Commutative character sheaves and geometric types for supercuspidal representations
Annales Henri Lebesgue, Volume 4 (2021), pp. 1389-1420.

Metadata

Keywordsfunction-sheaf dictionary, commutative character sheaves, types for supercuspidal representations

Abstract

We show that some types for supercuspidal representations of tamely ramified p-adic groups that appear in Jiu-Kang Yu’s work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu, assuming that the character vanishes on a certain derived subgroup. To define geometric types, we combine commutative character sheaves with Gurevich and Hadani’s geometrization of the Weil representation and Lusztig’s character sheaves.


References

[Adl98] Adler, Jeffrey D. Refined anisotropic K-types and supercuspidal representations, Pac. J. Math., Volume 185 (1998) no. 1, pp. 1-32 | DOI | MR | Zbl

[Ber70] Bertin, Jean-Étienne Exposé VI B  : Généralités sur les préschémas en groupes, Propriétés générales des schémas en groupes. Séminaire de Géométrie Algébrique du Bois Marie, Schémas en groupes I (SGA3) (Lecture Notes in Mathematics), Volume 151 (1970), pp. 318-410 | DOI

[BGA18] Bertapelle, Alessandra; González-Avilés, Cristian D. The Greenberg functor revisited, Eur. J. Math., Volume 4 (2018) no. 4, pp. 1340-1389 | DOI | MR | Zbl

[BK98] Bushnell, Colin J.; Kutzko, Philip C. Smooth representations of reductive p-adic groups: structure theory via types, Proc. Lond. Math. Soc., Volume 77 (1998) no. 3, pp. 582-634 | DOI | MR | Zbl

[BL94] Bernstein, Joseph; Lunts, Valery Equivariant sheaves and functors, Lecture Notes in Mathematics, 1578, Springer, 1994 | MR | Zbl

[CR18] Cunningham, Clifton; Roe, David From the function-sheaf dictionary to quasicharacters of p-adic tori, J. Inst. Math. Jussieu, Volume 17 (2018) no. 1, pp. 1-37 | DOI | MR | Zbl

[Del77] Deligne, Pierre Cohomologie étale. Séminaire de géométrie algébrique du Bois-Marie SGA 4 1/2 par P. Deligne, avec la collaboration de J.F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, Lecture Notes in Mathematics, 569, Springer, 1977 | Zbl

[Del80] Deligne, Pierre La conjecture de Weil. II, Publ. Math., Inst. Hautes Étud. Sci. (1980) no. 52, pp. 137-252 | DOI | Numdam | MR | Zbl

[FGI + 05] Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; L., Kleimand Steven; Nitsure, Nitin; Vistoli, Angelo Fundamental algebraic geometry, Mathematical Surveys and Monographs, 123, American Mathematical Society, 2005 | MR | Zbl

[Fin18] Fintzen, Jessica Types for tame p-adic groups (2018) (http://arxiv.org/abs/1810.04198) | Zbl

[Fin19] Fintzen, Jessica On the construction of tame supercuspidal representations (2019) (http://arxiv.org/abs/1908.09819)

[Gab70] Gabriel, Pierre Exposé VI A : Généralités sur les groupes algébriques, Propriétés générales des schémas en groupes. Séminaire de Géométrie Algébrique du Bois Marie, Schémas en groupes I (SGA3) (Lecture Notes in Mathematics), Volume 151 (1970), pp. 287-317 | DOI

[GH07] Gurevich, Shamgar; Hadani, Ronny The geometric Weil representation, Sel. Math., New Ser., Volume 13 (2007) no. 3, pp. 465-481 | DOI | MR | Zbl

[GH11] Gurevich, Shamgar; Hadani, Ronny The categorical Weil representation (2011) (http://arxiv.org/abs/1108.0351)

[Gro67] Grothendieck, Alexandre Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas. IV, Publ. Math., Inst. Hautes Étud. Sci. (1967) no. 32, pp. 1-361 (rédigé avec la collaboration de Jean Dieudonné) | Numdam | Zbl

[Hak18] Hakim, Jeffrey Constructing tame supercuspidal representations, Represent. Theory, Volume 22 (2018), pp. 45-86 | DOI | MR | Zbl

[Kal19] Kaletha, Tasho Regular supercuspidal representations, J. Am. Math. Soc., Volume 32 (2019), pp. 1071-1170 | DOI | MR | Zbl

[Kam09] Kamgarpour, Masoud Stacky abelianization of algebraic groups, Transform. Groups, Volume 14 (2009) no. 4, pp. 825-846 | DOI | MR | Zbl

[Kim07] Kim, Ju-Lee Supercuspidal representations: an exhaustion theorem, J. Am. Math. Soc., Volume 20 (2007) no. 2, pp. 273-320 | MR | Zbl

[Lus85] Lusztig, George Character sheaves. I, Adv. Math., Volume 56 (1985), pp. 193-237 | DOI | MR | Zbl

[Lus86] Lusztig, George Character sheaves. V, Adv. Math., Volume 61 (1986) no. 2, pp. 103-155 | DOI | MR | Zbl

[Lus04] Lusztig, George Character sheaves on disconnected groups. IV, Represent. Theory, Volume 8 (2004), pp. 145-178 | DOI | MR | Zbl

[Ree08] Reeder, Mark Supercuspidal L-packets of positive depth and twisted Coxeter elements, J. Reine Angew. Math., Volume 620 (2008), pp. 1-33 | DOI | MR | Zbl

[RY14] Reeder, Mark; Yu, Jiu-Kang Epipelagic representations and invariant theory, J. Am. Math. Soc., Volume 27 (2014) no. 2, pp. 437-477 | DOI | MR | Zbl

[Yu01] Yu, Jiu-Kang Construction of tame supercuspidal representations, J. Am. Math. Soc., Volume 14 (2001) no. 3, pp. 579-622 | MR | Zbl

[Yu15] Yu, Jiu-Kang Smooth models associated to concave functions in Bruhat–Tits theory, Autour des schémas en groupes – École d’Été franco-asiatique de géométrie algébrique et de théorie des nombres. Volume III (Panoramas et Synthèses), Volume 47 (2015), pp. 227-258 (http://smf4.emath.fr/Publications/PanoramasSyntheses/2016/47/html/smf_pano-synth_47_227-258.php) | MR | Zbl