Local theta correspondences between supercuspidal representations

Loke, Hung Yean; Ma, Jia-Jun

doi:10.24033/asens.2369

Local theta correspondences between supercuspidal representations
[Correspondances thêta locales entre les représentations supercuspidales]

Loke, Hung Yean ; Ma, Jia-Jun

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 927-991.

Résumé
Abstract

Par les travaux de Yu, Kim et Hakim-Murnaghan, on a une paramétrisation et une construction de toutes les représentations supercuspidales d'un groupe réductif $p$ -adique en termes de données supercuspidales, quand $p$ est suffisamment grand. Dans cet article, nous définirons une correspondance entre les données supercuspidales par l'intermédiaire d'applications moments et de correspondances thêta sur des corps finis. Ensuite, nous montrerons que les correspondances thêta locales entre les représentations supercuspidales sont complètement décrites par cette notion. Dans l'Appendice B, nous fournissons une courte démonstration d'un résultat de Pan sur la « préservation de la profondeur ».

By the works of Yu, Kim and Hakim-Murnaghan, we have a parameterization and construction of all supercuspidal representations of a reductive $p$ -adic group in terms of supercuspidal data, when $p$ is sufficiently large. In this paper, we will define a correspondence of supercuspidal data via moment maps and theta correspondences over finite fields. Then we will show that local theta correspondences between supercuspidal representations are completely described by this notion. In Appendix B, we give a short proof of a result of Pan on “depth preservation”.

MR Zbl

DOI : 10.24033/asens.2369

Classification : 22E46, 22E47
Keywords: Local theta correspondence, moment maps, supercuspidal representations.
Mot clés : Correspondances thêta locales, application moment, représentations supercuspidales.

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     author = {Loke, Hung Yean and Ma, Jia-Jun},
     title = {Local theta correspondences between supercuspidal representations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {927--991},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
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     year = {2018},
     doi = {10.24033/asens.2369},
     mrnumber = {3861566},
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Loke, Hung Yean; Ma, Jia-Jun. Local theta correspondences between supercuspidal representations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 927-991. doi : 10.24033/asens.2369. http://www.numdam.org/articles/10.24033/asens.2369/

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