Supercuspidal unipotent representations: L-packets and formal degrees

Yongqi Feng; Eric Opdam; Maarten Solleveld

doi:10.5802/jep.138

Supercuspidal unipotent representations: L-packets and formal degrees
[Représentations unipotentes supercuspidales : L-paquets et degrés formels]

Yongqi Feng ¹ ; Eric Opdam ² ; Maarten Solleveld ³

¹ Department of Mathematics, Shantou University Daxue Road 243, 515063 Shantou, China
² Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam Science Park 105-107, 1098 XG Amsterdam, The Netherlands
³ Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit Nijmegen Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1133-1193.

Résumé
Abstract

Soit $K$ un corps local non archimédien et soit $G$ un $K$ -groupe connexe, réductif et deployé sur une extension non ramifiée de $K$ . Nous étudions des représentations unipotentes supercuspidales du groupe $G (K)$ . Nous établissons une bijection entre l’ensemble de telles $G (K)$ -représentations irréductibles et l’ensemble des L-paramètres étendus pour $G (K)$ , qui sont triviaux sur le sous-groupe d’inertie du groupe de Weil de $K$ . Le bijection est caractérisée par quelques propriétés simples et une comparaison des degrés formels des représentations avec des facteurs $γ$ adjoints des L-paramètres.

On peut considérer cela comme une correspondance de Langlands locale pour toutes les représentations unipotentes supercuspidales. Nous comptons les L-paquets résultants en termes de données déduites du diagramme de Dynkin affine de $G$ . Finalement, nous prouvons que notre bijection satisfait à la conjecture de Hiraga, Ichino et Ikeda sur les degrés formels des représentations.

Let $K$ be a non-archimedean local field and let $G$ be a connected reductive $K$ -group which splits over an unramified extension of $K$ . We investigate supercuspidal unipotent representations of the group $G (K)$ . We establish a bijection between the set of irreducible $G (K)$ -representations of this kind and the set of cuspidal enhanced L-parameters for $G (K)$ , which are trivial on the inertia subgroup of the Weil group of $K$ . The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $γ$ -factors of L-parameters.

This can be regarded as a local Langlands correspondence for all supercuspidal unipotent representations. We count the ensuing L-packets, in terms of data from the affine Dynkin diagram of $G$ . Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.

Reçu le : 2019-11-20
Accepté le : 2020-10-03
Publié le : 2020-10-27

DOI : 10.5802/jep.138

Classification : 22E50, 11S37, 20G25, 43A99
Keywords: Reductive

p

-adic group, cuspidal representation, formal degree, local Langlands correspondence
Mots-clés : Groupes réductifs

p

-adiques, représentation cuspidale, degré formel, correspondance de Langlands locale

Affiliations des auteurs :

Yongqi Feng ¹ ; Eric Opdam ² ; Maarten Solleveld ³

¹ Department of Mathematics, Shantou University Daxue Road 243, 515063 Shantou, China
² Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam Science Park 105-107, 1098 XG Amsterdam, The Netherlands
³ Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit Nijmegen Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2020__7__1133_0,
     author = {Yongqi Feng and Eric Opdam and Maarten Solleveld},
     title = {Supercuspidal unipotent representations: {L-packets} and formal degrees},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1133--1193},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.138},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.138/}
}

TY  - JOUR
AU  - Yongqi Feng
AU  - Eric Opdam
AU  - Maarten Solleveld
TI  - Supercuspidal unipotent representations: L-packets and formal degrees
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2020
SP  - 1133
EP  - 1193
VL  - 7
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.138/
DO  - 10.5802/jep.138
LA  - en
ID  - JEP_2020__7__1133_0
ER  -

%0 Journal Article
%A Yongqi Feng
%A Eric Opdam
%A Maarten Solleveld
%T Supercuspidal unipotent representations: L-packets and formal degrees
%J Journal de l’École polytechnique — Mathématiques
%D 2020
%P 1133-1193
%V 7
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.138/
%R 10.5802/jep.138
%G en
%F JEP_2020__7__1133_0

Yongqi Feng; Eric Opdam; Maarten Solleveld. Supercuspidal unipotent representations: L-packets and formal degrees. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1133-1193. doi : 10.5802/jep.138. https://jep.centre-mersenne.org/articles/10.5802/jep.138/

Bibliographie
Cité par

[ABPS17] A.-M. Aubert, P. Baum, R. Plymen & M. Solleveld - “Conjectures about $p$ -adic groups and their noncommutative geometry”, in Around Langlands correspondences, Contemp. Math., vol. 691, American Mathematical Society, Providence, RI, 2017, p. 15-51 | DOI | MR | Zbl

[AMS18] A.-M. Aubert, A. Moussaoui & M. Solleveld - “Generalizations of the Springer correspondence and cuspidal Langlands parameters”, Manuscripta Math. 157 (2018) no. 1-2, p. 121-192 | DOI | MR | Zbl

[Bor79] A. Borel - “Automorphic $L$ -functions”, in Automorphic forms, representations and $L$ -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, American Mathematical Society, Providence, RI, 1979, p. 27-61 | Zbl

[DR09] S. DeBacker & M. Reeder - “Depth-zero supercuspidal $L$ -packets and their stability”, Ann. of Math. (2) 169 (2009) no. 3, p. 795-901 | DOI | Zbl

[Fen19] Y. Feng - “A note on the spectral transfer morphism for affine Hecke algebras”, J. Lie Theory 29 (2019) no. 4, p. 901-926 | MR | Zbl

[FO20] Y. Feng & E. Opdam - “On a uniqueness property of cuspidal unipotent representations”, Adv. Math. (2020), to appear | DOI

[GG99] B. H. Gross & W. T. Gan - “Haar measure and the Artin conductor”, Trans. Amer. Math. Soc. 351 (1999) no. 4, p. 1691-1704 | DOI | MR | Zbl

[GM16] M. Geck & G. Malle - “Reductive groups and the Steinberg map”, 2016 | arXiv | Zbl

[GR10] B. H. Gross & M. Reeder - “Arithmetic invariants of discrete Langlands parameters”, Duke Math. J. 154 (2010) no. 3, p. 431-508 | DOI | MR | Zbl

[Gro97] B. H. Gross - “On the motive of a reductive group”, Invent. Math. 130 (1997) no. 2, p. 287-313 | DOI | MR | Zbl

[Hai14] T. J. Haines - “The stable Bernstein center and test functions for Shimura varieties”, in Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, p. 118-186 | DOI | MR | Zbl

[HII08] K. Hiraga, A. Ichino & T. Ikeda - “Formal degrees and adjoint $γ$ -factors”, J. Amer. Math. Soc. 21 (2008) no. 1, p. 283-304, Correction: Ibid., p. 1211–1213 | DOI | MR | Zbl

[Kac90] V. G. Kac - Infinite-dimensional Lie algebras, Cambridge Univ. Press, Cambridge, 1990 | DOI | Zbl

[Kot84] R. E. Kottwitz - “Stable trace formula: cuspidal tempered terms”, Duke Math. J. 51 (1984) no. 3, p. 611-650 | DOI | MR | Zbl

[Kot97] R. E. Kottwitz - “Isocrystals with additional structure. II”, Compositio Math. 109 (1997) no. 3, p. 255-339 | DOI | MR | Zbl

[Lus78] G. Lusztig - Representations of finite Chevalley groups, CBMS Regional Conf. Series in Math., vol. 39, American Mathematical Society, Providence, RI, 1978 | MR | Zbl

[Lus84] G. Lusztig - “Intersection cohomology complexes on a reductive group”, Invent. Math. 75 (1984) no. 2, p. 205-272 | DOI | MR | Zbl

[Lus95] G. Lusztig - “Classification of unipotent representations of simple $p$ -adic groups”, Internat. Math. Res. Notices (1995) no. 11, p. 517-589 | DOI | MR | Zbl

[Lus02] G. Lusztig - “Classification of unipotent representations of simple $p$ -adic groups. II”, Represent. Theory 6 (2002), p. 243-289 | DOI | MR | Zbl

[Mor96] L. Morris - “Tamely ramified supercuspidal representations”, Ann. Sci. École Norm. Sup. (4) 29 (1996) no. 5, p. 639-667 | DOI | Numdam | MR | Zbl

[Opd16] E. Opdam - “Spectral transfer morphisms for unipotent affine Hecke algebras”, Selecta Math. (N.S.) 22 (2016) no. 4, p. 2143-2207 | DOI | MR | Zbl

[PR08] G. Pappas & M. Rapoport - “Twisted loop groups and their affine flag varieties”, Adv. Math. 219 (2008) no. 1, p. 118-198, with an appendix by T. Haines and M. Rapoport | DOI | MR | Zbl

[Pra82] G. Prasad - “Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits”, Bull. Soc. math. France 110 (1982) no. 2, p. 197-202 | DOI | Numdam | MR | Zbl

[Ree94] M. Reeder - “On the Iwahori-spherical discrete series for $p$ -adic Chevalley groups; formal degrees and $L$ -packets”, Ann. Sci. École Norm. Sup. (4) 27 (1994) no. 4, p. 463-491 | DOI | Numdam | MR | Zbl

[Ree00] M. Reeder - “Formal degrees and $L$ -packets of unipotent discrete series representations of exceptional $p$ -adic groups”, J. reine angew. Math. 520 (2000), p. 37-93, with an appendix by F. Lübeck | DOI | MR | Zbl

[Ree10] M. Reeder - “Torsion automorphisms of simple Lie algebras”, Enseign. Math. (2) 56 (2010) no. 1-2, p. 3-47 | DOI | MR | Zbl

[Ser79] J.-P. Serre - Local fields, Graduate Texts in Math., vol. 67, Springer-Verlag, New York-Berlin, 1979 | Zbl

[Sol18] M. Solleveld - “A local Langlands correspondence for unipotent representations”, 2018 | arXiv | Zbl

[Spr09] T. A. Springer - Linear algebraic groups, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009 | Zbl

[Ste68] R. Steinberg - Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., vol. 80, American Mathematical Society, Providence, RI, 1968 | MR | Zbl

[Tat79] J. Tate - “Number theoretic background”, in Automorphic forms, representations and $L$ -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, American Mathematical Society, Providence, RI, 1979, p. 3-26 | Zbl

[Thǎ11] N. Q. Thǎńg - “On Galois cohomology and weak approximation of connected reductive groups over fields of positive characteristic”, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011) no. 10, p. 203-208 | DOI | MR | Zbl

Cité par Sources :