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Jean-Loup Waldspurger

Publications and source records attributed to Jean-Loup Waldspurger.

31 records · Page 2Linked to original sources
arXiv2014

Stabilisation de la formule des traces tordue IV: transfert spectral archimédien

It is one of a series of papers whose goal is to stabilize the twisted trace formula. We consider here a "twisted space" over the real field. We prove in this twisted situation the results obtained by Arthur in his Selecta's paper. That is the existence of transfer of tempered representations. We use the Paley-Wiener's theorems of Renard and Delorme-Mezo, the work of Shelstad about transfer of functions and a recent result of Moeglin about super-tempered representations. As corollaries, we obtain a stable version of the Paley-Wiener theorem and the fact that transfer of functions preserves K-finitude.

math.RT↗
arXiv2014

Stabilisation de la formule des traces tordue III: intégrales orbitales et endoscopie sur un corps local non-archimédien; réductions et preuves

This is one of a series of articles, in collaboration with C. Moeglin, whose goal is to stabilize the twisted trace formula. In the previous article, we have stated several assertions about the stabilization of weighted orbital integrals over a non-archimedean field. Those integrals depend on a conjugacy class in our twisted space. Here we deduce all these assertions from a unique one, where the conjugacy class is assumed to be strongly regular.

math.RT↗
arXiv2014

Stabilisation de la formule des traces tordue II: intégrales orbitales et endoscopie sur un corps local non-archimédien; définitions et énoncés des résultats

It is the second article of a series whose final goal is the stabilization of the twisted trace formula. Over a non archimedean field, we define (after Arthur) wheighted orbital integrals and their stable and endoscopic versions. We state the main theorems that will be proved in the following articles. We begin to study the local properties of our integrals, for instance expansion in Shalika germs.

math.RT↗
arXiv2014

Stabilisation de la formule des traces tordue I: endoscopie tordue sur un corps local

This is the first of a series of papers devoted to the stabilization of the twisted trace formula. It is just an introduction. We present the local theory of twisted endoscopy, following the fundamental works of Kottwitz-Shelstad, Labesse and Shelstad. Our presentation is partly new because we try to give canonical definitions of the basic objects, which are more usually defined using auxiliary data. We prove also some results about the image of endoscopic transfer and about the dual transfer of stable distributions whose support is a finite set of stable conjugacy classes.

math.RT↗
arXiv2012

La formule des traces locale tordue

The local trace formula gives strong relations between two types of invariant distributions on a reductive group defined over a local field: orbital integrals and characters of representations. For connected reductive groups, the formula was proved by J. Arthur. We extend the formula to twisted spaces, that is, roughly speaking, to non-connected reductive groups. The formula has the same consequences as in the connected case. In particular, we compute the elliptic product of tempered elliptic twisted representations.

math.RT↗
arXiv2012

Calcul d'une valeur d'un facteur epsilon par une formule intégrale

Let d and m be two natural numbers of distinct parities. Let $π$ be an admissible irreducible tempered representation of GL(d,F), where F is a p-adic field. We assume that $π$ is self-dual. Then we can extend $π$ as a representation $\tildeπ$ of a non-connected group $GL(d,F)\rtimes \{1,θ\}$. Let $ρ$ be a representation of GL(m,F). We assume that it has similar properties as $π$. Jacquet, Piatetski-Shapiro and Shalika have defined the factor $ε(s,π\timesρ,ψ)$. We prove that we can compute $ε(1/2,π\timesρ,ψ)$ by an integral formula where occur the characters of $\tildeπ$ and $\tildeρ$. It's similar to the formula which, for special orthogonal groups, computes the multiplicities appearing in the local Gross-Prasad conjecture.

math.RT↗
arXiv2009

La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux

We prove the local Gross-Prasad conjecture for tempered representations of special orthogonal groups. Roughly speaking, the conjecture says that, if sigma is an irreducible representation of SO(n) and rho is an irreducible representation of SO(n-1), rho appears as quotient of the restriction of sigma to SO(n-1) with a multiplicity m(sigma,rho) that can be computed in terms of epsilon-factors. Our proof uses results of a previous papers which computes m(sigma,rho) and the epsilon-factors by integral formulas.

math.RT↗
arXiv2009

Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2ème partie: extension aux représentations tempérées

Let $F$ be a non-archimedean local field, of characteristic 0. Let $V$ be a finite dimensional vector space over $F$ and $q$ be a non-degenerate quadratic form on $V$. Denote $G$ the special orthogonal group of $(V,q)$. Let $W$ a non-degenerate hyperplane of $V$, denote $H$ the special orthogonal group of $W$. Let $π$, resp. $σ$, an admissible irreducible representation of $G(F)$, resp. $H(F)$. Denote $m(σ,π)$ the dimension of the complex space $Hom_{H(F)}(π_{| H(F)},σ)$. It's know that $m(σ,π)=0$ or 1. In a first paper, we have defined another term $m_{geom}(σ,π)$. It's an explicit sum of integrals of functions that can be deduced from the characters of $σ$ and $π$. Assume that $π$ and $σ$ are tempered. Then we prove the equality $m(σ,π)=m_{geom}(σ,π)$. This generalize the result of the first paper, where $π$ was supercuspidal. As in this paper, the previous equality implies as corollary (assuming certain properties of tempered $L$-packets) a weak form of the local Gross-Prasad conjecture, now for pairs of tempered $L$-packets.

math.RT↗
arXiv2009

Une formule intégrale reliée à la conjecture locale de Gross-Prasad

Let F be a non-archimedean local field, of characteristic 0. Let V be a finite dimensional vector space over F and q be a non-degenerate quadratic form on V. Denote d the dimension of V and G=SO(d) the special orthogonal group of (V,q). Let v_{0}\in V such that q(v_{0})\not=0, denote W the subspace of V orthogonal to v_{0} and H=SO(d-1) the special orthogonal group of W. Let π, resp. σ, an admissible irreducible representation of G(F), resp. H(F). Denote m(σ,π) the dimension of the complex space Hom_{H(F)}(π_{| H(F)},σ). By a theorem of Aizenbud, Gourevitch, Rallis and Schiffmann, we know that m(σ,π)=0 or 1. We define another term m_{geom}(σ,π). It's an explicit sum of integrals of functions that can be deduced from the characters of σand π. Assume that πis supercuspidal. Then we prove the equality m(σ,π)=m_{geom}(σ,π). Now, let Π, resp. Σ, an L-packet of tempered representations of G(F), resp. H(F). We use the sophisticated notion of L-paquet due to Vogan: the representations in the packets can be representations of inner forms of G(F), resp. H(F). We assume that certain conjectural properties of tempered L-packets are true. Assume that all elements of Πare supercuspidal. Then our integral formula implies the weak form of the Gross-Prasad conjecture: there exist a unique pair σ\times π\in Σ\times Πsuch that m(σ,π)=1.

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