arXiv subjects

David Renard

Publications and source records attributed to David Renard.

13 recordsLinked to original sources

arXiv2026

Representations of $p$-adic reductive groups

This book presents a part of the (complex) representation theory of $p$-adic reductive groups. Starting from a basis accessible to graduate students, it culminates with the theory of the "Bernstein center" and the Langlands classification of irreducible smooth representations. \vspace{0.5cm} \noindent The book consists of seven chapters, Chapters VI and VII constituting the core of the book. Chapter VI is devoted to the study of the category of smooth representations of a $p$-adic reductive group, establishing among other things the Bernstein decomposition theorem and the description of its center. Chapter VII deals with square-integrable and tempered representations, and the Langlands classification theorem is proved there. \vspace{0.5cm} \noindent The first four chapters are placed in a more general framework and tackle, in order, the study of algebras with idempotents, totally disconnected locally compact spaces and groups, smooth representations of the latter, and particular classes of representations (compact, unitary, square-integrable). Chapter V is a review of the structure results of $p$-adic reductive groups. Appendices give the elements of category theory and some results in algebra used in the text.

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arXiv2023

Generic irreducibility of parabolic induction for real reductive groups

Let $G$ be a real reductive linear group in the Harish-Chandra class. Suppose that $P$ is a parabolic subgroup of $G$ with Langlands decomposition $P=MAN$. Let $\pi$ be an irreducible representation of the Levi factor $L=MA$. We give sufficient conditions on the infinitesimal character of $\pi$ for the induced representation $i_P^G(\pi)$ to be irreducible. In particular, we prove that if $\pi_M$ is an irreducible representation of $M$, then for a generic character $\chi_\nu$ of $A$, the induced representation $i_P^G(\pi_M\boxtimes \chi_\nu)$ is irreducible. Here the parameter $\nu$ is in $\mathfrak{a}^*=(\mathrm{Lie}(A)\otimes_\mathbb R \mathbb C)^*$ and generic means outside a countable, locally finite union of hyperplanes which depends only on the infinitesimal character of $\pi$. Notice that there is no other assumption on $\pi$ or $\pi_M$ than being irreducible, so the result is not limited to generalised principal series or standard representations, for which the result is already well known.

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arXiv2019

S\'eries discr\`etes des espaces sym\'etriques et paquets d'Arthur

We check Sakellaridis-Venkatesh conjectures giving a description of the discrete spectrum of a spherical variety $\mathscr{X}=G/H$ in the Langlands-Arthur formalism when $G$ is a classical real group and $\mathscr{X}$ is a symmetric space. Then, we compute explicitly the representations in the relevant Arthur paquets which appear in the discrete spectrum, and we establish some multiplicity one results.

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arXiv2019

Sur les paquets d'Arthur de $\mathbf{Sp}(2n,\mathbb{R})$ contenant des modules unitaires de plus haut poids, scalaires

Soit $π$ un module de plus haut poids unitaire du groupe $G=Sp(2n,\mathbb R)$. On s'intéresse aux paquets d'Arthur contenant $π$. Lorsque le plus haut poids est scalaire, on détermine les paramètres de ces paquets, on établit la propriété de multiplicité un de $π$ dans le paquet, et l'on calcule le caractère $ρ_π$ (du groupe des composantes connexes du centralisateur du paramètre dans le groupe dual) associé à $π$ et qui joue un grand rôle dans la théorie d'Arthur. On fait de même pour certains modules de plus haut poids unitaires unipotents $σ_{n,k}$. Let $π$ be an irreducible unitary highest weight module for $G=Sp(2,\mathbb R)$. We would like to determine the Arthur packets containing $π$. When the highest weight is scalar, we determine the Arthur parameter of these packets, we establish the multiplicity one property of $π$ in the packet and we compute the character $ρ_π$ (of the group of connected components of the centralizer of $ψ$ in the dual group) associated to $π$ which plays an important role in Arthur's theory. We also deal with the case of some unipotent unitary highest weight modules $σ_{n,k}$.

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arXiv2019

Sur les paquets d'Arthur des groupes classiques réels

This article is part of a project which consists of investigating Arthur packets for real classical groups. Our goal is to give an explicit description of these packets and to establish the multiplicity one property (which is known to hold for $p$-adic and complex groups). The main result in this paper is a construction of packets from unipotent packets on $c$-Levi factors using cohomological induction. An important tool used in the argument is a statement of commutativity between cohomological induction and spectral endoscopic transfer.

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arXiv2018

Sur les paquets d'Arthur des groupes unitaires et quelques conséquences pour les groupes classiques

We give an explicit construction of Arthur packets for real unitary groups by cohomological and parabolic induction and following an idea communicated to us by P. Trapa, we show that they satisfy the multiplicity one property. In particular, we show the irreducibility of some parabolically induced representations for unitary groups, and use this to give the proof of analogous statements made in our work on Arthur packets of classical groups. Nous donnons une construction explicite des paquets d'Arthur des groupes unitaires réels par induction cohomologique et induction parabolique et en suivant une idée communiquée par P. Trapa, nous établissons la propriété de multiplicité un de ceux-ci. Nous montrons en particulier des résultats d'irréductibilité de certaines induites paraboliques pour les groupes unitaires, ce qui nous permet de compléter les démonstrations d'énoncés analogues annoncés dans nos travaux sur les paquets d'Arthur des groupes classiques.

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arXiv2018

Sur les paquets d'Arthur des groupes classiques et unitaires non quasi-déployés

Nous étendons aux groupes orthogonaux et unitaires non quasi-déployés sur un corps local des résultats de J. Arthur et de la première auteure établis dans le cas quasi-déployé. En particulier, nous obtenons une classification de Langlands complète pour les représentations tempérées dans le cas $p$-adique. Nous en déduisons en utilisant l'involution d'Aubert-Schneider-Stuhler un résultat de multiplicité un dans les paquets unipotents, et par des méthodes globales, le même résultat pour les paquets unipotents dans le cas archimédien. We extend to non quasi-split orthogonal and unitary groups over a local field some results of J. Arthur and the first author established in the quasi-split case. In particular, we obtain a full Langlands classification for tempered representations in the $p$-adic case. Using Aubert-Schneider-Stuhler involution, we deduce from this a multiplicity one result for unipotent packets, and by global methods, the same result for unipotent packets in the archimedean case.

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arXiv2017

Sur les paquets d'Arthur aux places réelles, translation

This article is part of a project which aims to describe as explicitly as possible the Arthur packets of classical real groups and to prove a multiplicity one result for them. Let $G$ be a symplectic or special orthogonal real group, and $ψ: W_{\mathbb R}\times \mathbf{SL}_2(\mathbb C)\rightarrow {}^LG$ be an Arthur parameter for $G$. Let $A(ψ)$ the component group of the centralizer of $ψ$ in $\hat G$. Attached to $ψ$ is a finite length unitary representation $π^A(ψ)$ of $G\times A(ψ)$, which is characterized by the endoscopic identities (ordinary and twisted) it satisfies. In [arXiv:1703.07226] we gave a description of the irreducible components of $π^A(ψ)$ when the parameter $ψ$ is "very regular, with good parity". In the present paper, we use translation of infinitesimal character to describe $π^A(ψ)$ in the general good parity case from the representation $π^A(ψ_+)$ attached to a very regular, with good parity, parameter $ψ_+$ obtained from $ψ$ by a simple shift.

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arXiv2017

Paquets d'Arthur des groupes classiques complexes

Nous décrivons explicitement les paquets d'Arthur des groupes classiques complexes, ainsi que leur paramétrisation interne par les caractères du groupe des composantes connexes du centralisateur de leur paramètre. Nous montrons d'abord qu'ils sont obtenus par induction parabolique préservant l'irréductibilité à partir des paquets unipotents de "bonne parité". Pour ceux-ci, nous montrons qu'ils coïncident avec les paquets définis par Barbasch-Vogan. Nous utilisons des résultats profonds de Barbasch entrant dans sa classification du dual unitaire de ces groupes. We describe explicitly Arthur packets for complex classical groups, as well as their internal parametrization by the group of characters of the component group of the stabilizer of their parameter. We first show that they are obtained by parabolic induction preserving irreducibility from unipotent packets of "good parity". For these, we show that they coincide with the packets defined by Barbasch and Vogan. We use deep results of Barbasch entering his classification of the unitary dual of these groups.

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arXiv2017

Paquets d'Arthur des groupes classiques et unitaires

Let $G=\mathbf{G}(\mathbb{R})$ be the group of real points of a quasi-split connected reductive algebraic group defined over $\mathbb{R}$. Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on $G$ supported by these packets to twisted $\mathbf{GL}_N$ in terms of standard modules and show that it coincides with the twisted trace prescribed by Arthur.

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arXiv2014

Euler-Poincaré pairing, Dirac index and elliptic pairing for Harish-Chandra modules

Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of equal rank, and let $\mathscr M$ be the category of Harish-Chandra modules for $G$. We relate three differentely defined pairings between two finite length modules $X$ and $Y$ in $\mathscr M$ : the Euler-Poincaré pairing, the natural pairing between the Dirac indices of $X$ and $Y$, and the elliptic pairing. (The Dirac index is a virtual finite dimensional representation of $\widetilde K$, the spin double cover of $K$.) Analogy with the case of Hecke algebras and a formal (but not rigorous) computation lead us to conjecture that the first two pairings coincide. In the second part of the paper, we show that they are both computed as the indices of Fredholm pairs (defined here in an algebraic sense) of operators acting on the same spaces. We construct index functions $f_X$ for any finite length Harish-Chandra module $X$. These functions are very cuspidal in the sense of Labesse, and their orbital integrals on elliptic elements coincide with the character of $X$. From this we deduce that the Dirac index pairing coincide with the elliptic pairing. These results are the archimedean analog of results of Schneider-Stuhler for $p$-adic groups.

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arXiv2013

Level one algebraic cusp forms of classical groups of small ranks

We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL_n over Q of any given infinitesimal character, for essentially all n <= 8. For this, we compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO_7, SO_8, SO_9 (and G_2) and determine Arthur's endoscopic partition of these spaces in all cases. We also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GL_n with trivial infinitesimal character, and to vector valued Siegel modular forms of genus 3. A part of our results are conditional to certain expected results in the theory of twisted endoscopy.

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arXiv2005

Dirac operators and Lie algebra cohomology

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves \cite{V2}, \cite{HP1}, \cite{Kdircoh}. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the corresponding nilpotent Lie algebra cohomology in many cases, while in general it has better algebraic behavior and it is more accessible for calculation.

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