Skip to main content
SpringerLink
Log in
Book cover

Arbeitstagung Bonn 2013 pp 17–41Cite as

Birkhäuser

Depth and the Local Langlands Correspondence

  • Chapter
  • First Online:

Part of the book series: Progress in Mathematics ((PM,volume 319))

Abstract

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. A.-M. Aubert, P. Baum, R.J. Plymen, M. Solleveld, Geometric structure and the local Langlands conjecture (2013). arXiv:1211.0180

    Google Scholar 

  2. A.-M. Aubert, P.F. Baum, R.J. Plymen, M. Solleveld, The local Langlands correspondence for inner forms of SL n . Res. Math. Sci. (2013, To appear). Preprint

    Google Scholar 

  3. A.I. Badulescu, Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle. Ann. Sci. Éc. Norm. Sup. (4) 35, 695–747 (2002)

    Google Scholar 

  4. A.I. Badulescu, Un théorème de finitude (with an appendix by P. Broussous). Compos. Math. 132, 177–190 (2002)

    Google Scholar 

  5. A. Borel, Automorphic L-functions. Proc. Symp. Pure Math 33 (2), 27–61 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Broussous, Minimal strata for GL m (D). J. Reine Angew. Math. 514, 199–236 (1999)

    MathSciNet  MATH  Google Scholar 

  7. P. Broussous, B. Lemaire, Building of GL(m, D) and centralizers. Transform. Groups 7 (1), 15–50 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. C.J. Bushnell, A. Frölich, Gauss Sums and p-adic Division Algebras. Lecture Notes in Mathematics, vol. 987 (Springer, Berlin, 1983)

    Google Scholar 

  9. C.J. Bushnell, A. Frölich, Non-abelian congruence Gauss sums and p-adic simple algebras. Proc. Lond. Math. Soc. (3) 50, 207–264 (1985)

    Google Scholar 

  10. C.J. Bushnell, G. Henniart, The essentially tame local Langlands correspondence, I. J. Am. Math. Soc. 18.3, 685–710 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. C.J. Bushnell, G. Henniart, The Local Langlands Conjecture for GL(2). Grundlehren der mathematischen Wissenschaften, vol. 335 (Springer, Berlin, 2006)

    Google Scholar 

  12. C.J. Bushnell, P.C. Kutzko, The admissible dual of SL(N) I. Ann. Scient. Éc. Norm. Sup. (4) 26, 261–280 (1993)

    Google Scholar 

  13. T.H. Chen, M. Kamgarpour, Preservation of depth in local geometric Langlands correspondence (2014). arXiv: 1404.0598

    Google Scholar 

  14. S. DeBacker, M. Reeder, Depth zero supercuspidal L-packets and their stability. Ann. Math. 169 (3), 795–901 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Deligne, D. Kazhdan, M.-F. Vigneras, Représentations des algèbres centrales simples p-adiques, in Représentations des Groupes réductifs sur un Corps Local (Travaux en cours, Hermann, 1984), pp. 33–117

    Google Scholar 

  16. R. Ganapathy, The local Langlands correspondence for GSp 4 over local function fields. Am. J. Math. 137 (6), 1441–1534 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Godement, H. Jacquet, Zeta Functions of Simple Algebras. Lecture Notes in Mathematics, vol. 260 (Springer, New York, 1972)

    Google Scholar 

  18. M. Grabitz, A.J. Silberger, W. Zink, Level zero types and Hecke algebras for local central simple algebras. J. Number Theory 91, 92–125 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. B.H. Gross, M. Reeder, “Arithmetic invariants of discrete Langlands parameters”, Duke Math. J. 154 (2010), 431–508.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Harris, R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton NJ, 2001)

    Google Scholar 

  21. G. Henniart, On the local Langlands conjecture for GL(n): the cyclic case. Ann. Math. 123, 143–203 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Henniart, Une preuve simple de conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math. 139 (2000), 439–455.

    Article  MathSciNet  Google Scholar 

  23. K. Hiraga, H. Saito, On L-packets for inner forms of SL n . Mem. Am. Math. Soc. 215 (1013) (2012)

    Google Scholar 

  24. H. Jacquet, Principal L-functions of the linear group. Proc. Symp. Pure Math. 33 (2), 63–86 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Jacquet, R. Langlands, Automorphic Forms on GL(2). Lecture Notes in Mathematics, vol. 114 (Springer, New York, 1970)

    Google Scholar 

  26. R.E. Kottwitz, Stable trace formula: cuspidal tempered terms. Duke Math. J. 51 (3), 611–650 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Lansky, A. Raghuram, On the correspondence of representations between GL(n) and division algebras. Proc. Am. Math. Soc. 131 (5), 1641–1648 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Laumon, M. Rapoport, U. Stuhler, \(\mathcal{D}\)-elliptic sheaves and the Langlands correspondence. Invent. Math. 113, 217–238 (1993)

    Google Scholar 

  29. G. Lusztig, Classification of unipotent representations of simple p-adic groups. Int. Math. Res. Not. 11, 517–589 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. G. Lusztig, Classification of unipotent representations of simple p-adic groups II. Represent. Theory 6, 243–289 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. C. Mœglin, Stabilité en niveau 0, pour les groupes orthogonaux impairs p-adiques. Doc. Math. 9, 527–564 (2004)

    MathSciNet  Google Scholar 

  32. L. Morris, Tamely ramified intertwining algebras. Invent. Math. 114 (1), 1–54 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Moy, G. Prasad, Jacquet functors and unrefined minimal K-types. Comment. Math. Helv. 71, 98–121 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Reeder, Supercuspidal L-packets of positive depth and twisted Coxeter elements. J. Reine Angew. Math. 620, 1–33 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Reeder, J.-K. Yu, Epipelagic representations and invariant theory. J. Am. Math. Soc. 27, 437–477 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. V. Sécherre, S. Stevens, Représentations lisses de GL m (D) IV: représentations supercuspidales. J. Inst. Math. Jussieu 7 (3), 527–574 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. J.-P. Serre, Corps Locaux (Hermann, Paris, 1962)

    MATH  Google Scholar 

  39. M. Tadić, Induced representations of GL(n, A) for p-adic division algebras A. J. Reine Angew. Math. 405, 48–77 (1990)

    Google Scholar 

  40. J. Tate, Number theoretic background. Proc. Symp. Pure Math 33 (2), 3–26 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  41. D. Vogan, The local Langlands conjecture, in Representation Theory of Groups and Algebras. Contemporary Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 1993), pp. 305–379

    Google Scholar 

  42. A. Weil, Exercices dyadiques. Invent. Math. 27, 1–22 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  43. J.-K. Yu, Bruhat-Tits theory and buildings, in Ottawa Lectures on Admissible Representations of Reductive p-adic Groups. Fields Institute Monographs (American Mathematical Society, Providence, RI, 2009), pp. 53–77

    Google Scholar 

  44. J.-K. Yu, On the local Langlands correspondence for tori, in Ottawa Lectures on Admissible Representations of Reductive p-adic Groups. Fields Institute Monographs (American Mathematical Society, Providence, RI, 2009), pp. 177–183

    Google Scholar 

  45. A.V. Zelevinsky, Induced representations of reductive p-adic groups II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4) 13 (2), 165–210 (1980)

    Google Scholar 

Download references

Acknowledgements

The authors wish to thank Vincent Sécherre for some helpful explanations on the construction of simple types for GL m (D), Mark Reeder for pointing out some examples where the depth is not preserved, Wilhelm Zink for explaining properties of the Artin reciprocity map and Paul Broussous for providing the reference [BaBr], which has allowed a substantial simplification of the proof of Proposition 2.6 from a previous version. The second author “Paul Baum” was partially supported by NSF grant DMS-1200475.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C., 4 place Jussieu, 75005, Paris, France

    Anne-Marie Aubert

  2. Mathematics Department, Pennsylvania State University, University Park, PA, 16802, USA

    Paul Baum

  3. School of Mathematics, Southampton University, Southampton, SO17 1BJ, UK

    Roger Plymen

  4. School of Mathematics, Manchester University, Manchester, M13 9PL, UK

    Roger Plymen

  5. Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525AJ, Nijmegen, The Netherlands

    Maarten Solleveld

Authors
  1. Anne-Marie Aubert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Baum
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roger Plymen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Maarten Solleveld
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Baum .

Editor information

Editors and Affiliations

  1. Max Planck Institute for Mathematics, Bonn, Germany

    Werner Ballmann

  2. Max Planck Institute for Mathematics, Bonn, Germany

    Christian Blohmann

  3. Max Planck Institute for Mathematics, Bonn, Germany

    Gerd Faltings

  4. Max Planck Institute for Mathematics, Bonn, Germany

    Peter Teichner

  5. Max Planck Institute for Mathematics, Bonn, Germany

    Don Zagier

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Aubert, AM., Baum, P., Plymen, R., Solleveld, M. (2016). Depth and the Local Langlands Correspondence. In: Ballmann, W., Blohmann, C., Faltings, G., Teichner, P., Zagier, D. (eds) Arbeitstagung Bonn 2013. Progress in Mathematics, vol 319. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43648-7_2

Download citation

Publish with us

Policies and ethics

Search

Navigation