Remote Access Transactions of the American Mathematical Society
Green Open Access

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. Collingwood, D. H., Representations of rank one Lie groups, Research Notes in Mathematics, 137. Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 853731 (88c:22014)
  • 3. Gelfand, I. M.; Tsetlin, M. L.: Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR 71 (1950), 825-828 (Russian). English transl. in: I. M. Gelfand, Collected Papers, vol. II, Springer-Verlag, Berlin, 1988. MR 0035774 (12:9j)
  • 4. Itoh, M.; Umeda, T., On Central Elements in the Universal Enveloping Algebras of the Orthogonal Lie Algebras. Compositio Math. 127 (2001), 333-359. MR 1845042 (2002d:17011)
  • 5. Knapp, W. A., Representation theory of semisimple groups. An overview based on examples. Princeton Mathematical Series, 36. Princeton University Press, Princeton, NJ, 1986. xviii+774 pp. MR 855239 (87j:22022)
  • 6. Kostant, B., On Whittaker vectors and representation theory. Invent. Math. 48 (1978), no. 2, 101-184. MR 507800 (80b:22020)
  • 7. Kraljević, H., Representations of the universal covering group of the group $ SU(n,1)$, Glas. Mat. Ser. III 8(28) No. 1 (1973), 23-72. MR 0330355 (48:8692)
  • 8. Lynch, T. E., Generalized Whittaker vectors and representation theory, Thesis, MIT, 1979.
  • 9. Matumoto, H., Boundary value problems for Whittaker functions on real split semisimple Lie groups. Duke Math. J. 53 (1986), no. 3, 635-676. MR 860664 (88b:22010)
  • 10. Matumoto, H., Whittaker vectors and the Goodman-Wallach operators, Acta Math. 161 (1988), 183-241. MR 971796 (90d:22018)
  • 11. Matumoto, H., $ C^{-\infty }$-Whittaker vectors corresponding to a principal nilpotent orbit of a real reductive linear Lie group, and wave front sets, Compositio Math. 82 (1992), 189-244. MR 1157939 (93c:22026)
  • 12. Oshima, T., Boundary value problems for systems of linear partial differential equations with regular singularities. Group representations and systems of differential equations (Tokyo, 1982), 391-432, Adv. Stud. Pure Math. 4, North-Holland, Amsterdam, 1984. MR 810637 (87c:58121)
  • 13. Taniguchi, K., Discrete Series Whittaker Functions of $ SU(n,1)$ and $ Spin(2n,1)$, J. Math. Sci. Univ. Tokyo 3 (1996), 331-377. MR 1424434 (97m:22003)
  • 14. Vogan, D. A., Representations of real reductive Lie groups. Progress in Mathematics, 15. Birkhäuser, Boston, Mass., 1981. MR 632407 (83c:22022)
  • 15. Wallach, N. R., Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I, 287-369, Lecture Notes in Math., 1024, Springer-Verlag, Berlin, 1983. MR 727854 (85g:22029)
  • 16. N. R. Wallach, Lie Algebra Cohomology and Holomorphic Continuation of Generalized Jacquet Integrals. Representations of Lie groups, Kyoto, Hiroshima, 1986, 123-151, Adv. Stud. Pure Math., 14, Academic Press, Boston, MA, 1988. MR 1039836 (91d:22014)

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.

American Mathematical Society