Acta Mathematica

, Volume 161, Issue 1, pp 183–241

Whittaker vectors and the Goodman-Wallach operators

  • Hisayosi Matumoto
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References

  1. [BV]Barbasch, D. &Vogan Jr., D. A., The local structure of characters.J. Funct. Anal., 37 (1980), 27–55.CrossRefMathSciNetGoogle Scholar
  2. [BW]Borel, A. &Wallach, N. R.,Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Annals of Mathematics Studies, Princeton University Press, Princeton N.J., 1980.Google Scholar
  3. [CC1]Casian, L. G. &Collingwood, D. A., Complex geometry and the asymptotics of Harish-Chandra modules for real reductive Lie groups I.Trans Amer. Math. Soc., 300 (1987), 73–107.CrossRefMathSciNetGoogle Scholar
  4. [CC2]—, Complex geometry and the asymptotics of Harish-Chandra modules for real reductive Lie groups II.Invent. Math., 86 (1986), 255–286.CrossRefMathSciNetGoogle Scholar
  5. [CC3]Casian, L. G. & Collingwood, D. A., Weight filtrations for induced representations of real reductive Lie groups (preprint, 1986).Google Scholar
  6. [CO]Casselman, W. &Osborne, M. S., The restriction of admissible representations to II.Math. Ann., 233 (1978), 193–198.CrossRefMathSciNetGoogle Scholar
  7. [Gd1]Goodman, R., Differential operators of infinite order on a Lie group I.J. Math. Mech., 19 (1970), 879–894.MATHMathSciNetGoogle Scholar
  8. [Gd2]—, Differential operators of infinite order on a Lie group II.Indiana Univ. Math. J., 21 (1971), 383–409.CrossRefMATHMathSciNetGoogle Scholar
  9. [Gd3]—, Horospherical functions on symmetric spaces.Canadian Mathematical Society Conference Proceedings, 1 (1981), 125–133.MATHGoogle Scholar
  10. [GG]Gelfand, I. M. &Graev, M. I., Construction of irreducible representations of simple algebraic group over a finite field.Soviet Math. Dokl., 3 (1962), 1646–1649.Google Scholar
  11. [Gv]Gevrey, M., Sur la nature analytique des solutions des équations aux dérivées partielles.Ann. Sci. École Norm. Sup., 35 (1918), 129–190.MATHMathSciNetGoogle Scholar
  12. [GW]Goodman, R. &Wallach, N. R., Whittaker vectors and conical vectors.J. Funct. Anal., 39 (1980), 199–279.CrossRefMathSciNetGoogle Scholar
  13. [Ha1]Hashizume, M., Whittaker models for semisimple Lie groups.Japan J. Math., 5 (1979), 349–401.MATHMathSciNetGoogle Scholar
  14. [Ha2]Hashizume, M., Whittaker models for representations with highest weights.Lectures on harmonic analysis on Lie groups and related topics (Strasbourg, 1979), 45–50; Lectures in Math., 14, Kinokuniya Book Store, Tokyo, 1982.Google Scholar
  15. [Ha3]—, Whittaker functions on semisimple Lie groups.Hiroshima Math. J., 13 (1982), 259–293.MathSciNetGoogle Scholar
  16. [He]Helgason, S., A duality for symmetric spaces with applications to group representations.Adv. in Math., 5 (1970), 1–154.CrossRefMATHMathSciNetGoogle Scholar
  17. [Ho]Howe, R., Wave front sets of representations of Lie groups, inAutomorphic Forms, Representation Theory, and Arithmetic. Bombay, 1981.Google Scholar
  18. [HS]Hecht, H. &Schmid, W., Characters, asymptotics and n-homology of Harish-Chandra modules.Acta Math., 151 (1983), 49–151.MathSciNetGoogle Scholar
  19. [Jc]Jacquet, H., Fonction de Whittaker associées aux groupes de Chevalley.Bull. Soc. Math. France, 95 (1967), 243–309.MATHMathSciNetGoogle Scholar
  20. [JL]Jacquet, H. &Langlands, R. P.,Automorphic forms on GL(2). Lecture Notes in Mathematics, No. 114, Springer-Verlag, Berlin-Heidelberg-New York, 1970.Google Scholar
  21. [Jo]Joseph,A., Goldie rank in the enveloping algebra of a semisimple Lie algebra II.J. Algebra, 65 (1980), 284–306.MATHMathSciNetGoogle Scholar
  22. [Ka1]Kawanaka, N., Generalized Gelfand-Graev representations and Ennola duality, inAlgebraic Groups and Related topics. Adv. Stud. in Pure Math. Kinokuniya Book Store and North-Holland, 1985, 175–206.Google Scholar
  23. [Ka2]—, Generalized Gelfand-Graev representations of exceptional simple algebraic groups over a finite field I.Invent. Math., 84 (1986), 575–616.CrossRefMATHMathSciNetGoogle Scholar
  24. [Ka3]Kawanaka, N., Shintani lifting and Gelfand-Graev representations. To appear inProc. Symp. Pure Math. (Proc. AMS Summer Institute at Arcata, 1986).Google Scholar
  25. [Km]Komatsu, H., Ultradistributions. I: Structure theorems and a characterization.J. Fac. Sci. Univ. Tokyo, Sect. IA, 20 (1973), 25–105.MATHMathSciNetGoogle Scholar
  26. [Ko]Kostant, B., On Whittaker vectors and representation theory.Invent. Math., 48 (1978), 101–184.CrossRefMATHMathSciNetGoogle Scholar
  27. [KV]Kashiwara, M. &Vergne, M.,K-types and the singular spectrum, inNon-commutative Harmonic Analysis. Lecture Notes in Mathematics, No. 728, Springer-Verlag, Berlin-Heiderlberg-New York, 1979.Google Scholar
  28. [Le]Lepowski, J., Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism.J. Algebra, 49 (1977), 470–495.CrossRefMathSciNetGoogle Scholar
  29. [Ly]Lynch, T. E.,Generalized Whittaker vectors and representation theory. Thesis, M.I.T., 1979.Google Scholar
  30. [Ma1]Matumoto, H., Boundary value problems for Whittaker functions on real split semisimple Lie groups.Duke Math. J., 53 (1986), 635–676.CrossRefMATHMathSciNetGoogle Scholar
  31. [Ma2]—, Whittaker vectors and associated varieties.Invent. Math., 89 (1987), 219–224.CrossRefMATHMathSciNetGoogle Scholar
  32. [Mc]McConnell, J. C., The K-theory of filtered rings and skew Laurent extensions, inSéminaire d’Algebre Paul Dubreil et Marie-Paule Malliavin. Lecture Notes in Mathematics. No. 1146, 288–298. Springer-Verlag, Berlin-Heidelberg-New York.Google Scholar
  33. [MŒ]Mœglin, C., Modéles de Whittaker et idéaux primitifs complètement premiers dans les algèbres enveloppantes.C. R. Acad. Sci Paris, 303 (1986), 845–848.Google Scholar
  34. [No]Northcott, D. G.,Finite Free Resolutions. Cambridge University Press, Cambridge-London-New York-Melbourne, 1976.Google Scholar
  35. [Qu]Quillen, D. G., Higher K-theory I, inAlgebraic K-theory I; Higher K-theories. Lecture Notes in Mathematics, No. 341, Springer-Verlag, Berlin-Heideberg-New York, 1973.Google Scholar
  36. [Ra]Raêvskiî, P. K., Associative hyper-envelopes of Lie algebras, their regular representations and ideals.Trans. Moscow Math. Soc., 15 (1966), 1–54.Google Scholar
  37. [Ro]Roumieu, C., Sur quelques extensions de la notion de distribution.Ann. Sci. École Norm. Sup., 77 (1960), 41–121.MATHMathSciNetGoogle Scholar
  38. [Sch]Schiffmann, G., Intégrales d’entrelacement et fonctions de Whittaker.Bull. Soc. Math. France, 99 (1971), 3–72.MATHMathSciNetGoogle Scholar
  39. [Shd]Shahidi, F., Whittaker models for real groups.Duke Math. J., 47 (1980), 99–125.CrossRefMATHMathSciNetGoogle Scholar
  40. [Shl]Shalika, J., The multiplicity one theorem forGL(n).Ann. of Math., 100 (1974), 171–193.CrossRefMATHMathSciNetGoogle Scholar
  41. [Shp]Shapovolov, N. N., On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra.Functional Anal. Appl., 6 (1972), 307–312.CrossRefGoogle Scholar
  42. [SS]Springer, T. A. &Steinberg, R., Conjugacy classes, inSeminar on Algebraic Group and Related Finite Groups. Lecture Notes in Mathematics, No. 131, Part E. Springer-Verlag, Berlin-Heiderberg-New York, 1970.Google Scholar
  43. [V1]Vogan Jr., D. A., Gelfand-Kirillov dimension for Harish-Chandra modules,Invent. Math., 48 (1978), 75–98.CrossRefMATHMathSciNetGoogle Scholar
  44. [V2]Vogan, Jr., D. A., The orbit method and primitive ideals for semisimple Lie algebras. Canadian Mathematical Society Conference Proceedings Vol 5,Lie Algebras And Related Topics, (1986), 281–316.Google Scholar
  45. [Wa]Wallach, N. R.,Harmonic Analysis on Homogeneous Spaces. Dekker, New York, 1973.Google Scholar
  46. [Y1]Yamashita, H., On Whittaker vectors for generalized Gelfand-Graev representations of semisimple Lie groups.J. Math. Kyoto Univ., 26 (1986), 263–298.MATHMathSciNetGoogle Scholar
  47. [Y2]Yamashita, H., Multiplicity one theorems for generalized Gelfand-Graev representations of semisimple Lie groups and Whittaker models for the discrete series (preprint, 1987).Google Scholar

Copyright information

© Almqvist & Wiksell 1988

Authors and Affiliations

  • Hisayosi Matumoto
    • 1
    • 2
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute for Advanced StudyPrincetonUSA

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