the categorical geometric Langlands conjecture for GL(2), as formulated in reference [AG], modulo a number of more tractable statements that we call Quasi-Theorems.

Motivation & Objective

• To reduce the geometric Langlands conjecture for GL(2) to two conjectures: one involving the extended Whittaker category and another on the spectral side.
• To resolve the issue that Eisenstein series functors on the spectral side do not preserve compactness by introducing a modified category of ind-coherent sheaves.
• To establish a fully faithful comparison between the automorphic and spectral sides via a gluing construction and extended Whittaker coefficients.
• To show that the cuspidal category on the automorphic side arises as a quotient of the spectral side via irreducible local systems.
• To demonstrate that the conjectural geometric Langlands equivalence is compatible with Eisenstein series, Hecke functors, and Kac-Moody localization.

Proposed method

• Embed the automorphic side into the extended Whittaker category via the extended Whittaker coefficient functor, conjectured to be fully faithful.
• Embed the spectral side into a category denoted Glue(Ǧ)spec using a gluing construction based on ind-coherent sheaves with nilpotent singular support.
• Use the functor Glue(CTenh_spec) to relate the modified spectral side to Glue(Ǧ)spec, which is fully faithful by Theorem 9.3.8.
• Construct a canonical fully faithful functor L_G,G^Whit^ext from Glue(Ǧ)spec to the extended Whittaker category, enabling comparison.
• Apply adjunctions and orthogonality to show that the cuspidal category vanishes under the extended Whittaker functor, proving vanishing of cokernel maps.
• Use the interdependence of conjectures to show that Conjectures 8.2.9 and 10.2.8 follow from the main conjecture (Conjecture 3.4.2), assuming its validity.

Experimental results

Research questions

• RQ1 How can the failure of Eisenstein series functors to preserve compactness on the spectral side be resolved within the geometric Langlands framework?
• RQ2 What is the correct replacement for the naive spectral side category QCoh(LocSys_Ǧ) that ensures compatibility with automorphic Eisenstein functors?
• RQ3 Can the geometric Langlands equivalence be established via a comparison of categories embedded into a common, more tractable category like the extended Whittaker category?
• RQ4 What is the precise relationship between the cuspidal automorphic category and the spectral category of irreducible local systems?
• RQ5 How does the Kac-Moody representation category relate to the automorphic category under the geometric Langlands correspondence?

Key findings

• The spectral side is corrected by replacing QCoh(LocSys_Ǧ) with IndCoh_Nilp^glob_Ǧ(LocSys_Ǧ), which resolves the compactness issue in Eisenstein series functors.
• The extended Whittaker coefficient functor coeff^ext_{G,G} is conjectured to be fully faithful, embedding D-mod(Bun_G) into the extended Whittaker category.
• The functor Glue(CT^enh_spec) is fully faithful, embedding the modified spectral side into Glue(Ǧ)_spec, as established by Theorem 9.3.8.
• The cuspidal category on the automorphic side is identified as the image of QCoh(LocSys_Ǧ^irred) under a composition of j_* and Ξ_Ǧ, under Conjecture 3.4.2.
• The automorphic cuspidal category is shown to be isomorphic to QCoh(LocSys_Ǧ^irred) ⊗_QCoh(LocSys_Ǧ) D-mod(Bun_G), confirming a key duality structure.
• Assuming Conjecture 10.5.10, the localization functor Loc_G identifies the homotopy category of the tempered automorphic category with a Verdier quotient of the Kac-Moody category KL(G,κ)_Ran(X).