Institut des Hautes Etudes Scientifiques (IHES) · Video summary

Sam Raskin - 3/6 Some Aspects of the Geometric Langlands Program

Sam Raskin explains the geometric Langlands program, focusing on Eisenstein series, functors, and their spectral vs geometric perspectives, emphasizing the importance of compactification and adjointness.

Institut des Hautes Etudes Scientifiques (IHES) · March 13, 2026 · 127-min episode · 9-min read

Sam Raskin - 3/6 Some Aspects of the Geometric Langlands Program

Key takeaways

🎯The existence of functors like Eisenstein series relies on properness and compactification, connecting geometric representation theory with algebraic geometry.03:29
🚀Focus on compactification techniques, like the Drinfeld compactification, to control the existence and properties of key functors.03:34
💡Understanding parabolic induction and the mystery of cuspidal parts is crucial for decomposing categories in geometric Langlands.10:05
📈The spectral counterpart involves lifting semisimple local systems to parabolics, a process invisible in classical spectral theory but geometric in nature.17:18
🔍Key to many proofs is showing functorial diagrams commute, often reduced to local properties at a single point, illustrating the importance of local-to-global principles.31:28
🚀Utilize local properties at points to prove global categories and diagrams commute, leveraging local-to-global principles effectively.31:28

Chapters

0:10🎯 Introduction and Overview
2:00🎯 Eisenstein Series and Functorial Formalism
5:21🎯 Adjoint Functors and Formalism in D-Modules
10:29🎯 Spectral and Geometric Parallel Constructions
1:31:42🎯 Geometric Langlands Conjecture and Main Strategy
1:31:52🎯 Outline of Strategy for Equivalence Proofs
1:56:04🎯 Single Point and the Role of Local Calculations
2:03:02🎯 Global versus Local Perspective and Geometric Arguments
2:03:33🎯 Summary and Final Remarks

The digest

High-Level Overview

This lecture presents critical structures and mechanisms underlying the Geometric Langlands Program, with a focus on the theory of Eisenstein series functors, their spectral counterparts, and key categorical constructions linking automorphic and spectral sides. It expounds on the existence, adjunctions, and geometric properties of Eisenstein series functors in the $D$-module and coherent sheaf contexts, while addressing foundational compatibility results and conjectural frameworks that underpin geometric Langlands equivalences. The lecture also discusses compact generation of categories involved, tempered subcategories, and the pivotal algebraic object controlling the Langlands correspondence. The presentation is complex and aimed at researchers familiar with geometric representation theory, algebraic geometry, and categorical aspects of Langlands duality.

# Structured Deep Dive

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The full 9-minute article, 22 more sections, and every new episode from Institut des Hautes Etudes Scientifiques (IHES) and the creators you follow.

1. Eisenstein Series Functors and Their Adjoints [00:02–07:20]
2. Analogues in $\ell$-adic and Betti Settings [07:20–10:00]
3. The Spectral Side: Functors on Local Systems [10:00–15:00]
4. Interpretation of the Spectral Eisenstein Operation [15:00–19:00]
5. Drinfeld Compactification and Technical Foundations [19:00–25:00]
6. Outline of the Proof of Diagram Commutativity [25:00–38:00]
7. Second Adjointness Theorem [36:00–39:00]
8. Geometry of the Zastava Space and Local-Global Reduction [40:00–47:00]
9. Compactified Eisenstein Series and Comparison to Classical Eisenstein Series [47:00–50:30]
10. Subtleties of Cohomological Shifts and Normalizations [51:00–56:00]
11. Generators and Compact Generation of the Automorphic Category [1:08:30–1:14:00]
12. Algebra Object $\mathcal{A}_G$ and Internal Endomorphisms [1:23:20–1:31:00]
13. Tempered Subcategories and Their Equivalence [1:38:30–1:42:00]
14. Calculation of Global Sections of $\mathcal{A}_G$ in the $D$-Module Setting [1:48:56–2:00:20]
15. Temperedness Characterization via Hecke Functors [2:04:00–2:06:50]
Eisenstein Series Functor ($\mathrm{Eis}_P$)
Drinfeld Compactification
Spectral Eisenstein Functor
Bernstein-Style Second Adjointness
Zastava Space
Algebra Object $\mathcal{A}_G$
Tempered Subcategories and Functorial Equivalences