Notes on Geometric Langlands

This page will contain a collection of .pdf files intended to share with the community my current understanding of Geometric Langlands
(global and local, classical and quantum).

Not all papers that appear in the list of references are ready to be posted, but most of them exist in one form or another, and you're
welcome to request a .pdf file from me. Otherwise, I'll be posting them gradually.

Needless to say, comments and corrections of any kind are extremely welcome!

Note on the status of the assertions:

If a statement is designated as a "Theorem", this means that the proof exists (but may have not been written down yet).

If a statement is designated as a "Quasi-Theorem", this means that I see a way how to prove it, but haven't thought through the details.
If you are a graduate student or a post-doc interested in the area, you are welcome to supply a proof and claim the result.

Papers:

The substance of the Langlands conjecture

• Outline of the proof of the geometric Langlands conjecture for GL(2) (an "official paper", last updated Nov. 12, 2014)
• Singular support of coherent sheaves, and the geometric Langlands conjecture (an "official paper", joint with D. Arinkin, last updated Oct. 31, 2014)
• A generalized vanishing conjecture (last updated Oct. 18, 2010)
• Quantum Langlands correspondence (last updated Nov 13, 2007)
• From geometric to function-theoretic Langlands (or how to invent shtukas) (last updated June 30, 2016)
• Recent progress in geometric Langlands theory (last updated June 30, 2016)
• The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support
  (an "official paper", joint with D. Arinkin, D.~Kazhdan, S. Raskin, N. Rozenblyum, Y. Varshavsky, last updated April, 2022)

The automorphic side

• Asymptotics of geometric Whittaker coefficients (joint with D. Arinkin, last updated March 16, 2015)
• Compact generation of the category of D-modules on Bun(G) (an "official" paper, joint with V. Drinfeld, last updated May 1, 2015)
• Contractibility of the space of rational maps (an "official paper", last updated Jan. 17, 2012)
• The Atiyah-Bott formula for the cohomology of Bun(G) (an "official paper", last updated June 24, 2019)
• The extended Whittaker category (last updated Dec. 2, 2010)
• Categories over the Ran space (last updated Dec. 2, 2010)
• Localization and the long intertwining operator for representations of affine Kac-Moody algebras (joint with S.~Arkhipov, last updated May 15, 2015)

The spectral side

• The category of singularities as a crystal and global Springer fibers (an "official" paper, joint with D.Arinkin, last updated May 12, 2015)

Eisenstein series

• Geometric constant term functor(s) (an "official" paper, joint with V. Drinfeld, last updated Nov. 8, 2013)
• A "strange" functional equation for Eisenstein series and miraculous duality on the moduli stack of bundles (an "official" paper, last updated Aug. 6, 2014)
• Eisenstein series and quantum groups (an "official" paper, last updated May 23, 2015)
• What acts on geometric Eisenstein series (last updated July 4, 2011)

Local Langlands and FLE

• Notes on Factorizable Sheaves (last updated Feb. 26, 2008)
• Local and global Whittaker categories (an "official" paper, last updated June 19, 2020)
• The semi-infinite intersection cohomology sheaf (an "official" paper, last updated Aug. 29, 2019)
• The semi-infinite intersection cohomology sheaf-II: the Ran space version (an "official" paper, last updated Sept. 3, 2019)
• A conjectural extension of the Kazhdan-Lusztig euivalence (an "official" paper, last updated Oct. 6, 2018)
• Parameters and duality for the metaplectic geometric Langlands theory (an "official" paper, last updated April 28, 2019)
• Metaplectic Whittaker category and quantum groups: the "small" FLE (an "official" paper, joint with S.~Lysenko, last updated March 8, 2019)
• On factorization algebras arising in the quatum geometric Langlands theory (an "official" paper, last updated May 29, 2020)

Foundational stuff

• Generalities on DG categories (last updated Dec. 30, 2012)
• Filtered colimits of categories, by N.~Rozenblyum (last updated Dec. 30, 2012)
• Functors given by kernels, adjunctions and duality (an "official" paper, last updated Nov. 23, 2015)
• Stacks (last updated Aug. 14, 2011)
• Quasi-coherent sheaves on stacks (last updated Aug. 18, 2011)
• Sheaves of categories and the notion of 1-affineness (an ``official" paper, last updated Aug. 9, 2014)
• Ind-coherent sheaves (an "official" paper, last updated Oct. 14 , 2012)
• DG Indschemes (an "official" paper, joint with N. Rozenblyum, last updated June 23, 2013)
• Crystals and D-modules (an "official" paper, joint with N. Rozenblyum, last updated Oct. 1, 2014)
• On some finiteness questions for algebraic stacks (an "official" paper, joint with V. Drinfeld, last updated Oct. 26, 2012)
• Staibility of the ULA property under taking perverse cohomologies (last updated March 18, 2016)

Book project 'A study in derived algebraic geometry' by D. Gaitsgory and N. Rozenblyum (preliminary version; last updated April 13, 2022)

This is the version with the initial ordering and numeration of chapters (used as reference in papers that were written before 2016)

• Preface
• Introduction to Part I (Preliminaries)
• Chapter I.1: Some higher algebra (an introduction to oo-categories and review of Lurie's books)
• Chapter I.2: Basics of derived algebraic geometry (introduces (derived) prestcks, stacks, schemes and Artin stacks)
• Chapter I.3: Quasi-coherent sheaves on prestacks (introduces and studies the basic properties of the category of quasi-coherent sheaves on a prestack)
• Introduction to Part II (Ind-coherent sheaves)
• Chapter II.1: Ind-coherent sheaves on schemes (introduces and studies elementary properties of IndCoh on schemes)
• Chapter II.2: IndCoh as a functor out of the category of correspondences (studies IndCoh as a functor of the category of correspondences)
• Chapter II.3: QCoh and IndCoh (discusses the relationship between QCoh and IndCoh in the framework of correspondences)
• Introduction to Part III (Inf-schemes)
• Chapter III.1: Deformation theory (sets up deformation theory, the latter being necessary for introducing inf-schemes)
• Chapter III.2: (Ind)-inf-schemes (defines inf-schemes, which are algebro-geometric objects that include DG schemes and de Rham prestacks)
• Chapter III.3: Ind-coherent sheaves on (ind)-inf-schemes (extends the formalism of IndCoh to inf-schemes and discusses its functoriality)
• Chapter III.4: An application: crystals (explains how the "formalism of 6 operations" for D-modules follows from the theory of IndCoh on inf-schemes)
• Introduction to Part IV (Formal geometry)
• Chapter IV.1: Formal moduli (reinterprets Lurie's theory of formal moduli problems using the language of inf-schemes)
• Chapter IV.2: Lie algebras and co-commutative co-algebras (sets up the theory of Le algebras from the point of view of Koszul-Quillen duality)
• Chapter IV.3: Formal groups and Lie algebras (explains how the pass between Lie algebras and formal groups within the framework of DAG)
• Chapter IV.4: Lie algebroids (initiates the study of Lie algebroids and modules over them)
• Chapter IV.5: Infinitesimal differential geometry (studies various aspects of infinitesimal geometry, such as the n-th infinitesimal neighborhood)
• Introduction to Part V (Categories of correspondences)
• Chapter V.1: The (oo,2)-category of correspondences (introduces the formalism of correspondences)
• Chapter V.2: Extension theorems for category of correspondences (extends the theory of IndCoh from schemes to inf-schemes)
• Chapter V.3: The (symmetric) monoidal structure on the category of correspondences (shows how the formalism of correspondences encodes Serre duality)
• Introduction to Part A (appendix on (oo,2)-categories)
• Chapter A.1: Basics of (oo,2)-categories (defines (oo,2)-categories and introduces some basic constructions)
• Chapter A.2: Straightening and Yoneda (oo,2)-categories (constructs the straightening/unstraightening procedures and the Yoneda embedding)
• Chapter A.3: Adjunctions in (oo,2)-categories (studies the procedure of passing to the adjoint 1-morphism)
• References
• Glossary
• Index of Notation

This is the version that corresponds more closely to published version (with minor corrections and fixed typos)

• Volume 1, Correspondences and duality
• Volume 2, Deformations, Lie theory and formal geometry