On the Proof of the Geometric Langlands Correspondance

Dennis Gaitsgory, holder of the Israel Gelfand Chair in Mathematics at IHES, supported by Squarepoint Foundation, and Sam Raskin, a regular visitor at IHES, lead a team of mathematicians who recently provided a proof of the unramified geometric Langlands conjecture in a series of five papers, totaling over 800 pages. Part of this work was carried out during numerous simultaneous visits by the authors to IHES. In the text below, Dennis Gaitsgory explains certain aspects of the geometric Langlands correspondence.

“The Langlands program (both classical and geometric) stems from a principle first articulated by Israel Gelfand, who identified basic symmetries—i.e., simple Lie groups (or, more generally, reductive groups)—as one of the pillars of modern mathematics.

Gelfand realized that pairing instances of these basic symmetries with a field of dimension 1, a basic input in number theory, often generates intriguing mathematical objects and questions.

One such pairing is that of a reductive group with a function field (or number field), resulting in an object known as the automorphic space. In the 1960s, Robert Langlands discovered an involution defined on the set of basic symmetries, allowing for a new way of coupling these data. This pairing is now known as the classical Langlands correspondence, which relates number theory to geometry and representation theory.

The geometric Langlands correspondence, on the other hand, arises from pairing a reductive group with a Riemann surface (or its algebraic-geometric incarnation, i.e., an algebraic curve over a given ground field)—a geometric object that can be visualized as a sphere with a certain number of handles.

There are three different settings for the geometric Langlands correspondence, which are closely related but distinct:

1. Over the algebraic closure of a finite field with l-adic coefficients (in this case, there is a direct link to the classical Langlands correspondence over function fields);
2. Over the complex numbers, in the Betti setting;
3. Over an arbitrary field of characteristic 0, in the de Rham setting.

In all these settings, the geometric Langlands correspondence gives rise to the following two objects.

On the automorphic side, we get the moduli space of principal G-bundles on the Riemann surface (or the algebraic curve), while on the spectral side, we get the moduli space of local systems (with respect to the Langlands dual group) on the same Riemann surface (or algebraic curve).

These two geometric objects are not isomorphic—there is no direct relation between them. The goal of the geometric Langlands program is to establish a relationship between these objects. This is achieved via an equivalence of certain categories, which I will now describe.

On one side, we consider the category of D-modules on the moduli space of principal G-bundles on the Riemann surface (or algebraic curve). On the other side, we consider the category of quasi-coherent sheaves on the moduli space of local systems (with respect to the Langlands dual group) on the Riemann surface (or algebraic curve).

In our work, we proved that, in the unramified case, these two categories are equivalent.

In the setting most closely related to the classical Langlands program—working over the algebraic closure of a finite field with l-adic coefficients—we discovered that the appropriate category to consider is the category of l-adic sheaves with nilpotent singular support. The significance of this category was recognized early on by Gérard Laumon in 1987, but it remained dormant for a long time and was rediscovered by David Ben-Zvi, David Nadler and Zhiwei Yun in the topological context, which our work was inspired by.

On the spectral side, the challenge was to make sense of the stack of local systems; a priori it is not clear what it means to vary l-adic local systems in algebro-geometric families.

The equivalence between these two categories has not been shown in full generality, but if one accepts it, we have shown that one can work one’s way back to the classical setting taking the trace of the Frobenius endofunctor. This gives an isomorphism of vector spaces, which is in some sense the best wish of Langlands as it allows one to completely describe the space of automorphic functions in terms of the spectral side. It is the space of sections of the dualizing sheaf on the moduli space (and not simply the space of functions, as one might have naively guessed).

Let me now say a few words about the techniques we have developed in the last 30 years while working on the proof the geometric Langlands correspondence.

These tools emerged as we tried to encompass two historical approaches to proving the geometric Langlands correspondence.

On one side, Vladimir Drinfeld discovered an explicit way to pass from the spectral side to the automorphic side, arriving at what is known as the Whittaker model, from which one could hope to descend to the automorphic space. This approach was further developed by Laumon.

On the other side, Alexander Beilinson and Drinfeld attempted to prove the geometric Langlands correspondence using techniques of localization of Kac-Moody representations onto the moduli space of bundles, drawing on ideas from conformal field theory and geometric representation theory.

In our work, we sought to generalize and integrate both approaches, constructing categories that encompass the classical objects considered initially.

Ultimately, we played both approaches against each other to provide a proof of the (unramified) geometric Langlands conjecture. This idea came to me about a year before Laumon’s 60th birthday, and I first presented my strategy of proof at his birthday conference. Around that time, in a joint work with Dima Arinkin, we gained a clear understanding of the exact statements needed to prove the geometric Langlands correspondence.

In the future, my team and I hope to develop the ramified version, which would provide insight into the relationship one might expect in the number field case.”

Photo credit for the cover picture: The Oberwolfach Photo Collection