Automorphic Forms
Org: Stephen Kudla (Toronto) and Colette Moeglin (CNRSIMJ) [PDF]
 IOAN BADULESCU, Université de Poitiers, Dept. Math. 86000 Poitiers, France
Global JacquetLanglands and automorphic representations
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We first explain the local JacquetLanglands transfer for all unitary representations. Using this local correspondence one may define a global JacquetLanglands correspondence and prove it thanks to the work of Arthur and Clozel on the trace formula for simple algebras. As a consequence one may transfer the results of MoeglinWaldspurger and JacquetShalika to inner forms of the linear group. In particular we obtain the classification of automorphic representations of the adele group of the group of invertible elements of a central simple algebra of finite dimension over a global field. All fields here are supposed to be of characteristic zero.
 PASCAL BOYER, Université Paris 6
Faisceaux pervers des cycles évanescents de quelques variétés de Shimura unitaires
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La preuve de la correspondance locale de Langlands en inégales caractéristiques donnée par Harris et Taylor, repose sur l'étude du groupe de cohomologie en degré médian du modèle local dit de DeligneCarayol. Il se trouve que les autres groupes de cohomologie n'apportent aucune contribution supercuspidale de sorte que l'on peut tout aussi bien étudier la somme alternée de ces groupes laquelle est reliée, via le théorème de SerreTate, à la somme alternée des groupes de cohomologie de variétés de Shimura associées à certains groupes unitaires. Cette dernière peut alors être calculée selon un procédé désormais classique de comptage de points via la formule des traces.
Le thème principal de cet exposé sera d'expliquer comment, en utilisant la structure perverse du complexe des cycles évanescents, obtenir chacun des groupes de cohomologie ciavant, ce qui au passage fournira une preuve de la conjecture de monodromiepoids pour les variétés de Shimura étudiées.
 PIERREHENRI CHAUDOUARD, CNRS, Université ParisSud, Mathématique, Bât. 425, F91405 Orsay Cedex, France
The truncated Hitchin fibration and the weighted fundamental lemma
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The main geometric terms of the ArthurSelberg trace formula are the weighted orbital integrals. According to Arthur, to stabilize the full trace formula, one needs the weighted fundamental lemma which is a family of complicated identities between padic weighted orbital integrals. This is a generalization of the fundamental lemma of LanglandsShelstad. Ngô Bao Châu recently proved the fundamental lemma of LanglandsShelstad by a cohomological study of the Hitchin fibration.
My talk will be based on the proof of the weighted fundamental lemma I obtained with Gérard Laumon. I shall introduce a truncated version of the Hitchin fibration such that a weighted orbital integral of Arthur computes the number of rational points of a truncated fiber. I shall explain how to extend the results of Ngô to our situation and how to deduce the weighted fundamental lemma of Arthur.
 CLIFTON CUNNINGHAM, University of Calgary
Character sheaves of algebraic groups defined over local fields
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At the beginning of the paper introducing character sheaves of connected reductive algebraic groups, George Lusztig wrote:
This paper is an attempt to construct a geometric theory of characters of a reductive algebraic group G defined over an algebraically closed field. We are seeking a theory which is as close as possible to the theory of irreducible (complex) characters of the corresponding groups G(F_{q}) over a finite field F_{q}, and yet it should have a meaning over algebraically closed fields. The basic objects in the theory are certain irreducible (ladic) perverse sheaves ... on G; they are the analogues of the irreducible (ladic) representations of G(F_{q}) and are called the character sheaves of G.
Making use of the dictionnaire fonctionsfaisceaux, Lusztig then showed that there is indeed a close relation between certain character sheaves of connected reductive algebraic groups G defined over finite fields F_{q} and characters of representations of the group G(F_{q}).
In this talk we introduce machinery which likewise establishes that there is a close relation between certain character sheaves of connected reductive algebraic groups G defined over local fields K and characters of certain representations of the group G(K). We do this by considering a family of integral models for G and applying the corresponding vanishing cycles functors to character sheaves of G. The main results of the talk concern techniques by which the resulting families of vanishing cycles of perverse sheaves can be determined.
 FRANCOIS DAT, Paris XIII

 BROOKE FEIGON, Toronto

 PAUL MEZO, Carleton University, Ottawa
Twisted endoscopic character identities
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Let G be a connected real reductive group. The theory of endoscopy attaches to G a set of groups whose harmonic analysis is related to G. The precise relationship takes the form of identities involving orbital integrals, or identities involving representation characters. It has been proven by Shelstad. One may generalize this theory by twisting the endoscopic data with respect to an automorphism and quasicharacter of G. The resulting identities of twisted orbital integrals have been established in some cases by Renard. We shall present the conjectured twisted character identities.
 SOPHIE MOREL, IAS

 FIONA MURNAGHAN, Toronto

 MARC HUBERT NICOLE, Jussieu

 VINCENT SECHERRE, Université de la MéditerranéeAix Marseille 2
lmodular representations of inner forms of GL(n) over a padic field with l different from p
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Let G be an inner form of GL(n) over a padic field, and let l be a prime number different from p. Our aim is to give a classification of all irreducible smooth lmodular representations of G in terms of the discrete series of its Levi subgroups. Such a classification has been obtained for GL(n) by Zelevinsky (for complex representations) and by Vigneras (for lmodular representations). This work relies in particular on the study of the reduction modulo l of irreducible ladic representations of G. It appears that, unlike the GL(n) case, integral cuspidal ladic representations of G may not reduce into irreducible lmodular representations. We study and explain this phenomenon by using the theory of simple types.
Joint work with Alberto Minguez.
