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This course is an introduction to the LanglandsShelstad fundamental lemma and its solution by B.C. Ngô. The fundamental lemma is a combinatorial identity between orbital integrals. It seems to be an essential ingredient when one wants to prove Langlands functorialities of endoscopic type or express HasseWeil zeta functions of Shimura varieties in terms of automorphic Lfunctions.
First we will review its statement and its geometric interpretation in terms of Affine Springer fibers. We will give the relation between orbital integrals and the Hitchin fibration. We will also introduce Arthur's weighted orbital integrals and their geometric counterparts : truncated Affine Springer fibers and truncated Hitchin fibration.
Then we will explain Ngô's solution which is based on a deep cohomological study of the elliptic part of the Hitchin fibration. We will also explain how one can extend Ngô's main cohomological result to the hyperbolic part of the truncated Hitchin fibration. It's the key of the solution by Laumon and myself of the weighted fundamental lemma (which is also an essential ingredient for endoscopic functorialities).
References:
Le lemme fondamental pour les algèbres de Lie. (Ngo Bao Chau)
http://arxiv.org/abs/0801.0446
Fibration de Hitchin et endoscopie. (Ngo Bao Chau)
Invent. Math. 164 (2006), no. 2, 399453.
Le lemme fondamental pondéré. II. Énoncés cohomologiques
(Chaudouard,Laumon) arXiv:0912.4512
Le lemme fondamental pondéré I : constructions géométriques
(Chaudouard, Laumon) arXiv:0902.2684
Fixed point varieties on affine flag manifolds. (Kazhdan, Lusztig)
Israel J. Math. 62 (1988), no. 2, 129168.
Homology of affine Springer fibers in the unramified case.
(Goresky, Kottwitz, Macpherson)
Duke Math. J. 121 (2004), no. 3, 509561.
In this course I will introduce the notion of the Hitchin system for groups GL_n, SL_n and PGL_n over a Riemann surface of genus g. Motivated by some mirror symmetry phenomena between certain Hodge numbers of the SL_n and PGL_n Hitchin systems we will discuss approaches to get information on the cohomology of the total space of the Hitchin system. One will be an arithmetic technique by using the character tables of SL_n(F_q) and PGL_n(F_q) to get formulas for the virtual SerreHodge polynomials of certain associated character varieties which are diffeomorphic with the total space of the corresponding Hitchin systems. If time permits I will sketch some relationships between our mirror symmetry formulas and the work of Ngo on the fundamental lemma in the Langlands program.
References: related survey paper, more recent work.Tthe course will be similar to the one in Barcelona; slides.
Exercise sheet: pdf
In these introductory lectures I'd like to explain basic properties of the moduli space of vector bundles on curves, its relation to the affine Graßmannian and the moduli space of Higgs bundles. For both moduli problems there is a notion of stability and an (adelic) description of the points of the spaces similar to the description of line bundles on a curve using divisors.
Plan: Moduli of vector bundles on curves. Definition and basic properties: Weil's description of the points of the moduli space. This can be made geometric using the affine Graßmannian as a variety parameterizing modifications of bundles. Smoothness and connected components of the moduli space. Stability of bundles and how this defines a compact moduli space. The moduli space of Higgs bundles. Basic properties, description of the points of the space. Relation to line bundles on spectral curves. Stability of Higgs bundles.
We begin with the formalism of tstructures and explain how to glue them. We then introduce the category of perverse sheaves, discuss simple objects and important properties. The theory culminates in the decomposition theorem. Finally, we provide descriptions and applications of perverse sheaves.
Prerequisites: Triangulated categories, derived categories of sheaves on topological spaces and the formalism of the six functors.
Exercise sheet: pdf
Origin of Hitchin's equation: ASDequations, dimensional reduction. Parabolic structures, logarithmic singularities, stability. De Rham and Dolbeault moduli spaces, HermiteEinstein metric. Nonabelian Hodge correspondence, deformation theory. Lambdaconnections, hyperKaehler structure on the moduli space. Spectral curves and sheaves, Hitchinmap. Apparent singularities of differential equations.
Prerequisites: Basic knowledge of Differential and Complex Algebraic Geometry, sheaf cohomology and Lietheory will be sufficient to follow this course.
References
 N. Hitchin: The SelfDuality Equations on a Riemann Surface,
Proc. London Math. Soc. 55 (1987) pp. 59126.
 C. Simpson: Harmonic Bundles on Noncompact Curves, J. Amer.
Math. Soc. 3 (1990), no. 3, pp. 713770.
 O. Biquard, P. Boalch: Wild Nonabelian Hodge Theory on Curves,
Compos. Math. 140 (2004), no. 1, pp. 179204.
Exercise sheet: pdf