Talbot 2005: The Geometric Langlands Program

Mentored by David Ben-Zvi

February 27 - March 4, 2005
North Conway, New Hampshire

References for the Talbot workshop on geometric Langlands:

* Moduli of vector bundles on surfaces. Algebraic stacks.
-Sorger: Lectures on moduli of principal G-bundles over algebraic curves. Available from ICTP.

* Tannakian categories. Reconstructing an algebraic group from its tensor category of finite-dimensional representations.
-Breen: Tannakian categories. (in the big Motives books)
-Deligne: Categories tannakeniennes.
-Deligne and Milne: Tannakian categories.
and as Tannakian categories relate to geometric Langlands for reductive groups other than GL(n):
-Mirkovic and Vilonen: Geometric Langlands duality and representations of algebraic groups over commutative rings.
-Ginzburg: Perverse sheaves on a loop group and Langlands duality.

* D-modules. Beilinson-Bernstein localization (giving an equivalence between Harish-Chandra modules and sheaves of D-modules on the flag variety).
-Bernstein: Course on D-modules. Available from Arinkin's webpage.
-Milicic: Algebraic D-modules and representation theory of semisimple Lie groups. Available from Milicic's webpage.
-Milicic: Localization and represention theory of reductive Lie groups. Available from Milicic's webpage.
-Borel: Algebraic D-modules.

* Perverse sheaves. Riemann-Hilbert correspondence with D-modules. Intersection cohomology. l-adic perverse sheaves.
-Goresky and MacPherson: various
-Rietsch: An introduction to perverse sheaves.
-Massey: Notes on perverse sheaves and vanishing cycles.
-Kietag and Friehl: Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform.
-Borel: Intersection cohomology.
-Kashiwara and Schapira: Sheaves on manifolds.
-Beilinson, Bernstein and Deligne: Faisceaux Pervers.

* Fourier-Mukai transform. Equivalences of derived categories
-Mukai: derived categories of sheaves on abelian varieties.
-Rothstein: Sheaves with connection on abelian varieties.
-Bondal and Orlov: Derived categories of coherent sheaves.

* Representations of affine Kac-Moody algebras.
-Ben-Zvi and Frenkel: Vertex algebras and algebraic curves.
-Beilinson and Drinfeld: Chiral algebras.

* Integrable systems, quantizations of, and Hitchin's integrable system.
-Hitchin: Stable bundles and integrable systems.
-Donagi and Markman: Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles.
-Semenov-Tian-Shansky: Quantum and classical integrable systems.
-Beilinson and Drinfeld: Quantization of Hitchin's integrable system and Hecke eigensheaves. Available from Arinkin's webpage.

* Langlands program. Hecke algebras. There are surveys by Gelbart and by Knapp.
-Haines, Kottwitz and Prasad: Iwahori-Hecke algebras.
-Kazhdan and Lusztig: Representations of Coxeter groups and Hecke algebras.
--, Schubert varieties and Poincare duality. (Proc Symp AMS...)
Lusztig: Singularities, character formulas and a q-analogue for weight multiplicities. (Asterisque 1982)

* Opers.
-Beilinson and Drinfeld: Opers.
--, Quantization of Hitchin's integrable system and Hecke eigensheaves. Available from Arinkin's webpage.

-Frenkel: Recent advances in the Langlands program.
-Gaitsgory, Frenkel and Vilonen: On the geometric Langlands conjecture.
-Bernstein and Gelbart (eds.): An introduction to the Langlands program.

Other references may be found from the geometric Langlands webpages of Vilonen and of Arinkin.