Représentations et quasi-caractères de niveau 0 ; endoscopie
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 193-278.

Soient F une extension finie de p et G un groupe réductif connexe défini sur F. On suppose que p est grand relativement à G. Soit G 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 (F), il associe une combinaison linéaire de représentations admissibles et irréductibles de G(F). On note transfert cet homomorphisme. Notons p 0,G le projecteur de Bernstein tel que, pour une représentation admissible et irréductible π de G(F), on a p 0,G (π)=π si π est de niveau 0 et p 0,G (π)=0 si π est de niveau strictement positif. On définit de même p 0,G . On démontre que p 0,G préserve l’espace des combinaisons linéaires stables de représentations admissibles et irréductibles de G (F) et que p 0,G transfert=transfertp 0,G .

Let F be a finite extension of p and let G be a connected reductive group over F. We assume that p is large relatively to G. Let G 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 (F), associates a finite linear combination of irreducible admissible representations of G(F). Let p 0,G be the Bernstein’s projector such that, for an irreducible admissible representation π of G(F), we have p 0,G (π)=π if π has level 0 and p 0,G (π)=0 if π has strictly positive level. Define similarly p 0,G . We prove that p 0,G preserves the space of stable finite linear combinations of irreducible admissible representations of G (F) and that p 0,G transfert=transfertp 0,G .

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DOI : https://doi.org/10.5802/jep.145
Classification : 22E50
Mots clés : Représentations de niveau 0, transfert endoscopique
@article{JEP_2021__8__193_0,
     author = {Jean-Loup Waldspurger},
     title = {Repr\'esentations et quasi-caract\`eres de~niveau~$0$~; endoscopie},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {193--278},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.145},
     language = {fr},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.145/}
}
Jean-Loup Waldspurger. Représentations et quasi-caractères de niveau $0$ ; endoscopie. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 193-278. doi : 10.5802/jep.145. https://jep.centre-mersenne.org/articles/10.5802/jep.145/

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