Skip to main content
SpringerLink
Log in

Langlands’ Conjecture on Plancherel Measures for p-Adic Groups

  • Chapter
Harmonic Analysis on Reductive Groups

Part of the book series: Progress in Mathematics ((PM,volume 101))

Abstract

One of the major achievements of Harish-Chandra was a derivation of the Plancherel formula for real and p-adic groups [9,10]. To have an explicit formula, one will have to compute the measures appearing in the formula; the so called Plancherel measures and formal degrees [12]. (For reasons stemming from L-indistinguishability, we would like to distinguish between the formal degrees for discrete series and the Plancherel measures for non-discrete tempered representations, cf. Proposition 9.3 of [29].) While for real groups the Plancherel measures are completely understood [1, 9, 22], until recently little was known in any generality for p-adic groups [29] (except for their rationality and general form due to Silberger [39]). On the other hand any systematic study of the non-discrete tempered spectrum of a p-adic group would very likely have to follow the path of Knapp and Stein [20, 21] and their theory of 72-groups. Since the basic reducibility theorems for p-adic groups are available [40, 41], it is the knowledge of Plancherel measures which would be necessary to determine the R-groups. This is particularly evident from the important and the fundamental work of Keys [16, 17, 18] and the work of the author [29, 30, 31].

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Arthur, Intertwining operators and residues I: weighted characters, J. Funct. Anal. 84 (1989), 19–84.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Arthur On some problems suggested by the trace formula, Lie Group Representations II, Lecture Notes in Math, Vol. 1041, Springer-Verlag, Berlin-Heidelberg-NewYork, 1983, pp. 1–49.

    Google Scholar 

  3. J. Arthur Unipotent automorphic representations: Conjectures, Orbites Unipotenteset Repŕesentations, II. Groupes p-adic et réels, Societe Mathematique de France, Astérisque, Nos. 171–172, 1989, pp. 13–71.

    Google Scholar 

  4. J. Arthur Unipotent automorphic representations: Global motivations, Proceedings of the Conference on Automorphic Forms, Shimura Varieties, and L-functions, Vol.I (L. Clozel and J.S. Milne, eds.), in Perspectives in Mathematics, vol. 10, AcademicPress, 1990, pp. 1–75.

    Google Scholar 

  5. J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Studies, Vol. 120, Princeton University Press, Princeton, 1989.

    MATH  Google Scholar 

  6. I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive p-adic groups I, Ann. Scient. Éc. Norm. Sup. 10 (1977), 441–472.

    MathSciNet  MATH  Google Scholar 

  7. A. Borel, Automorphic L-functions, Proc. Sympos. Pure Math., AMS, 33, II (1979), 27–61.

    Google Scholar 

  8. S. Gelbart and H. Jacquet, A relation between automorphic representations ofGL(2) and GL(3), Ann. Scient. Éc. Norm Sup. 11 (1978), 471–542.

    MathSciNet  MATH  Google Scholar 

  9. Harish-Chandra, Harmonic analysis on real reductive groups HI. The Maass-Selberg relation and the PI an cherel formula, Annals of Math. 104 (1976), 117–201.

    Article  MathSciNet  MATH  Google Scholar 

  10. Harish-Chandra The Plancherel formula for reductive p-adic groups, reprinted in Collected Papers, Springer Verlag, Berlin-Heidelberg-New York, 1984, pp. 353–367.

    Google Scholar 

  11. R. Howe and A. Moy, Hecke algebra isomorphisms for GL n over a p-adic field, Jour. of Algebra 131 (1990), 388–424.

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Jabon, D. Keys, and A. Moy, An explicit Plancherel formula for U(2,1), preprint.

    Google Scholar 

  13. H. Jacquet, I.I. Piatetski-Shapiro, and J.A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), 367–464.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Jacquet, I.I. Piatetski-Shapiro, and J.A. Shalika Conducteur des representations du groupe linéaire, Math. Ann. 256 (1981), 199–214.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Jacquet and J.A. Shalika, Exterior square L-functions, Proceedings of the Conference on Automorphic Forms, Shimura Varieties, and L-functions, Vol. II (L. Clozel and J.S. Milne, eds.), in Perspectives in Mathematics, vol. 11, Academic Press, 1990, pp. 143–226.

    Google Scholar 

  16. D. Keys, On the decomposition of reducible principal series representation of p-adic Chevalley groups, Pacific J. Math. 101 (1982), 351–388.

    MathSciNet  MATH  Google Scholar 

  17. D. Keys Principal series representations of special unitary groups over local field, Comp. Math. 51 (1984), 115–130.

    MathSciNet  MATH  Google Scholar 

  18. D. Keys L-indistinguishability and R-groups for quasi-split groups: Unitary groups of even dimension, Ann. Scient. Éc. Norm. Sup. 20 (1987), 31–64.

    MathSciNet  MATH  Google Scholar 

  19. D. Keys and F. Shahidi, Artin L-functions and normalization of intertwining oper-ators, Ann. Scient. Éc. Norm. Sup. 21 (1988), 67–89.

    MathSciNet  MATH  Google Scholar 

  20. A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Annals of Math. 93 (1971), 489–578.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. W. Knapp and E. M. SteinIntertwining operators for semisimple groups, II, Invent. Math. 60 (1980), 9–84.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. W. Knapp and G. J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Annals of Math. 116 (1982), 389–455.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. P. Langlands, Euler Products, Yale University Press, New Haven, 1971.

    MATH  Google Scholar 

  24. R. P. Langlands On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Math., Vol 544, Springer-Verlag, Berlin-Heidelberg-New York, 1976.

    MATH  Google Scholar 

  25. R. P. Langlands Problems in the theory of automorphic forms, Lee. Notes in Math., Vol.170, Springer-Verlag, Berlin-Heidelberg-New York, pp. 18–86.

    Google Scholar 

  26. R. P. Langlands On Artin’s L-functions, Rice University Studies 56 (1970), 23–28.

    Google Scholar 

  27. R. P. Langlands On the classification of irreducible representations of real algebraic groups, Representation Theory and Harmonic Analysis on Semisimple Lie Groups (P.J. Sally, Jr. and D.A. Vogan, eds.), Mathematical Surveys and Monographs, AMS, Vol 31, 1989, pp. 101–170.

    Google Scholar 

  28. P. J. Sally and M. Tadic, On representations of non-archimedean symplectic groups, preprint.

    Google Scholar 

  29. F. Shahidi, A proof of Lang lands conjecture on Plancherel measures; complementary series for p-adic groups, Annals of Math. 132 (1990), 273–330.

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Shahidi, Some results on L-indistinguishability for SL(r), Canadian J. Math. 35(1983), 1075–1109.

    Article  MathSciNet  MATH  Google Scholar 

  31. F. Shahidi, Third symmetric power L-functions for GL(2), Comp. Math. 70 (1989), 245–273.

    MathSciNet  MATH  Google Scholar 

  32. F. Shahidi, Fourier transforms of intertwining operators and Plancherel measures for GL(n), Amer. J. Math. 106 (1984), 67–111.

    Article  MathSciNet  MATH  Google Scholar 

  33. F. Shahidi, Local coefficients and normalization of intertwining operators for GL(n), Comp. Math. 48 (1983), 271–295.

    MathSciNet  MATH  Google Scholar 

  34. F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297–356.

    Article  MathSciNet  MATH  Google Scholar 

  35. F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions, Annals of Math. 127 (1988), 547–584.

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Shahidi, On multiplicativity of local factors, in Festschrift in honor of I.I. Piatet-ski-Shapiro, Part II, Israel Mathematical Conference Proceedings 3 (1990), 279–289.

    Google Scholar 

  37. F. Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 973–1007.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Shelstad, L-indistinguishability for real groups, Math. Ann. 259 (1982), 385–430.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Silberger, Special representations of reductive p-adic groups are not integrable, Annals of Math. 111 (1980), 571–587.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Silberger, The Knapp-Stein dimension theorem for p-adic groups, Proc. Amer. Math.Soc. 68 (1978), 243–246.

    MathSciNet  MATH  Google Scholar 

  41. A. Silberger, Introduction to Harmonic Analysis on Reductive p-adic Groups Math.Notes, Vol. 23, Princeton University Press, Princeton, 1979.

    MATH  Google Scholar 

  42. D. Vogan, Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75–98.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. L. Waldspurger, Un exercice sur GSp(4,F) et les representations de Weil, Bull. Soc. Math. France 115 (1987), 35–69.

    MathSciNet  MATH  Google Scholar 

  44. A. V. Zelevinsky, Induced representations of reductive p-adic groups II: on irreducible representations of GL(n), Ann. Scient. Éc. Norm. Sup. 13 (1980), 165–210.

    MathSciNet  MATH  Google Scholar 

  45. W. Casselman, On some results of Atkin and Lehner, Math. Ann. 206 (1973), 311–318.

    Article  MathSciNet  MATH  Google Scholar 

  46. G. I. Olšanskii, Intertwining operators and complementary series, Math. USSR Sbornik. 22 (1974), 217–255.

    Article  Google Scholar 

  47. B. Tamir, On L-functions and intertwining operators for unitary groups, preprint (1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Freydoon Shahidi

Authors
  1. Freydoon Shahidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Bowdoin College, Brunswick, ME, 04011, USA

    William H. Barker

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Paul J. Sally Jr.

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer Science+Business Media New York

About this chapter

Cite this chapter

Shahidi, F. (1991). Langlands’ Conjecture on Plancherel Measures for p-Adic Groups. In: Barker, W.H., Sally, P.J. (eds) Harmonic Analysis on Reductive Groups. Progress in Mathematics, vol 101. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0455-8_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0455-8_14

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6768-3

  • Online ISBN: 978-1-4612-0455-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation