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.

@article{ASENS_2018__51_4_927_0,
     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},
     number = {4},
     year = {2018},
     doi = {10.24033/asens.2369},
     mrnumber = {3861566},
     zbl = {1418.22009},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2369/}
}

TY  - JOUR
AU  - Loke, Hung Yean
AU  - Ma, Jia-Jun
TI  - Local theta correspondences between supercuspidal representations
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2018
SP  - 927
EP  - 991
VL  - 51
IS  - 4
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://archive.numdam.org/articles/10.24033/asens.2369/
DO  - 10.24033/asens.2369
LA  - en
ID  - ASENS_2018__51_4_927_0
ER  -

%0 Journal Article
%A Loke, Hung Yean
%A Ma, Jia-Jun
%T Local theta correspondences between supercuspidal representations
%J Annales scientifiques de l'École Normale Supérieure
%D 2018
%P 927-991
%V 51
%N 4
%I Société Mathématique de France. Tous droits réservés
%U http://archive.numdam.org/articles/10.24033/asens.2369/
%R 10.24033/asens.2369
%G en
%F ASENS_2018__51_4_927_0

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://archive.numdam.org/articles/10.24033/asens.2369/

Bibliographie
Cité par

Adler, J. D.; DeBacker, S. Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive $p$ -adic group, Michigan Math. J., Volume 50 (2002), pp. 263-286 | DOI | MR | Zbl

Aubert, A.-M. Conservation de la ramification modérée par la correspondance de Howe, Bull. Soc. Math. France, Volume 117 (1989), pp. 297-303 | DOI | Numdam | MR | Zbl

Bushnell, C. J.; Kutzko, P. C., 129, Princeton Univ. Press, 1993 | MR | Zbl

Broussous, P.; Stevens, S. Buildings of classical groups and centralizers of Lie algebra elements, Journal of Lie Theory, Volume 19 (2009), pp. 55-78 | MR | Zbl

Bruhat, F.; Tits, J. Schémas en groupes et immeubles des groupes classiques sur un corps local, II. Groupes unitaires, Bull. Math. Soc. France, Volume 115 (1987), pp. 141-195 | DOI | Numdam | MR | Zbl

DeBacker, S. Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of math., Volume 156 (2002), pp. 295-332 | DOI | MR | Zbl

Dieudonné, J., Springer, 1963 | MR | Zbl

DeBacker, S.; Reeder, M. Depth-zero supercuspidal $L$ -packets and their stability, Ann. of math., Volume 169 (2009), pp. 795-901 | DOI | MR | Zbl

Gérardin, P. Weil representations associated to finite fields, J. of Algebra, Volume 46 (1977), pp. 54-101 | DOI | MR | Zbl

Gan, W. T.; Kim, J.-L. Tame Types of Nonlinear Covering Groups (in preparation)

Hakim, J.; Murnaghan, F. Distinguished Tame Supercuspidal Representations, Int. Math. Res. Pap., Volume 2008 (2008) | MR | Zbl

Howe, R.; Moy, A., CBMS Regional Conference Series in Mathematics, 59, Amer. Math. Soc., 1985 | MR | Zbl

Howe, R. Tamely ramified supercuspidal representations of ${Gl}_{n}$ , Pacific J. Math., Volume 73 (1973), pp. 437-460 http://projecteuclid.org/euclid.pjm/1102810618 | DOI | MR | Zbl

Howe, R. $θ$ -series and invariant theory, Automorphic Forms, Representations and $L$ -functions (Proc. Symp. Pure Math), Volume 33 (1979), pp. 275-285 | DOI | MR | Zbl

Howe, R. Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (Piatetski-Shapiro, I. et al., eds.) (Isr. Math. Conf. Proc.), Volume 8, Ramat-Gan: Bar-Ilan University (1995), pp. 1-182 | MR | Zbl

Howard, T. K.; Weissman, M. H. Depth-Zero Representations of Nonlinear Covers of $p$ -Adic Groups, Int. Math. Res. Not., Volume 21 (2009), pp. 3979-3995 | DOI | MR | Zbl

Kim, J.-L. Supercuspidal Representations: An Exhaustion Theorem, J. Amer. Math. Soc., Volume 20 (2007), pp. pp. 273-320 | DOI | MR | Zbl

Kim, J.-L.; Murnaghan, F. Character Expansions and Unrefined Minimal $K$ -Types, Amer. J. Math., Volume 125 (2003), pp. pp. 1199-1234 | DOI | MR | Zbl

Kim, J.-L.; Murnaghan, F. $K$ -types and $Γ$ -asymptotic expansions, J. reine angew. Math., Volume 592 (2006), pp. 189-236 | DOI | MR | Zbl

Li, J.-S. Minimal representations and reductive dual pairs, Proceedings of Representation theory of Lie groups, Park City, UT, 1998 (IAS/Park City Math. Ser.), Volume 8, Amer. Math. Soc. (2000), pp. 293-340 | MR | Zbl

Li, J.-S. Singular unitary representations of classical groups, Invent. math., Volume 97 (1989), pp. 237-255 | DOI | MR | Zbl

Loke, H. Y.; Ma, J.-j.; Savin, G. Local theta correspondences between epipelagic supercuspidal representations, Math. Z., Volume 283, pp. 169-196 | DOI | MR | Zbl

Moy, A.; Prasad, G. Unrefined minimal K-types for p-adic groups, Invent. math., Volume 116 (1994), pp. 393-408 | DOI | MR | Zbl

Moy, A.; Prasad, G. Jacquet functors and unrefined minimal $K$ -types, Comment. Math. Helvetici, Volume 71 (1996) | MR | Zbl

Mœglin, C.; Vignéras, M.-F.; Waldspurger, J.-L., Lecture Notes in Math., 1291, Springer | MR | Zbl

Pan, S.-Y. Supercuspidal Representations and Theta Correspondence (preprint http://ir.lib.nthu.edu.tw/handle/987654321/52348 )

Pan, S.-Y. Splittings of the metaplectic covers of some reductive dual pairs, Pacific J. Math., Volume 199 (2001), pp. 163-226 | DOI | MR | Zbl

Pan, S.-Y. Depth preservation in local theta correspondence, Duke Math. J., Volume 113 (2002), pp. 531-592 | DOI | MR | Zbl

Pan, S.-Y. Local theta correspondence of depth zero representations and theta dichotomy, J. Math. Soc. Japan, Volume 54 (2002), pp. 793-845 | DOI | MR | Zbl

Pan, S.-Y. Local theta correspondence and minimal $K$ -types of positive depth, Israel J. Math., Volume 138 (2003), pp. 317-352 | DOI | MR | Zbl

Pan, S.-Y. Supercuspidal representations and preservation principle of theta correspondence, J. reine angew. Math. (2016) | DOI | MR | Zbl

Stevens, S. Intertwining and Supercuspidal Types for $p$ -adic Classical Groups, Proc. London Math. Soc., Volume 83 (2001), pp. 120-140 | DOI | MR | Zbl

Stevens, S. Semisimple characters for p-adic classical groups, Duke Math. J., Volume 127 (2005), pp. 123-173 | DOI | MR | Zbl

Stevens, S. The supercuspidal representations of p-adic classical groups, Invent. math., Volume 172 (2008), pp. 289-352 | DOI | MR | Zbl

Sun, B.; Zhu, C. Conservation relations for local theta correspondence, J. Amer. Math. Soc, Volume 28 (2015), pp. 939-983 | DOI | MR | Zbl

Waldspurger, J.-L. Démonstration d'une conjecture de dualité de Howe dans le cas $p$ -adique, $p \neq 2$ , Israel Math. Conf. Proc., Volume 2 (1990), pp. 267-324 | MR | Zbl

Yu, J.-K. Construction of tame supercuspidal representations, J. Amer. Math. Soc., Volume 14 (2001), pp. 579-622 | DOI | MR | Zbl

Yu, J.-K. Bruhat-Tits Theory and Buildings, Ottawa lectures on admissible representations of reductive $p$ -adic groups, Volume 26, Amer. Math. Soc. (2009) | MR | Zbl

Cité par Sources :