This year's topic is the Langlands correspondence for general reductive groups over function fields. We will study Vincent Lafforgue's fundamental paper Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale, which attaches a semisimple Langlands parameter to every cuspidal automorphic representation of a reductive group G over the field of functions of a smooth projective algebraic curve X over a finite field. If time permits, we will also consider Lafforgue's work with Alain Genestier, which uses the global construction to define a local Langlands parametrization.
Assume G is a split reductive group; then a Langlands parameter for G is a homomorphism from the arithmetic fundamental group of X (minus a finite set of points) to the Langlands dual group. Lafforgue breaks with the method used by Drinfel'd and Laurent Lafforgue to prove the global Langlands correspondence for GL(n), which (like the method pioneered by Langlands and Kottwitz to study the zeta functions of Shimura varieties) is based on a comparison of the Arthur-Selberg trace formula with the Grothendieck-Lefschetz trace formula. Instead, Lafforgue constructs combinatorial data, which we call -pseudocharacters, that contain enough information to define semisimple representations, by defining a new algebra of operators, called excursion operators, that incorporate the action of the global Galois group on the moduli spaces (stacks) of shtukas with paws. The excursion operators generalize the familiar Hecke correspondences, and Lafforgue's main theorem is a decomposition of the space of cusp forms (of fixed level) over characters of the algebra of excursion operators and the identification of these characters with global Langlands parameters.
The bulk of Lafforgue's paper is concerned with a close analysis of the geometry of these moduli stacks, specifically with showing how the geometric Satake correspondence interprets the action of the unramified Hecke correspondences in such a way as to identify the restriction of the global Langlands parameter at an unramified place with what is predicted by the arithmetic Satake correspondence. In so doing, Lafforgue introduces methods of geometric representation theory into arithmetic in an utterly novel way. The scope of these new methods promises to be much broader than the applications to the global Langlands correspondence that motivated Lafforgue's work. For example, Peter Scholze has outlined a program to interpret some of Lafforgue's constructions in the setting of perfectoid geometry, and to apply them to the local Langlands program for p-adic groups.
Lafforgue's paper is available at the arXiv link indicated above; there are also shorter versions (in English as well as French) on Lafforgue's home page. The course is provisionally scheduled to meet on Tuesdays and Thursdays from 1:10-2:25, but this may change, and the scheduling will be flexible to accomodate travel schedules of the instructor as well as students. Dates on the tentative schedule below in black correspond to the official schedule. Dates in red through Mid-February are for background in automorphic forms, to be held at 6:30 PM on Wednesdays, in a room to be determined. Subsequent dates in red will be discussed later.
Tentative weekly schedule
Week 1, January 26, 27, 28 Statement of main results of Vincent Lafforgue
- (a) Introduction to the results of Vincent Lafforgue
- (b) automorphic representations, L-group, Hecke algebra and classical Satake isomorphism
- (c) moduli stacks of G-shtukas, paws, excursion operators, pseudocharacters
Week 2, February 2, 3, 4 Theory of pseudocharacters, following Richardson and Lafforgue
- (a) Definition of pseudo-representations
- (b) Automorphic representations, cusp forms, finiteness properties
- (c) Relation between pseudocharacters and representations, introduction to Hecke stacks
- G. Harder, Chevalley Groups Over Function Fields and Automorphic Forms, Annals of Math. 1974
- A. Borel, Automorphic forms on reductive groups, informal notes
Week 3, February 9 (no class February 11) Moduli of G-torsors
- (a) Cusp forms and moduli stack of shtukas with no paws
C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves
Week 4, February 16, 17, 18
Moduli of G-shtukas
- (a) Moduli stacks of vector bundles and G-shtukas
- (b) to be determined
- (c) Moduli stacks of G-shtukas and Hecke correspondences
Week 5, February 23, 25 Loop groups and affine Grassmannians
(a) Affine Grassmannians as ind-schemes, the case of GL(n) (b) Affine Grassmannians in general References: U. Görtz, Affine Springer fibers and affine Deligne-Lusztig varieties
Week 6, March 1, 3 Geometric Satake isomorphism
(a) Beilinson-Drinfeld affine Grassmannian, geometric Satake isomorphism, 1 (b) Geometric Satake isomorphism, 2 References: T. Richarz, A new approach to the geometric Satake equivalence
There will be no class on March 8 and 10th, but on March 11, W. Zhang's talk in the automorphic forms seminar will be closely related to the topics of this course.
Week 7, March 22, 24 Uniformization 2
(a) Geometric Satake isomorphism, 3; local structure of moduli spaces of shtukas (March 23 there will be a book talk at Book Culture, on a topic unrelated to this course.) (b) Coalescence, creation, and annihilation operators
Week 8, March 29, 30, 31 Excursion operators
(a) Definitions of excursion operators, 1 (b) Perverse sheaves [NOTE CHANGE!], with a proof of semisimplicity of the Satake category (c) Properties of excursion operators References (for (b)): G. Williamson, An illustrated guide to perverse sheaves
Week 9, April 5, 7 Drinfeld's lemma and applications
(a) Hecke correspondences (b) Partial Frobenius operators and end of the definition of excursion operators
Week 10-11, April 12, 21 Langlands parametrization [NO CLASS 4/14 or 4/19!]
(a) Potential automorphy, 1 (b) Potential automorphy, 2
Week 12, April 26, 28, Compatibility with geometric Satake
(a) Compatibility with geometric Satake (minuscule case) (b) Compatibility with geometric Satake (general case)