arXiv subjects

Thomas Lanard

Publications and source records attributed to Thomas Lanard.

7 recordsLinked to original sources

arXiv2025

An algorithm for Aubert-Zelevinsky duality \`a la M{\oe}glin-Waldspurger

Let $F$ be a locally compact non-Archimedean field of characteristic $0$, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$. The goal of this paper is to give an explicit description of the Aubert-Zelevinsky duality for $G$ in terms of Langlands parameters. We present a new algorithm, inspired by the Moeglin-Waldspurger algorithm for $\mathrm{GL}_n(F)$, which computes the dual Langlands data in a recursive and combinatorial way. Our method is simple enough to be carried out by hand and provides a practical tool for explicit computations. Interestingly, the algorithm was discovered with the help of machine learning tools, guiding us toward patterns that led to its formulation.

math.RT↗

arXiv2024

Modulo $\ell$ distinction problems

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$. We start by proving a general theorem allowing to lift supercuspidal $\overline{\mathbb{F}}_{\ell}$-representations of $\mathrm{GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline{\mathbb{Q}}_{\ell}$-representation. Given a quadratic field extension $E/F$ and an irreducible $\overline{\mathbb{F}}_{\ell}$-representation $π$ of $\mathrm{GL}_n(E)$, we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $ϕ_π$ is irreducible and conjugate-self-dual, then $π$ is either $\mathrm{GL}_n(F)$-distinguished or $(\mathrm{GL}_n(F),ω_{E/F})$-distinguished (where $ω_{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case $p=2$. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$. After that, we give a complete classification of the $\mathrm{GL}_2(F)$-distinguished representations of $\mathrm{GL}_2(E)$. Using this classification we discuss a modular version of the Prasad conjecture for $\mathrm{PGL}_2$. We show that the "classical" Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil-Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\mathrm{SL}_2(F)$-distinguished modular representations of $\mathrm{SL}_2(E)$.

math.RT↗

arXiv2022

Unipotent $\ell$-blocks for simply-connected $p$-adic groups

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 study the unipotent $\ell$-blocks of $G$, for $\ell \neq p$. To that end, 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 $G$. We also describe the stable $\ell$-block decomposition of the depth zero category of an unramified classical group.

math.RT↗

arXiv2022

Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$

We consider the category of depth $0$ representations of a $p$-adic quasi-split reductive group with coefficients in $\overline{\mathbb{Z}}[\frac{1}{p}]$. We prove that the blocks of this category are in natural bijection with the connected components of the space of tamely ramified Langlands parameters for $G$ over $\overline{\mathbb{Z}}[\frac{1}{p}]$. As a particular case, this depth $0$ category is thus indecomposable when the group is tamely ramified. Along the way we prove a similar result for finite reductive groups. As an application, we deduce that the semi-simple local Langlands correspondence $\pi\mapsto \varphi_{\pi}$ constructed by Fargues and Scholze takes depth $0$ representations to tamely ramified parameters, using a motivic version of their construction recently announced by Scholze. We also bound the restriction of $\varphi_{\pi}$ to tame inertia in terms of the Deligne-Lusztig parameter of $\pi$ and show, in particular, that $\varphi_{\pi}$ is unramified if $\pi$ is unipotent.

math.RT↗

arXiv2021

Equivalence of categories between coefficient systems and systems of idempotents

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 subcategory 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.

math.RT↗

arXiv2020

Sur les $\ell$-blocs de niveau zéro des groupes $p$-adiques II

Let $G$ be a $p$-adic group which splits over an unramified extension and $Rep_Λ^{0}(G)$ the abelian category of smooth level $0$ representations of $G$ with coefficients in $Λ=\overline{\mathbb{Q}}_{\ell}$ or $\overline{\mathbb{Z}}_{\ell}$. We study the finest decomposition of $Rep_Λ^{0}(G)$ into a product of subcategories that can be obtained by the method introduced in an article of Lanard (arXiv:1703.08689), which is currently the only one available when $Λ=\overline{\mathbb{Z}}_{\ell}$ 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 with parabolic induction and restriction or the compatibility with the local Langlands correspondence. The factors of this decomposition are not blocks, but we show how to group them to obtain "stable" blocks.

math.RT↗

arXiv2017

Sur les $\ell$-blocs de niveau zéro des groupes $p$-adiques

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $Rep_Λ^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $Λ=\overline{\mathbb{Q}}_{\ell}$ 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.

math.RT↗