Institut des Hautes Etudes Scientifiques (IHES) · Video summary

Sam Raskin - 2/6 Some Aspects of the Geometric Langlands Program

Sam Raskin discusses key aspects of the Geometric Langlands Program, focusing on spectral actions, categories of sheaves, and the role of Whitaker models in reformulating the conjecture.

Institut des Hautes Etudes Scientifiques (IHES) · March 12, 2026 · 108-min episode · 7-min read

Sam Raskin - 2/6 Some Aspects of the Geometric Langlands Program

Key takeaways

💡The Whitaker coefficients serve as a categorified analog of Fourier coefficients, sufficing to characterize automorphic categories and local Langlands correspondence.01:16
⚡Categorical approach simplifies the spectral problem, with key functors like LG-contracting the complex web of sheaves into manageable invariants, establishing categorical analogs of classical automorphic theory.01:31
📚The category of D-modules on the stack of bundles is compactly generated, enabling categorical reconstructions of the Langlands functor via universal properties.01:39
🎯The spectral action factors through categories like quasi-coherent sheaves, connecting local systems to sheaf categories over moduli stacks.05:09
🔍Local-to-global principles are encoded in functors like the local systems evaluation, linking local data at points to global sheaves and automorphic forms.07:17
🚀Examine categories of sheaves with no potent singular support to understand the maximal subcategories where spectral functors act faithfully.38:45

Chapters

0:10🎯 Introduction and Recap of Previous Lecture
4:31🎯 Spectral Actions and Spaces of Local Systems
7:44🎯 Local Systems and Evaluation Bundles
20:21🎯 The Spectral Problem and Its Formulation
31:04🎯 Categories of Nilpotent Support and Maximal Subcategories
57:26🎯 Whitaker Coefficients and Normalization
1:08:10🎯 Duality and Micro-Localization in the Framework
1:18:35🎯 Geometric Langlands Functor and Its Construction
1:42:02🎯 Comparison with Classical and Non-Abelian Settings
1:47:25🎯 Summary and Conclusion

The digest

High-Level Overview

This lecture delves into central aspects of the geometric Langlands program, focusing on the interplay between different categorical frameworks (D-module, Betti, and étale settings), the construction and significance of spectral actions, and the geometric incarnation of Hecke operators. Sam Raskin outlines how these structures interconnect via functors between categories of sheaves and quasi-coherent sheaves on moduli spaces of local systems. The talk culminates in a discussion of the Whitaker coefficient functor, its geometric analogue, and the formulation of a unique "geometric Langlands functor." This exposition is fundamental for researchers aiming to grasp the categorical and spectral framework underpinning geometric Langlands and for those exploring its equivalences and conjectures.

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Structured Deep Dive
1. Recap of the Three Geometric Settings ([00:53]–[03:30])
2. Spectral Actions and Local Systems ([04:00]–[14:00])
3. The Spectral Problem and Its Resolution ([20:00]–[31:50])
4. Generators and Structural Description ([25:20]–[32:00])
5. Relationship Between Different Settings and Equivalences ([51:00]–[56:50])
6. Whitaker Coefficients in the Geometric Setting ([57:00]–[01:16:40])
7. Construction and Characterization of the Geometric Langlands Functor ([01:18:00]–[01:35:50])
8. Further Remarks and Open Questions ([01:35:50]–[End])
Key Frameworks and Mental Models
Spectral Action Functor
Nilpotent Singular Support Condition
Whitaker Coefficient Functor (Geometric Analog of Fourier Coefficients)
Duality of Quasi-Coherent Sheaves on Algebraic Stacks
Practical Implications & Takeaways
Action-Oriented Summary

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