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Bertrand Lemaire

Publications and source records attributed to Bertrand Lemaire.

11 recordsLinked to original sources
arXiv2025

Induction automorphe: représentations unitaires et spectre résiduel

Let $E/F$ be a finite cyclic extension of local fields of characteristic zero, of degree $d$, and $κ$ be a character of $F^\times$ whose kernel is $\mathrm{N}_{E/F}(E^\times)$. For $m\in \mathbb{N}^*$, we prove that every irreducible unitary representation of $\mathrm{GL}_m(E)$ has a $κ$-lift to $\mathrm{GL}_{md}(F)$, given by a character identity as in Henniart-Herb [HH]. Let ${\bf E}/{\bf F}$ be a finite cyclic extension of number fields, of degree $d$, and $\mathfrak{K}$ be a character of $\mathbb{A}_{\bf F}^\times$ whose kernel is ${\bf F}^\times \mathrm{N}_{{\bf E}/{\bf F}}(\mathbb{A}_{\bf E}^\times)$. We prove that every automorphic discrete representation of $\mathrm{GL}_m(\mathbb{A}_{\bf E})$ has a (strong) $\mathfrak{K}$-lift to $\mathrm{GL}_{md}(\mathbb{A}_{\bf F})$, i.e. compatible with the local lifting maps. We describe the image and the fibres of these local and global lifting maps. Locally, we also treat the elliptic representations.

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arXiv2024

Développement fin de la contribution unipotente à la formule des traces sur un corps global de caractéristique p>0, I

For a field $F$ and a connected reductive group $G$ defined over $F$, we develop a theory of Kempf-Rousseau-Hesselink unipotent $F$-strata in $G(F)$ that should allow us to attack open problems in positive characteristic. As an application, we use this theory to establish the fine expansion of the unipotent contribution to the (non-twisted) trace formula over a global field of characteristic $p>0$. The unipotent $F$-strata play here the role of the unipotent geometric orbits in Arthur's work over a number field. The expansion in terms of products of local distributions is not discussed here; it will be the subject of further work.

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arXiv2024

Une mesure de Radon invariante sur les $F$-strates unipotentes

Let $F$ be a non-Archimedean locally compact field and $G$ a connected reductive group defined over $F$. To any unipotent element $u$ in $G(F)$, we have associated in [L] an $F$-stratum $\boldsymbol{\mathfrak{Y}}_{F,u}$ which is a (possibly infinite) union of unipotent $G(F)$-orbits. We define here a "canonical" non-zero positive $G(F)$-invariant Radon measure on $\boldsymbol{\mathfrak{Y}}_{F,u}$. Under additional assumptions, we deduce the convergence of the orbital integral associated to the $G(F)$-orbit of $u$. The construction, valid in any characteristic, generalizes the one of Deligne-Ranga Rao [RR] and also applies to nilpotent strata in $\textrm{Lie}(G)(F)$.

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arXiv2019

Matching of orbital integrals (transfer) and Roche Hecke algebra isomorphisms

Let $F$ be a non-Archimedan local field, $G$ a connected reductive group defined and split over $F$, and $T$ a maximal $F$-split torus in $G$. Let $χ_0$ be a depth zero character of the maximal compact subgroup $\mathcal{T}$ of $T(F)$. It gives by inflation a character $ρ$ of an Iwahori subgroup $\mathcal{I}$ of $G(F)$ containing $\mathcal{T}$. From Roche, $χ_0$ defines a split endoscopic group $G'$ of $G$, and there is an injective morphism of ${\Bbb C}$-algebras $\mathcal{H}(G(F),ρ) \rightarrow \mathcal{H}(G'(F),1_{\mathcal{I}'})$ where $\mathcal{H}(G(F),ρ)$ is the Hecke algebra of compactly supported $ρ^{-1}$-spherical functions on $G(F)$ and $\mathcal{I}'$ is an Iwahori subgroup of $G'(F)$. This morphism restricts to an injective morphism $ζ: \mathcal{Z}(G(F),ρ)\rightarrow \mathcal{Z}(G'(F),1_{\mathcal{I}'})$ between the centers of the Hecke algebras. We prove here that a certain linear combination of morphisms analogous to $ζ$ realizes the transfer (matching of strongly $G$-regular semisimple orbital integrals). If ${\rm char}(F)=p>0$, our result is unconditional only if $p$ is large enough.

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arXiv2019

Intégrales orbitales sur $GL(N,{\Bbb F}_q((t)))$

Let $F$ be a non--Archimedean local field of characteristic $\geq 0$, and let $G=GL(N,F)$, $N\geq 1$. An element $γ\in G$ is said to be quasi--regular if the centralizer of $γ$ in $M(N,F)$ is a product of field extensions of $F$. Let $G_{\rm qr}$ be the set of quasi--regular elements of $G$. For $γ\in G_{\rm qr}$, we denote by $\mathcal{O}_γ$ the ordinary orbital integral on $G$ associated with $γ$. In this paper, we replace the Weyl discriminant $\vert D_G\vert$ by a normalization factor $η_G: G_{\rm qr}\rightarrow {\Bbb R}_{>0}$ which allows us to obtain the same results as proven by Harish--Chandra in characteristic zero: for $f\in C^\infty_{\rm c}(G)$, the normalized orbital integral $I^G(γ,f)=η_G^{1\over 2}(γ)\mathcal{O}_γ(f)$ is bounded on $G$, and for $ε>0$ such that $N(N-1)ε<1$, the function $η_G^{-{1\over 2}-ε}$ is locally integrable on $G$.

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arXiv2016

Caractères tordus des représentations admissibles

Let $F$ be a non--Archimedean locally compact field (${\rm car}(F)\geq 0$), ${\bf G}$ be a connected reductive group defined over $F$, $θ$ be an $F$--automorphism of ${\bf G}$, and $ω$ be a character of ${\bf G}(F)$. We fix a Haar measure $dg$ on ${\bf G}(F)$. For a smooth irreducible $(θ,ω)$--stable complex representation $π$ of ${\bf G}(F)$, that is such that $π\circ θ\simeq π\otimes ω$, the choice of an isomorphism $A$ from $π\otimes ω$ to $π\circ θ$ defines a distribution $Θ_π^A$, called the \og ($A$--)twisted character of $π$\fg: for a compactly supported locally constant function $f$ on ${\bf G}(F)$, we put $Θ_π^A(f)={\rm trace}(π(fdg)\circ A)$. In this paper, we study these distributions $Θ_π^A$, without any restrictive hypothesis on $F$, ${\bf G}$ or $θ$. We prove in particular that the restriction of $Θ_π^A$ on the open dense subset of ${\bf G}(F)$ formed of those elements which are $θ$--quasi--regular is given by a locally constant function, and we describe how this function behaves with respect to parabolic induction and Jacquet restriction. This leads us to take up again the Steinberg theory of automorphisms of an algebraic group, from a rationnal point of view.

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arXiv2015

Le lemme fondamental pour l'endoscopie tordue: le cas o{ù} le groupe endoscopique non ramifi{é} est un tore

We prove the fundamental lemma for twisted endoscopy, for the unit elements of the spherical Hecke algebras, in the case of a non ramified elliptic endo- scopic datum whose underlying group is a torus. This implies that the fundamental lemma for twisted endoscopy is now proved, for all elements in the spherical Hecke algebras, in characteristic zero and any residue characteristic.

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arXiv2015

Le lemme fondamental pour l'endoscopie tordue: réduction aux éléments unités

We show here that the fundamental lemma for twisted endoscopy, now proved for the unit elements in the spherical Hecke algebras, implies the fundamental lemma for all elements of these Hecke algebras. The proof, whose idea is due to Arthur, uses the transfer, which is known as a consequence of the fundamental lemma for the units.

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arXiv2013

La transformée de Fourier pour les espaces tordus sur un groupe réductif $\mathfrak{p}$-adique I. Le théorème de Paley-Wiener

Let ${\boldsymbol{G}}$ be a connected reductive group defined over a non--Archimedean local field $F$. Put $G={\boldsymbol{G}}(F)$. Let $θ$ be an $F$--automorphism of ${\boldsymbol{G}}$, and let $ω$ be a smooth character of $G$. This paper is concerned with the smooth complex representations $π$ of $G$ such that $π^θ=π\circθ$ is isomorphic to $ωπ=ω\otimesπ$. If $π$ is admissible, in particular irreducible, the choice of an isomorphism $A$ from $ωπ$ to $π^θ$ (and of a Haar measure on $G$) defines a distribution $Θ_π^A={\rm tr}(π\circ A)$ on $G$. The twisted Fourier transform associates to a compactly supported locally constant function $f$ on $G$, the function $(π,A)\mapsto Θ_π^A(f)$ on a suitable Grothendieck group. Here we describe its image (Paley--Wiener theorem), and we reduce the description of its kernel (spectral density theorem) to a result on the discrete part of the theory.

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