arXiv subjects
Publications and source records attributed to Gerard Laumon.
We identify, up to homeomorphisms, the affine Springer fibers for GL(n) on a local field of equal characteristics with some coverings of compactified jacobians of singular projective curves. This allows us to prove an irreducibility property of these affine Springer fibers and to deform them.
Assuming the purity conjecture for the affine Springer fibers which has been formulated by Goresky, Kottwitz and MacPherson, we prove a geometric analog of the fundamental lemma for unitary groups. Our approach is similar to the one of Goresky, Kottwitz and MacPherson. Our main new ingredient is the link between affine Springer fibers and compactified Jacobians which is described in math.AG/0204109.
In their proof of the Drinfeld-Langlands correspondence, Frenkel, Gaitsgory and Vilonen make use of a geometric Fourier transformation. Therefore, they work either with l-adic sheaves in characteristic p>0, or with D-modules in characteristic 0. Actually, they only need to consider the Fourier transforms of homogeneous sheaves for which one expects a uniform geometric construction in any characteristic. In this note, we propose such a homogeneous geometric Fourier transformation. It extends the geometric Radon transformation which has been studied by Brylinski.
In 1967, Langlands conjectured a natural correspondence between automorphic representations and Galois representations, over number fields as well as over function fields. In 1983, Drinfeld discovered a geometric analog of the Langlands correspondence in the function field case, which extends the geometric class field theory of Lang and Rosenlicht. The so called Drinfeld-Langlands correspondence is a conjectural duality between two moduli spaces that are naturally associated to an algebraic curve X and a reductive group G. When X is projective and G is the full linear group GL(n), a large part of this correspondence has recently been established by E. Frenkel, D. Gaitsgory et K. Vilonen.
In this paper I construct a geometric transformation for generalized 1-motives which extends the Fourier-Mukai transformation for O-Modules on abelian varieties, the geometric Fourier transformation for D-Modules on vector spaces and the geometric Mellin transformation for D-Modules on tori. In particular, I construct an equivalence of triangulated categories between the derived category of quasi-coherent D-Modules on an abelian variety and the derived category of quasi-coherent O-Modules on the universal extension of the dual abelian variety. This equivalence has also been obtained by Mitchell Rothstein.
Drinfeld a associe a tout systeme local irreductible de rang 2 sur une surface de Riemann compacte un "faisceau automorphe". Ce faisceau automorphe vit sur l'espace de modules des fibres vectoriels de rang 2 sur cette meme surface de Riemann. En fait, Drinfeld a donne deux constructions de ce faisceau automorphe. La premiere est basee sur l'existence d'un modele de Whittaker pour toute representation automorphe cuspidale pour GL(2). Elle garde un sens en caracteristique positive et elle s'etend conjecturalement en rang n arbitraire. La seconde construction est basee sur un calcul explicite de l'anneau des sections globales d'un certain faisceau d'operateurs differentiels sur l'espace de modules des fibres vectoriels. Elle n'a pour l'instant de sens qu'en caracteristique nulle mais, par contre, elle est de portee plus generale que la premiere construction car elle a un sens non seulement pour GL(n) mais aussi pour un groupe reductif arbitraire. Ceci est le texte de deux exposes consacres a la premiere construction de Drinfeld et a son extension conjecturale en rang n arbitraire, que j'ai donnes a Luminy lors du Colloque "Fibres vectoriels sur les courbes et theorie de Langlands", en juin 1995.