Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the composition series of the standard Whittaker $ (\mathfrak{g},K)$-modules

Author: Kenji Taniguchi
Journal: Trans. Amer. Math. Soc. 365 (2013), 3899-3922
MSC (2010): Primary 22E46, 22E45
Published electronically: November 6, 2012
MathSciNet review: 3042608
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a real reductive linear Lie group $ G$, the space of Whittaker functions is the representation space induced from a non-degenerate unitary character of the Iwasawa nilpotent subgroup. Defined are the standard Whittaker $ (\mathfrak{g},K)$-modules, which are $ K$-admissible submodules of the space of Whittaker functions. We first determine the structures of them when the infinitesimal characters characterizing them are generic. As an example of the integral case, we determine the composition series of the standard Whittaker $ (\mathfrak{g},K)$-module when $ G$ is the group $ U(n,1)$ and the infinitesimal character is regular integral.

References [Enhancements On Off] (What's this?)

  • 1. Capelli, A., Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37 (1890), 1-37.
  • 2. David H. Collingwood, Representations of rank one Lie groups, Research Notes in Mathematics, vol. 137, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 853731
  • 3. I. M. Gel′fand and M. L. Cetlin, Finite-dimensional representations of the group of unimodular matrices, Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 825–828 (Russian). MR 0035774
  • 4. Minoru Itoh and Tôru Umeda, On central elements in the universal enveloping algebras of the orthogonal Lie algebras, Compositio Math. 127 (2001), no. 3, 333–359. MR 1845042, 10.1023/A:1017571403369
  • 5. Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239
  • 6. Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, 10.1007/BF01390249
  • 7. Hrvoje Kraljević, Representations of the universal convering group of the group 𝑆𝑈(𝑛,1), Glasnik Mat. Ser. III 8(28) (1973), 23–72 (English, with Serbo-Croatian summary). MR 0330355
  • 8. Lynch, T. E., Generalized Whittaker vectors and representation theory, Thesis, MIT, 1979.
  • 9. Hisayosi Matumoto, Boundary value problems for Whittaker functions on real split semisimple Lie groups, Duke Math. J. 53 (1986), no. 3, 635–676. MR 860664, 10.1215/S0012-7094-86-05335-4
  • 10. Hisayosi Matumoto, Whittaker vectors and the Goodman-Wallach operators, Acta Math. 161 (1988), no. 3-4, 183–241. MR 971796, 10.1007/BF02392298
  • 11. Hisayosi Matumoto, 𝐶^{-∞}-Whittaker vectors corresponding to a principal nilpotent orbit of a real reductive linear Lie group, and wave front sets, Compositio Math. 82 (1992), no. 2, 189–244. MR 1157939
  • 12. Toshio Ōshima, Boundary value problems for systems of linear partial differential equations with regular singularities, Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 391–432. MR 810637
  • 13. Kenji Taniguchi, Discrete series Whittaker functions of 𝑆𝑈(𝑛,1) and 𝑆𝑝𝑖𝑛(2𝑛,1), J. Math. Sci. Univ. Tokyo 3 (1996), no. 2, 331–377. MR 1424434
  • 14. David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407
  • 15. Nolan R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 287–369. MR 727854, 10.1007/BFb0071436
  • 16. Nolan R. Wallach, Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 123–151. MR 1039836

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 22E46, 22E45

Retrieve articles in all journals with MSC (2010): 22E46, 22E45

Additional Information

Kenji Taniguchi
Affiliation: Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1, Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258, Japan

Keywords: Whittaker modules
Received by editor(s): February 24, 2011
Received by editor(s) in revised form: January 26, 2012
Published electronically: November 6, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.