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Representations of Reductive p-adic Groups pp 175–190Cite as

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On the Unicity of Types for Toral Supercuspidal Representations

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Part of the book series: Progress in Mathematics ((PM,volume 328))

Abstract

For tame arbitrary-length toral, also called positive regular, supercuspidal representations of a simply connected and semisimple p-adic group G, constructed as per Adler-Yu, we determine which components of their restriction to a maximal compact subgroup are types. We give conditions under which there is a unique such component and then present a class of examples for which there is not, disproving the strong version of the conjecture of unicity of types on maximal compact open subgroups. We restate the unicity conjecture and prove it holds for the groups and representations under consideration under a mild condition on depth.

The second author’s research is supported by NSERC Canada Discovery Grant 06294.

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References

  1. J.D. Adler, Refined anisotropic \(K\)-types and supercuspidal representations. Pacific J. Math. 185(1), 1–32 (1998)

    Article  MathSciNet  Google Scholar 

  2. J. Bernstein, Le “centre” de Bernstein, Representations of reductive groups over a local field, ed. by P. Deligne (Travaux en Cours, Hermann, Paris, 1984), pp. 1–32

    Google Scholar 

  3. C. Breuil, A. Mézard, Multiplicités modulaires et représentations de \({\rm GL}_2({\bf Z}_p)\) et de \({\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)\) en \(l=p\), Duke Math. J. 115(2), 205–310 (2002). With an appendix by Guy Henniart

    Google Scholar 

  4. F. Bruhat, J. Tits, Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)

    Article  Google Scholar 

  5. F. Bruhat, J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Inst. Hautes Études Sci. Publ. Math. 60, 197–376 (1984)

    Google Scholar 

  6. J. Fintzen, On the Moy-Prasad filtration. Preprint arXiv:1511.00726v3 [math.RT] (2017)

  7. J. Fintzen, Types for tame \(p\)-adic groups. Preprint arXiv:1810.04198 [math.TR] (2018)

  8. J. Hakim, F. Murnaghan, Distinguished tame supercuspidal representations. Int. Math. Res. Pap. IMRP 2, Art. ID rpn005, 166 (2008)

    Google Scholar 

  9. F. Hurst, Primitive Tori in einfachen spaltenden Gruppen und ihre Fixpunkte im Bruhat-Tits Gebäude, Inaugural-Dissertation zur Erlangung der Doktorwürde der Naturwissenschaftlich-Mathematischen Gesamtfakultät der Ruprecht-Karls-Universität Heidelberg (2005), 164pp

    Google Scholar 

  10. T. Kaletha, Regular supercuspidal representations, Preprint arXiv:1602.03144v2 (2016)

  11. J.-L. Kim, Supercuspidal representations: an exhaustion theorem. J. Amer. Math. Soc. 20(2), 273–320 (electronic) (2007)

    Google Scholar 

  12. P. Latham, Unicity of types for supercuspidal representations of \(p\)-adic \(\rm SL_2\). J. Number Theor. 162, 376–390 (2016)

    Article  MathSciNet  Google Scholar 

  13. P. Latham, The unicity of types for depth-zero supercuspidal representations. Represent. Theor. 21, 590–610 (2017)

    Article  MathSciNet  Google Scholar 

  14. P. Latham, On the unicity of types in special linear groups. Manuscripta Math. 157(3–4), 445–465 (2018)

    Article  MathSciNet  Google Scholar 

  15. L. Morris, Some tamely ramified supercuspidal representations of symplectic groups. Proc. London Math. Soc. 63(3), 519–551 (1991)

    Article  MathSciNet  Google Scholar 

  16. A. Moy, G. Prasad, Unrefined minimal \(K\)-types for \(p\)-adic groups. Invent. Math. 116(1–3), 393–408 (1994)

    Article  MathSciNet  Google Scholar 

  17. F. Murnaghan, Parametrization of tame supercuspidal representations, in On certain\(L\)-functions. Clay Mathematical Proceedings 13, American Mathematical Society, Providence, RI, 2011, pp. 439–469

    Google Scholar 

  18. F. Murnaghan, Distinguished positive regular representations. Bull. Iranian Math. Soc. 43(4), 291–311 (2017)

    MathSciNet  MATH  Google Scholar 

  19. M. Nevins, Branching rules for supercuspidal representations of \(SL_2(k)\), for \(k\) a \(p\)-adic field. J. Algebra 377, 204–231 (2013)

    Article  MathSciNet  Google Scholar 

  20. M. Nevins, On branching rules of depth-zero representations. J. Algebra 408, 1–27 (2014)

    Article  MathSciNet  Google Scholar 

  21. V. Paskunas, Unicity of types for supercuspidal representations of \({\text{ GL }}_N\). Proc. Lond. Math. Soc. 91(3), 623–654 (2005)

    Article  MathSciNet  Google Scholar 

  22. T.A. Springer, Reductive groups, automorphic forms, representations and\(L\)-functions, in (Proceedings Symposium Pure Mathematical, Oregon State University, Corvallis, Ore, 1977), Part 1 (XXXIII, American Mathematical Society, Providence, R.I, Proceeding Symposium Pure Mathematical, 1979), pp. 3–27

    Google Scholar 

  23. J. Tits, Reductive groups over local fields, automorphic forms, representations and \(L\)-functions (Oregon State University, Corvallis, Ore, 1977), Part 1, Proceedings Symposium Pure Mathematical, XXXIII, American Mathematical Society, Providence, R.I., 1979, pp. 29–69

    Google Scholar 

  24. J.-K. Yu, Construction of tame supercuspidal representations. J. Amer. Math. Soc. 14(3), 579–622 (2001) (electronic)

    Google Scholar 

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Acknowledgements

The second author warmly thanks Anne-Marie Aubert, Manish Mishra, Alan Roche and Steven Spallone for the invitation to the excellent conference Representation theory of p-adic groups at IISER Pune, India. The stimulating environment of the workshop contributed significantly to this article; in particular, fellow participant Jeff Adler provided invaluable insight into tori and buildings, and he offered up the torus of Example 8.2.

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Authors and Affiliations

  1. Department of Mathematics, King’s College London, Strand, London, UK

    Peter Latham

  2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada

    Monica Nevins

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  1. Peter Latham
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  2. Monica Nevins
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Correspondence to Monica Nevins .

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Editors and Affiliations

  1. Institut de Mathématiques de Jussieu, Paris, France

    Anne-Marie Aubert

  2. Department of Mathematics, Indian Institute of Science Education and Research Pune, Pune, Maharashtra, India

    Manish Mishra

  3. Department of Mathematics, University of Oklahoma, Norman, OK, USA

    Alan Roche

  4. Department of Mathematics, Indian Institute of Science Education and Research Pune, Pune, Maharashtra, India

    Steven Spallone

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Latham, P., Nevins, M. (2019). On the Unicity of Types for Toral Supercuspidal Representations. In: Aubert, AM., Mishra, M., Roche, A., Spallone, S. (eds) Representations of Reductive p-adic Groups. Progress in Mathematics, vol 328. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-6628-4_6

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