Jean-François DAT

Professeur de Mathématiques à Sorbonne Université.
Chercheur à l'I.M.J. , projet Formes Automorphes.

Institut de Mathématiques de Jussieu
4, place Jussieu
75252 Paris cedex 05

Tel : +33 1 44 27 54 28 Fax : +33 1 44 27 78 18 Bureau : 15-25 412 e-mail : jean-francois.dat[at]imj-prg.fr
Research

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Statement

I am interested in connections between Representation Theory, Algebraic Geometry and Number Theory (Galois representations), in the general framework of Langlands' Program. My first work was in the field of smooth representation theory of p-adic groups. More recently, I have focused on the cohomology of certain moduli spaces that arise naturally in this context as some p-adic avatars of Shimura varieties. I am particularly interested in those phenomena which cannot be dealt with Trace Formula arguments only : non semi-simplicity, extensions, finite or integral coefficients. One of my current projects is to obtain a cohomological realization of the mod l local Langlands correspondence, due to Vignéras, and in particular an interpretation of the nilpotent part of this correspondence.



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On complex smooth representations of p-adic groups

These papers deal with K-theoretic aspects and applications to Hecke algebras. More recently with Bernstein blocks from a dual Langlands perspective.

On modular and integral smooth representations of p-adic groups

Here the techniques meet the modern approach to irreducible complex representations : use of type theory, dynamics on the building, etc... I was lead to introduce new tools such as : use of non-Archimedean asymptotic estimations of matrix coefficients, integral version of intertwining operators, parahoric functors, mu-functions for positive characteristic coefficients, etc... The main results are the noetherian properties for representations with values in any noetherian ring R in which p is invertible, the second adjunction property in the same context, the generic irreducibility for parabolically induced families with positive characteristic coefficients.

On non-abelian Lubin-Tate theory

Named after Lubin-Tate's pioneering works on an explicit construction of Artin's local reciprocity law, this theory studies the cohomology of certain moduli spaces for p-divisible groups -the broadest definition of which is due to Rapoport-Zink, aiming both at studying bad reduction of Shimura varieties and at providing explicit realizations of local Langlands functoriality. My main contribution has been to modify the usual way to such a realization by introducing a convenient equivariant derived category formalism ; this should be generally necessary as soon as one is interested in non-semisimple Galois actions and non-supercuspidal reductive action. Sofar, I have considered only the most famous examples : the Lubin-Tate and Drinfeld towers.

On p-adic period domains

These are non Archimedean analogues of Griffith's period spaces in complex Hodge theory. They are related to p-adic Hodge theory via Fontaine's theory of crystalline representations. In particular they are the natural target of the period maps defined on Rapoport-Zink spaces. A well known -- and the best understood-- example is Drinfeld's symmetric space. Beyond this example, the theory was mainly developed by Rapoport, Zink, Kottwitz, and then Orlik. This is the first monography on this topic. We start from the foundations and we emphasize the Harder-Narasimhan stratification of the boundary of a period space. Many proofs and some statements appear here for the first time.

On Deligne-Lusztig theory


Expository papers


Other material



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Credits. My research has been supported by the following agencies. [Up]

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