Supercuspidal unipotent representations: L-packets and formal degrees

Feng, Yongqi; Opdam, Eric; Solleveld, Maarten

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

Soit $K$ un corps local non archimédien et soit $\mathbf{G}$ un $K$-groupe connexe, réductif et deployé sur une extension non ramifiée de $K$. Nous étudions des représentations unipotentes supercuspidales du groupe $\mathbf{G}\left(K\right)$. Nous établissons une bijection entre l’ensemble de telles $\mathbf{G}\left(K\right)$-représentations irréductibles et l’ensemble des L-paramètres étendus pour $\mathbf{G}\left(K\right)$, 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 $\gamma$ 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 $\mathbf{G}$. 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 $\mathbf{G}$ be a connected reductive $K$-group which splits over an unramified extension of $K$. We investigate supercuspidal unipotent representations of the group $\mathbf{G}\left(K\right)$. We establish a bijection between the set of irreducible $\mathbf{G}\left(K\right)$-representations of this kind and the set of cuspidal enhanced L-parameters for $\mathbf{G}\left(K\right)$, 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 $\gamma$-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 $\mathbf{G}$. Finally, we prove that our bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal degrees of the representations.

Classification : 22E50, 11S37, 20G25, 43A99
Keywords: Reductive $p$-adic group, cuspidal representation, formal degree, local Langlands correspondence
Mot clés : Groupes réductifs $p$-adiques, représentation cuspidale, degré formel, correspondance de Langlands locale

[ABPS17] Aubert, Anne-Marie; Baum, Paul; Plymen, Roger; Solleveld, Maarten Conjectures about $p$-adic groups and their noncommutative geometry, Around Langlands correspondences (Contemp. Math.), Volume 691, American Mathematical Society, Providence, RI, 2017, pp. 15-51 | DOI | MR | Zbl

[AMS18] Aubert, Anne-Marie; Moussaoui, Ahmed; Solleveld, Maarten Generalizations of the Springer correspondence and cuspidal Langlands parameters, Manuscripta Math., Volume 157 (2018) no. 1-2, pp. 121-192 | DOI | MR | Zbl

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

[DR09] DeBacker, Stephen; Reeder, Mark Depth-zero supercuspidal $L$-packets and their stability, Ann. of Math. (2), Volume 169 (2009) no. 3, pp. 795-901 | DOI | Zbl

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

[FO20] Feng, Yongqi; Opdam, Eric On a uniqueness property of cuspidal unipotent representations, Adv. Math. (2020) (to appear) | DOI

[GG99] Gross, Benedict H.; Gan, Wee Teck Haar measure and the Artin conductor, Trans. Amer. Math. Soc., Volume 351 (1999) no. 4, pp. 1691-1704 | DOI | MR | Zbl

[GM16] Geck, Meinolf; Malle, Gunter Reductive groups and the Steinberg map, 2016 | arXiv

[GR10] Gross, Benedict H.; Reeder, Mark Arithmetic invariants of discrete Langlands parameters, Duke Math. J., Volume 154 (2010) no. 3, pp. 431-508 | DOI | MR | Zbl

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

[Hai14] Haines, Thomas J. The stable Bernstein center and test functions for Shimura varieties, Automorphic forms and Galois representations. Vol. 2 (London Math. Soc. Lecture Note Ser.), Volume 415, Cambridge Univ. Press, Cambridge, 2014, pp. 118-186 | DOI | MR | Zbl

[HII08] Hiraga, Kaoru; Ichino, Atsushi; Ikeda, Tamotsu Formal degrees and adjoint $\gamma$-factors, J. Amer. Math. Soc., Volume 21 (2008) no. 1, pp. 283-304 (Correction: Ibid., p. 1211–1213) | DOI | MR | Zbl

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

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

[Kot97] Kottwitz, Robert E. Isocrystals with additional structure. II, Compositio Math., Volume 109 (1997) no. 3, pp. 255-339 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

[Sol18] Solleveld, Maarten A local Langlands correspondence for unipotent representations, 2018 | arXiv

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

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

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

[Thǎ11] Thǎńg, Nguyêñ Quôć On Galois cohomology and weak approximation of connected reductive groups over fields of positive characteristic, Proc. Japan Acad. Ser. A Math. Sci., Volume 87 (2011) no. 10, pp. 203-208 http://projecteuclid.org/euclid.pja/1322748851 | DOI | MR | Zbl