Représentations et quasi-caractères de niveau $0$ ; endoscopie

Waldspurger, Jean-Loup

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 193-278.

Soient $F$ une extension finie de ${\mathbb{Q}}_p$ et $G$ un groupe réductif connexe défini sur $F$. On suppose que $p$ est grand relativement à $G$. Soit $G^{\prime }$ un groupe endoscopique de $G$. D’après Arthur, il existe un homomorphisme de transfert spectral. Grosso modo, à une combinaison linéaire stable de représentations admissibles et irréductibles de $G^{\prime }\left(F\right)$, il associe une combinaison linéaire de représentations admissibles et irréductibles de $G\left(F\right)$. On note $transfert$ cet homomorphisme. Notons $p^{0,G}$ le projecteur de Bernstein tel que, pour une représentation admissible et irréductible $\pi$ de $G\left(F\right)$, on a $p^{0,G}\left(\pi \right)=\pi$ si $\pi$ est de niveau $0$ et $p^{0,G}\left(\pi \right)=0$ si $\pi$ est de niveau strictement positif. On définit de même $p^{0,G^{\prime }}$. On démontre que $p^{0,G^{\prime }}$ préserve l’espace des combinaisons linéaires stables de représentations admissibles et irréductibles de $G^{\prime }\left(F\right)$ et que $p^{0,G}\circ transfert=transfert\circ p^{0,G^{\prime }}$.

Let $F$ be a finite extension of ${\mathbb{Q}}_p$ and let $G$ be a connected reductive group over $F$. We assume that $p$ is large relatively to $G$. Let $G^{\prime }$ be an endoscopic group of $G$. Following Arthur, we have, roughly speaking, a spectral transfer morphism, denoted by $transfert$, which, to a stable finite linear combination of irreducible admissible representations of $G^{\prime }\left(F\right)$, associates a finite linear combination of irreducible admissible representations of $G\left(F\right)$. Let $p^{0,G}$ be the Bernstein’s projector such that, for an irreducible admissible representation $\pi$ of $G\left(F\right)$, we have $p^{0,G}\left(\pi \right)=\pi$ if $\pi$ has level $0$ and $p^{0,G}\left(\pi \right)=0$ if $\pi$ has strictly positive level. Define similarly $p^{0,G^{\prime }}$. We prove that $p^{0,G^{\prime }}$ preserves the space of stable finite linear combinations of irreducible admissible representations of $G^{\prime }\left(F\right)$ and that $p^{0,G}\circ transfert=transfert\circ p^{0,G^{\prime }}$.

Classification : 22E50
Mot clés : Représentations de niveau $0$, transfert endoscopique
Keywords: Representations of depth $0$, endoscopic transfer

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