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ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF $p$-ADIC REDUCTIVE GROUPS

Part of: Representation theory of groups Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  13 January 2020

FLORIAN HERZIG ,
KAROL KOZIOŁ  and
MARIE-FRANCE VIGNÉRAS
FLORIAN HERZIG
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Room 6290, Toronto, ON M5S 2E4, Canada; [email protected]
KAROL KOZIOŁ
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI48109-1043, USA; [email protected]
MARIE-FRANCE VIGNÉRAS
Affiliation:
Institut de Mathématiques de Jussieu, 4 place Jussieu, 75005 Paris, France; [email protected]

Abstract

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Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_{p}$ and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 20C08: Hecke algebras and their representations
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e2
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Abdellatif, R., ‘Classification des représentations modulo p de SL(2, F)’, Bull. Soc. Math. France 142(3) (2014), 537–589.CrossRefGoogle Scholar
Abe, N., Henniart, G., Herzig, F. and Vignéras, M.-F., ‘Questions on mod $p$ representations of reductive $p$-adic groups’, unpublished note, available electronically at https://arxiv.org/abs/1703.02063.Google Scholar
Abe, N., Henniart, G., Herzig, F. and Vignéras, M.-F., ‘A classification of irreducible admissible mod p representations of p-adic reductive groups’, J. Amer. Math. Soc. 30(2) (2017), 495–559.CrossRefGoogle Scholar
Badulescu, A. I., ‘Jacquet–Langlands et unitarisabilité’, J. Inst. Math. Jussieu 6(3) (2007), 349–379.CrossRefGoogle Scholar
Badulescu, A. I., ‘Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations’, Invent. Math. 172(2) (2008), 383–438. With an appendix by Neven Grbac.CrossRefGoogle Scholar
Barnet-Lamb, T., Gee, T., Geraghty, D. and Taylor, R., ‘Local-global compatibility for l = p, II’, Ann. Sci. Éc. Norm. Supér. (4) 47(1) (2014), 165–179.CrossRefGoogle Scholar
Barnet-Lamb, T., Gee, T., Geraghty, D. and Taylor, R., ‘Potential automorphy and change of weight’, Ann. of Math. (2) 179(2) (2014), 501–609.CrossRefGoogle Scholar
Barthel, L. and Livné, R., ‘Irreducible modular representations of GL2 of a local field’, Duke Math. J. 75(2) (1994), 261–292.CrossRefGoogle Scholar
Bernstein, J. N., ‘Le ‘centre’ de Bernstein’, inRepresentations of Reductive Groups Over A Local Field, (ed. Deligne, P.) Travaux en Cours (Hermann, Paris, 1984), 1–32.Google Scholar
Beuzart-Plessis, R., ‘A short proof of the existence of supercuspidal representations for all reductive p-adic groups’, Pacific J. Math. 282(1) (2016), 27–34.CrossRefGoogle Scholar
Borel, A., ‘Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup’, Invent. Math. 35 (1976), 233–259.CrossRefGoogle Scholar
Borel, A., Linear Algebraic Groups, 2nd edn, Graduate Texts in Mathematics, 126 (Springer, New York, 1991).CrossRefGoogle Scholar
Borel, A. and Harder, G., ‘Existence of discrete cocompact subgroups of reductive groups over local fields’, J. Reine Angew. Math. 298 (1978), 53–64.Google Scholar
Borel, A. and Tits, J., ‘Groupes réductifs’, Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–150.CrossRefGoogle Scholar
Bourbaki, N., ‘Lie groups and Lie algebras. Chapters 4–6’, inElements of Mathematics (Berlin) (Springer, Berlin, 2002), Translated from the 1968 French original by Andrew Pressley.Google Scholar
Bourbaki, N., Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples (Springer, Berlin, 2012), Second revised edition of the 1958 edition.Google Scholar
Breuil, C., ‘Sur quelques représentations modulaires et p-adiques de GL2(Qp). I’, Compos. Math. 138(2) (2003), 165–188.CrossRefGoogle Scholar
Breuil, C. and Paškūnas, V., ‘Towards a modulo p Langlands correspondence for GL2’, Mem. Amer. Math. Soc. 216(1016) (2012), vi + 114.Google Scholar
Breuil, C. and Schneider, P., ‘First steps towards p-adic Langlands functoriality’, J. Reine Angew. Math. 610 (2007), 149–180.Google Scholar
Bushnell, C. J. and Henniart, G., ‘Explicit functorial correspondences for level zero representations of p-adic linear groups’, J. Number Theory 131(2) (2011), 309–331.CrossRefGoogle Scholar
Cabanes, M. and Enguehard, M., Representation Theory of Finite Reductive Groups, New Mathematical Monographs, 1 (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Casselman, B., ‘Introduction to the theory of admissible representations of $p$-adic reductive groups’, unpublished notes distributed by P. Sally, draft dated May 1, 1995, available electronically at https://www.math.ubc.ca/∼cass/research/publications.html.Google Scholar
Cheng, C., ‘Mod p representations of SL2(ℚp)’, J. Number Theory 133(4) (2013), 1312–1330.CrossRefGoogle Scholar
Clozel, L., ‘Motifs et formes automorphes: applications du principe de fonctorialité’, inAutomorphic Forms, Shimura Varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., 10 (Academic Press, Boston, MA, 1990), 77–159.Google Scholar
Clozel, L., ‘Représentations galoisiennes associées aux représentations automorphes autoduales de GL(n)’, Publ. Math. Inst. Hautes Études Sci. 73 (1991), 97–145.CrossRefGoogle Scholar
Clozel, L., Harris, M. and Taylor, R., ‘Automorphy for some l-adic lifts of automorphic mod l Galois representations’, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras.CrossRefGoogle Scholar
Conrad, B., Diamond, F. and Taylor, R., ‘Modularity of certain potentially Barsotti–Tate Galois representations’, J. Amer. Math. Soc. 12(2) (1999), 521–567.CrossRefGoogle Scholar
Deligne, P., Kazhdan, D. and Vignéras, M.-F., ‘Représentations des algèbres centrales simples p-adiques’, inRepresentations of Reductive Groups Over A Local Field, Travaux en Cours (Hermann, Paris, 1984), 33–117.Google Scholar
Emerton, M., Gee, T. and Herzig, F., ‘Weight cycling and Serre-type conjectures for unitary groups’, Duke Math. J. 162(9) (2013), 1649–1722.CrossRefGoogle Scholar
Gan, W. T. and Savin, G., ‘On minimal representations definitions and properties’, Represent. Theory 9 (2005), 46–93.CrossRefGoogle Scholar
Gee, T., Herzig, F. and Savitt, D., ‘General Serre weight conjectures’, J. Eur. Math. Soc. (JEMS) 20(12) (2018), 2859–2949.CrossRefGoogle Scholar
Harris, M. and Taylor, R., The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, 151 (Princeton University Press, Princeton, NJ, 2001), With an appendix by Vladimir G. Berkovich.Google Scholar
Henniart, G., ‘La conjecture de Langlands locale pour GL(3)’, Mém. Soc. Math. France (N.S.) 11–12 (1983), 186 pages.Google Scholar
Henniart, G., ‘Sur les représentations modulo p de groupes réductifs p-adiques’, inAutomorphic Forms and L-functions II. Local Aspects, Contemp. Math., 489 (American Mathematical Society, Providence, RI, 2009), 41–55.Google Scholar
Henniart, G., ‘Induction automorphe globale pour les corps de nombres’, Bull. Soc. Math. France 140(1) (2012), 1–17.CrossRefGoogle Scholar
Henniart, G. and Vigneras, M.-F., ‘Comparison of compact induction with parabolic induction’, Pacific J. Math. 260(2) (2012), 457–495.CrossRefGoogle Scholar
Henniart, G. and Vignéras, M.-F., ‘A Satake isomorphism for representations modulo p of reductive groups over local fields’, J. Reine Angew. Math. 701 (2015), 33–75.Google Scholar
Henniart, G. and Vignéras, M.-F., ‘Representations of a p-adic group in characteristic p’, inRepresentations of Reductive Groups, Proc. Sympos. Pure Math., 101 (American Mathematical Society, Providence, RI, 2019), 171–210.Google Scholar
Herzig, F., ‘The classification of irreducible admissible mod p representations of a p-adic GLn’, Invent. Math. 186(2) (2011), 373–434.CrossRefGoogle Scholar
Hu, Y., ‘Sur quelques représentations supersingulières de GL2(ℚp f)’, J. Algebra 324(7) (2010), 1577–1615.CrossRefGoogle Scholar
Kazhdan, D., ‘Cuspidal geometry of p-adic groups’, J. Analyse Math. 47 (1986), 1–36.CrossRefGoogle Scholar
Kottwitz, R. E., ‘Sign changes in harmonic analysis on reductive groups’, Trans. Amer. Math. Soc. 278(1) (1983), 289–297.CrossRefGoogle Scholar
Kottwitz, R. E., ‘Tamagawa numbers’, Ann. of Math. (2) 127(3) (1988), 629–646.CrossRefGoogle Scholar
Kozioł, K., ‘A classification of the irreducible mod-p representations of U(1, 1)(ℚp 2/ℚp)’, Ann. Inst. Fourier (Grenoble) 66(4) (2016), 1545–1582.CrossRefGoogle Scholar
Kozioł, K. and Xu, P., ‘Hecke modules and supersingular representations of U(2, 1)’, Represent. Theory 19 (2015), 56–93.CrossRefGoogle Scholar
Labesse, J.-P., ‘Changement de base CM et séries discrètes’, inOn the Stabilization of the Trace Formula, Stab. Trace Formula Shimura Var. Arith. Appl., 1 (Int. Press, Somerville, MA, 2011), 429–470.Google Scholar
Le, D., ‘On some nonadmissible smooth irreducible representations for GL2’, Math. Res. Lett. (to appear).Google Scholar
Lusztig, G., ‘Some examples of square integrable representations of semisimple p-adic groups’, Trans. Amer. Math. Soc. 277(2) (1983), 623–653.Google Scholar
Margulis, G. A., Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17 (Springer, Berlin, 1991).CrossRefGoogle Scholar
Morris, L., ‘Level zero G -types’, Compos. Math. 118(2) (1999), 135–157.CrossRefGoogle Scholar
Ollivier, R. and Vignéras, M.-F., ‘Parabolic induction in characteristic p’, Selecta Math. (N.S.) 24(5) (2018), 3973–4039.CrossRefGoogle Scholar
Paškūnas, V., ‘Coefficient systems and supersingular representations of GL2(F)’, Mém. Soc. Math. Fr. (N.S.) (99) (2004), vi + 84 pages.Google Scholar
Platonov, V. and Rapinchuk, A., Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139 (Academic Press, Boston, MA, 1994), Translated from the 1991 Russian original by Rachel Rowen.Google Scholar
Rogawski, J. D., Automorphic Representations of Unitary Groups in Three Variables, Annals of Mathematics Studies, 123 (Princeton University Press, Princeton, NJ, 1990).CrossRefGoogle Scholar
Serre, J.-P., Trees, Springer Monographs in Mathematics (Springer, Berlin, 2003), Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation.Google Scholar
Tits, J., ‘Reductive groups over local fields’, inAutomorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. 1977), Part 1, Proc. Sympos. Pure Math., XXXIII (American Mathematical Society, Providence, RI, 1979), 29–69.Google Scholar
Vignéras, M.-F., Représentations l-modulaires d’un groupe réductif p-adique avec l≠p, Progress in Mathematics, 137 (Birkhäuser Boston, Inc., Boston, MA, 1996).Google Scholar
Vignéras, M.-F., ‘Pro-p-Iwahori Hecke ring and supersingular Fp -representations’, Math. Ann. 331(3) (2005), 523–556.CrossRefGoogle Scholar
Vignéras, M.-F., ‘Représentations irréductibles de GL(2, F) modulo p’, inL-functions and Galois Representations, London Math. Soc. Lecture Note Ser., 320 (Cambridge University Press, Cambridge, 2007), 548–563.CrossRefGoogle Scholar
Vignéras, M.-F., ‘The pro-p-Iwahori-Hecke algebra of a reductive p-adic group, II’, Münster J. Math. 7(1) (2014), 363–379.Google Scholar
Vignéras, M.-F., ‘The pro-p-Iwahori Hecke algebra of a reductive p-adic group I’, Compos. Math. 152(4) (2016), 693–753.CrossRefGoogle Scholar
Vignéras, M.-F., ‘The pro-p-Iwahori Hecke algebra of a reductive p-adic group III (spherical Hecke algebras and supersingular modules)’, J. Inst. Math. Jussieu 16(3) (2017), 571–608.CrossRefGoogle Scholar