Thomas Lanard

About

Postdoc at Vienna University, Austria

Research Areas: Representation theory, Number theory, Algebra
Current interests: Representations of p-adic groups, Langlands program

Publications

  • Unipotent \(\ell\)-blocks for simply connected \(p\)-adic groups (pdf)
    Let \(F\) be a non-archimedean local field and \(G\) the \(F\)-points of a connected simply-connected reductive group over \(F\). In this paper, we construct the unipotent \(\ell\)-blocks of \(G\). To do that we introduce the notion of d-1-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and d-Harish-Chandra theory. The \(\ell\)-blocks are then constructed using these d-1-series, with d the order of q modulo \(\ell\), and consistent systems of idempotents on the Bruhat-Tits building of \(G\).
  • Équivalence de catégories entre systèmes de coefficients et d'idempotents (pdf,arXiv:1912.06566)
    The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of \(Rep_{R}(G)\), the category of smooth representations of a \(p\)-adic group \(G\) with coefficients in \(R\). In particular, they were used to construct level 0 decompositions when \(R=\overline{\mathbb{Z}}_{\ell}\), \(\ell \neq p\), by Dat for \(GL_n\) and the author for a more general group. Wang proved in the case of \(GL_n\) that the sub-category associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of \(GL_n\) and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field.
  • Sur les \(\ell\)-blocs de niveau zéro des groupes \(p\)-adiques II (pdf,arXiv:1806.09543)
    Let \(G\) be a \(p\)-adic group which splits over an unramified extension and \(Rep_{\Lambda}^{0}(G)\) the abelian category of smooth level 0 representations of \(G\) with coefficients in \(\Lambda=\overline{\mathbb{Q}}_{\ell}\) or \(\overline{\mathbb{Z}}_{\ell}\). We study the finest decomposition of \(Rep_{\Lambda}^{0}(G)\) into a product of subcategories that can be obtained by the method introduced by Lanard, which is currently the only one available when \(\Lambda=\overline{\mathbb{Z}}_{\ell}\) and \(G\) is not an inner form of \(GL_n\). We give two descriptions of it, a first one on the group side à la Deligne-Lusztig, and a second one on the dual side à la Langlands. We prove several fundamental properties, like for example the compatibility to parabolic induction and restriction or the compatibility to the local Langlands correspondence. The factors of this decomposition are not blocks, but we show how to group them to obtain "stable" blocks. Some of these results support a conjecture given by Dat.
  • Sur les \(\ell\)-blocs de niveau zéro des groupes \(p\)-adiques, Compos. Math. 154 (2018), no. 7 (pdf)
    Let \(G\) be a \(p\)-adic group that splits over an unramified extension. We decompose \(Rep_{\Lambda}^{0}(G)\), the abelian category of smooth level \(0\) representations of \(G\) with coefficients in \(\Lambda=\overline{\mathbb{Q}}_{\ell}\) or \(\overline{\mathbb{Z}}_{\ell}\), into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat-Tits building and Deligne-Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

  • Thesis : Sur les \(\ell\)-blocs de niveau zéro des groupes \(p\)-adiques (pdf)

    CV

    Education

    2019–
    Post-Doc in Mathematics,
    University of Vienna, Austria
    2015-2019

    PhD in Mathematics,
    Advisor : Jean-François Dat,
    Université Pierre et Marie Curie, IMJ-PRG, Paris

    Subject: On the l-locks of p-adic groups

    2014-2015

    Master (M2) of Mathematics,
    École Normale Supérieure de Lyon

    Subject: Introduction to the theory of L and zeta functions and their applications

    2013-2014
    Agrégation of Mathematics,
    École Normale Supérieure de Lyon
    Rank : 1st
    2012-2013
    Master (M1) of Mathematics, ERASMUS,
    Imperial College London / École Normale Supérieure de Lyon
    2011-2012
    Bachelor of Mathematics,
    École Normale Supérieure de Lyon

    Talks

    December 2019
    London Number Theory Seminar, London, United-Kingdom
    February 2019
    Colloquium GDR TLAG, Poitiers, France
    February 2019
    Seminare of the University of East Anglia, Norwich, United-Kingdom
    December 2018
    Seminare of the University of Vienna, Vienna, Austria
    November 2018
    Seminare of the Laboratoire de Mathématiques de Versailles, Versailles, France
    February 2018
    Seminare Groupes Réductifs et Formes Automorphes de l’IMJ-PRG, Paris, France
    November 2017
    Seminare Groupes, Représentations et Géométrie, Paris, France

    Research Internships

    2015
    Institut Mathématique de Jussieu, Jean François Dat
    Representations of p-adic groups and Langlands conjectures
    2013
    Imperial College London, Kevin Buzzard
    The Modularity Theorem
    2012
    École Polytechnique, Centre de Mathématiques Laurent Schwartz, Alain Plagne
    Waring’s Problem

    Teaching

    2019-2029
    Algebraic Number Theory
    Master, Teaching assistant of A. Mínguez
    University of Vienna
    2015-2019
    Mathematics in continuing education
    L3, Autonomous
    Polytech’ Paris
    2018-2019
    Group theory
    L3, Teaching assistant of B. Stroh
    Sorbonne Université
    2018-2019
    Arithmetic
    L2, Teaching assistant of S. Chemla
    Sorbonne Université
    2015-2018
    Fourier analysis and Distributions
    L3, Teaching assistant of L. Lazzarini
    Polytech’ Paris

    Computer science

    2011-2012
    Programming Theory,
    École Normale Supérieure de Lyon
    Validated with 19/20
    2011
    TIPE : Ant colony optimization algorithms
    Lycée Michel-Montaigne, Bordeaux, France
    2010
    TIPE : Artificial intelligence for the game of Go
    Lycée Michel-Montaigne, Bordeaux, France

    Programming language : C++, Python, Pascal, HTML, CSS, JavaScript