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International Mathematics Research Papers (2008) Vol. 2008 : article ID rpn005, 166 pages, doi:10.1093/imrp/rpn005 published on June 4, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Distinguished Tame Supercuspidal Representations

Jeffrey Hakim1 and Fiona Murnaghan2

1 Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue NW, Washington, DC 20016
2 Department of Mathematics, University of Toronto, 40 Saint George Street, Toronto, Canada M5S 2E4

Correspondence: Correspondence to be sent to: jhakim{at}american.edu

This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal representations of finite groups of Lie type are distinguished relative to particular quadratic characters. As an application of the main results, we obtain necessary and sufficient conditions for equivalence of two of Yu's supercuspidal representations associated to distinct G-data.



References

  1. Adler J. Refined anisotropic K-types and supercuspidal representations. Pacific Journal of Mathematics (1998) 185(no. 1):1–32.[ISI]
  2. Adler J., DeBacker S. Some applications of Bruhat–Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group. Michigan Mathematical Journal (2002) 50(no. 2):263–86.[CrossRef][ISI]
  3. Adler J., Roche A. An intertwining result for p-adic groups. Canadian Journal of Mathematics (2000) 52(no. 3):449–67.
  4. Anandavardhanan U. K., Prasad D. Distinguished representations for SL(2). Mathematical Research Letters (2003) 10(no. 5–6):867–78.[ISI]
  5. Bernstein I. N., Zelevinskii A. V. Representations of the group GL(n, F) where F is a non-archimedean local field. Russian Mathematical Surveys (1976) 31(no. 3):1–68.
  6. Bernstein J. p-Invariant Distributions on GL(N) and the Classification of Unitary Representations of GL(N) (Non-Archimedean Case). 50–102. Lecture Notes in Mathematics 1041. Berlin: Springer, 1983.
  7. Bourbaki N. General Topology. Chap. 1–4. New York: Springer, 1989.
  8. Bushnell C., Kutzko P. The Admissible Dual of GL(N) via Compact Open Subgroups. Annals of Mathematics Studies 119. Princeton, NJ: Princeton University Press, 1993.
  9. Bruhat F., Tits J. Groupes réductifs sur un corps local. Chap. 1. Publications mathématiques de Institut des Hautes Études Scientifiques (1972) 41:5–251.[CrossRef]
  10. Bruhat F., Tits J. Groupes réductifs sur un corps local. Chap. 2. Publications mathématiques de Institut des Hautes Études Scientifiques (1984) 60:197–376.
  11. Bruhat F., Tits J. Schémas en groupes et immeubles des groupes classiques sur un corps local 2: groupes unitaires. Bulletin de la Société Mathématique de France (1987) 115:141–95.[ISI]
  12. DeBacker S. Some applications of Moy-Prasad filtrations to harmonic analysis on a reductive p-adic group. Michigan Mathematical Journal (2002) 50:241–61.[CrossRef][ISI]
  13. Flath D. Decomposition of Representations into Tensor Products. Automorphic Forms, Representations, and L-Functions. part 1, 179–83. Proceedings of Symposia in Pure Mathematics 33. Providence, RI: American Mathematical Society, 1979.
  14. Gelfand I., Kazhdan D. Representations of GLn(K) where K is a Local Field. 95–118. Lie Groups and Their Representations. Budapest, Hungary: Halstead Press, 1975.
  15. Gérardin P. Weil representations associated to finite fields. Journal of Algebra (1977) 46(no. 1):54–101.[CrossRef][ISI]
  16. Goldman O., Iwahori N. The space of Formula -adic norms. Acta Mathematica (1963) 109:137–77.[CrossRef][ISI]
  17. Gross B. Some applications of Gelfand pairs to number theory. Bulletin of the American Mathematical Society (1991) 24(no. 2):277–301.[CrossRef][ISI]
  18. Hakim J. Supercuspidal Gelfand pairs. Journal of Number Theory (2003) 100:251–69.[CrossRef][ISI]
  19. Hakim J., Mao Z. Cuspidal representations associated to (GL(n),O(n)) over finite fields and p-adic fields. Journal of Algebra (1999) 213:129–43.[CrossRef][ISI]
  20. Hakim J., Mao Z. Supercuspidal representations of GL(N) distinguished by a unitary subgroup. Pacific Journal of Mathematics (1998) 185(no. 1):149–62.[ISI]
  21. Hakim J., Murnaghan F. Globalization of distinguished supercuspidal representations of GL(N). Canadian Mathematical Bulletin (2002) 45(no. 2):220–30.
  22. Hakim J., Murnaghan F. Tame supercuspidal representations of GL(N) distinguished by a unitary group. Compositio Mathematica (2002) 133(no. 2):199–244.[CrossRef][ISI]
  23. Hakim J., Murnaghan F. Two types of distinguished supercuspidal representations. International Mathematics Research Notices (2002) 2002(no. 35):1857–89.[Abstract/Free Full Text]
  24. Harish-Chandra. Admissible Invariant Distributions on Reductive p-Adic Groups. Preface and notes by Stephen DeBacker and Paul J. Sally, Jr. University Lecture Series, vol. 16. Providence, RI: American Mathematical Society, 1999.
  25. Heumos M., Rallis S. Symplectic-Whittaker models for GLn. Pacific Journal of Mathematics (1990) 146(no. 2):247–79.[ISI]
  26. Howe R. Tamely ramified supercuspidal representations of GLn. Pacific Journal of Mathematics (1977) 73(no. 2):437–60.[ISI]
  27. Jacquet H. Automorphic Spectrum of Symmetric Spaces. Representation Theory and Automorphic Forms (1997) 61:443–55. Special issue, Proceedings of Symposia in Pure Mathematics.
  28. Kable A. Asai L-functions and Jacquet's conjecture. American Journal of Mathematics (2004) 126:789–820.[CrossRef][ISI]
  29. Kim J.-L. Supercuspidal representations: An exhaustion theorem. Journal of the American Mathematical Society (2007) 20(no. 2):273–320.[CrossRef][ISI]
  30. Kim J.-L., Moy A. Involutions, classical groups and buildings. Journal of Algebra (2001) 242:495–515.[CrossRef][ISI]
  31. Kim J.-L., Murnaghan F. Character expansions and unrefined minimal K-types. American Journal of Mathematics (2003) 125:1199–1234.[CrossRef][ISI]
  32. Lusztig G. Symmetric Spaces over a Finite Field. The Grothendieck Festschrift—Cartier P., et al, eds. vol. 3:57–81. Boston: Birkhäuser, 1990.
  33. Moeglin C., Vigneras M.-F., Waldspurger J.-L. Correspondence de Howe sur un corps p-adique. Lecture Notes in Mathematics 1291. New York: Springer, 1987.
  34. Morris L. Level zero G-types. Compositio Mathematica (1999) 118:135–57.[CrossRef][ISI]
  35. Moy A. Local constants and the tame Langlands correspondence. American Journal of Mathematics (1991) 64:863–930.
  36. Moy A., Prasad G. Unrefined minimal K-types for p-adic groups. Inventiones Mathematicae (1994) 116:393–408.[CrossRef][ISI]
  37. Moy A., Prasad G. Jacquet functors and unrefined minimal K-types. Commentarii Mathematici Helvetici (1996) 71:98–121.[CrossRef][ISI]
  38. Murnaghan F., Repka J. Reducibility of some induced representations of p-adic unitary groups. Transactions of the American Mathematical Society (1999) 351:193–210.[CrossRef][ISI]
  39. Murnaghan F., Repka J. Reducibility of some induced representations of split classical groups. Compositio Mathematica (1998) 114:263–313.[CrossRef][ISI]
  40. Prasad D. On a conjecture of Jacquet about distinguished representations of GL(N). Duke Mathematical Journal (2001) 109:67–78.[CrossRef][ISI]
  41. Prasad G. Galois fixed points in the Bruhat-Tits building of a reductive group. Bulletin de la Société Mathématique de France (2001) 129:169–74.
  42. Prasad G., Yu. J.-K. On finite group actions on reductive groups and buildings. Inventiones Mathematicae (2002) 147(no. 3):545–60.[CrossRef][ISI]
  43. Rader C., Rallis C. Spherical characters on Formula -adic symmetric spaces. American Journal of Mathematics (1996) 118:91–178.[CrossRef][ISI]
  44. Richardson R. Orbits, invariants and representations associated to involutions of reductive groups. Inventiones Mathematicae (1982) 66:287–312.[CrossRef][ISI]
  45. Rousseau G. Immeubles des groupes réductifs sur les corps locaux. Thesis, University of Paris, 1977.
  46. Steinberg R. Endomorphisms of Linear Algebraic Groups. Memoirs of the American Mathematical Society 80. Providence, RI: American Mathematical Society, 1968.
  47. Tits J. Reductive Groups over Local Fields. Automorphic Forms, Representations, and L-Functions. part 1, 29–69. Proceedings of Symposia in Pure Mathematics 33. Providence, RI: American Mathematical Society, 1979.
  48. Yu J.-K. Construction of Tame Supercuspidal Representations. Journal of the American Mathematical Society (2001) 14:579–622.[CrossRef][ISI]
  49. Yu J.-K. Smooth models associated to concave functions in Bruhat-Tits theory. (2002) preprint, version 1.3, December 13, 2002.

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This Article
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