Skip to main content
SpringerLink
Log in

Distinction of Representations via Bruhat-Tits Buildings of p-Adic Groups

À la mémoire de mon ami François Courtès 1970–2016

  • Chapter
  • First Online:
Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2221))

  • 1120 Accesses

Abstract

Let G∕H be a symmetric space over a non-archimedean local field F: G is (the group of F-points of) a reductive group over F and H ⊂ G is the subgroup of (F-rational) points in G fixed by an involution.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Institutional subscriptions

Notes

  1. 1.

    A matrix is called monomial if each row or column exactly contains a single non-zero coefficient.

  2. 2.

    In fact easy results or routine details will be left to the reader.

  3. 3.

    The distance on the tree is normalized so that the length of an edge is 1.

  4. 4.

    For more details, cf. [Ca, §1].

  5. 5.

    Cf. [Ca, §3] for more details.

  6. 6.

    Also see [SS] for another construction of this involution.

  7. 7.

    The reader will forgive me to use the same symbol d to denote the dimension of X H and the coboundary map.

  8. 8.

    An \({\mathcal H}(H)\)-module M is non degenerate if \({\mathcal H}(H)\cdot M =M\).

  9. 9.

    The proof of this theorem is due to François Courtès; see the appendix of [BC].

References

  1. P. Abramenko, K.S. Brown, Buildings. Theory and Applications. Graduate Texts in Mathematics, vol. 248 (Springer, New York, 2008)

    Book  Google Scholar 

  2. A.-M. Aubert, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique. Trans. Am. Math. Soc. 347(6), 2179–2189 (1995)

    MATH  Google Scholar 

  3. A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35, 233–259 (1976)

    Article  MathSciNet  Google Scholar 

  4. R. Bott, An application of the Morse theory to the topology of Lie-groups. Bull. Soc. Math. France 84, 251–281 (1956)

    Article  MathSciNet  Google Scholar 

  5. N. Bourbaki, Elements de Mathématique, Groupes et algèbres de Lie, Chapitre 4 à 6 (Springer, Berlin, 2007)

    MATH  Google Scholar 

  6. P. Broussous, F. Courtès, Distinction of the Steinberg representation, with an appendix by François Courtès. Int. Math. Res. Not. 11, 3140–3157 (2014)

    Article  Google Scholar 

  7. C.J. Bushnell, P.C. Kutzko, Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77(3), 582–634 (1998)

    Article  MathSciNet  Google Scholar 

  8. G.E. Bredon, Sheaf Theory. Graduate Texts in Mathematics, vol. 170 (Springer, New York, 2012)

    Google Scholar 

  9. A. Borel, J.-P. Serre, Cohomologie à supports compacts des immeubles de Bruhat-Tits; applications à la cohomologie des groupes S-arithmétiques. C. R. Acad. Sci. Paris Sér. A-B 272, A110–A113 (1971)

    MATH  Google Scholar 

  10. F. Bruhat, J. Tits, Groupes réductifs sur un corps local I. Schémas en groupes. Existence d’une donnée radicielle valuée. Publ. Math. IHES 60, 1–376 (1984)

    MATH  Google Scholar 

  11. F. Bruhat, J. Tits, Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Soc. Math. France 112(2), 259–301 (1984)

    Article  MathSciNet  Google Scholar 

  12. W. Casselman, Introduction to admissible representations of p-adic groups. Notes, IAS (June 1974). Available at https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf

  13. W. Casselman, The Steinberg character as a true character, in Harmonic Analysis on Homogeneous Spaces. Proceedings of Symposia in Pure Mathematics, vol. XXVI (American Mathematical Society, Providence, 1973), pp. 413–417

    Google Scholar 

  14. F. Courtès, Distinction of the Steinberg representation II: an equality of characters. Forum Math. 27(6), 3461–3475 (2015)

    Article  MathSciNet  Google Scholar 

  15. F. Courtès, Distinction of the Steinberg representation III: the tamely ramified case (2014). Preprint

    Google Scholar 

  16. S. Dasgupta, J. Teitelbaum, The p-adic upper half plane, in p-Adic Geometry. University Lecture Series, vol. 45 (American Mathematical Society, Providence, 2008), pp. 65–121

    Google Scholar 

  17. A. Fröhlich, Local fields, in Algebraic Number Theory, ed. by J.W.S. Cassels, A. Fröhlich. Proceedings of the Instructional Conference Held at the University of Sussex, Brighton, September 1–17, 1965 (Harcourt Brace Jovanovich, London, 1986)

    Google Scholar 

  18. P. Garrett, Buildings and Classical Groups (Chapman & Hall, London, 1997)

    Book  Google Scholar 

  19. F.Q. Gouvêa, p-adic Numbers. An Introduction. Universitext (Springer, Berlin, 1993)

    Google Scholar 

  20. Harish-Chandra, Harmonic analysis on reductive p-adic groups, in Harmonic Analysis on Homogeneous Spaces. Proceedings of Symposia in Pure Mathematics, vol. XXVI (American Mathematical Society, Providence, 1973), pp. 413–417

    Google Scholar 

  21. N. Iwahori, H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of \({\mathfrak p}\)-adic Chevalley groups. Inst. Hautes Études Sci. Publ. Math. 25, 5–48 (1965)

    Google Scholar 

  22. W.S. Massey, Homology and Cohomology Theory. An Approach Based on Alexander-Spanier Cochains, vol. 46. Monographs and Textbooks in Pure and Applied Mathematics (Marcel Dekker Inc., New York, 1978)

    Google Scholar 

  23. D. Prasad, On a conjecture of Jacquet about distinguished representations of GL n. Duke J. Math 109, 67–78 (2001)

    Article  MathSciNet  Google Scholar 

  24. D. Prasad, A ‘relative’ local Langlands correspondence. Math. Arxiv (2015). arXiv:1512.04347

    Google Scholar 

  25. M. Ronan, Buildings: main ideas and applications. I. Main ideas. Bull. London Math. Soc. 24(1), 1–51 (1992)

    Article  MathSciNet  Google Scholar 

  26. M. Ronan, Buildings: main ideas and applications. II. Arithmetic groups, buildings and symmetric spaces. Bull. London Math. Soc. 24(2), 97–126 (1992)

    Article  MathSciNet  Google Scholar 

  27. J.-P. Serre, Arbres, amalgames, SL2. Astérisque, vol. 46 (SMF, Paris, 1977)

    Google Scholar 

  28. R. Steinberg, Endomorphisms of linear algebraic groups. Mem. Am. Math. Soc. 80, 1–108 (1968)

    MATH  Google Scholar 

  29. P. Schneider, U. Stuhler, Representation theory and sheaves on the Bruhat-Tits building. Inst. Hautes Études Sci. Publ. Math. 85, 97–191 (1997)

    Article  MathSciNet  Google Scholar 

  30. J. Tits, Reductive groups over local fields. Proc. Sympos. Pure Math. 33(part I), 29–69 (1979)

    Google Scholar 

  31. A. Weil, Basic Number Theory, Reprint of the second (1973) edition. Classics in Mathematics (Springer, Berlin, 1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, UMR 7348 du CNRS, Futuroscope Chasseneuil Cedex, France

    Paul Broussous

Authors
  1. Paul Broussous
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Broussous .

Editor information

Editors and Affiliations

  1. Aix-Marseille Université CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France

    Volker Heiermann

  2. School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India

    Dipendra Prasad

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Broussous, P. (2018). Distinction of Representations via Bruhat-Tits Buildings of p-Adic Groups. In: Heiermann, V., Prasad, D. (eds) Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms. Lecture Notes in Mathematics, vol 2221. Springer, Cham. https://doi.org/10.1007/978-3-319-95231-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation