Inventiones mathematicae

, Volume 117, Issue 1, pp 181–205

Discrete decomposability of the restriction ofAq(λ) with respect to reductive subgroups and its applications

  • Toshiyuki Kobayashi


LetG′⊂G be real reductive Lie groups and q a θ-stable parabolic subalgebra of Lie (G) ⊗ ℂ. This paper offers a sufficient condition on (G, G′, q) that the irreducible unitary representation\(\mathop {A_q }\limits^--- \) ofG with non-zero continuous cohomology splits into a discrete sum of irreducible unitary representations of a subgroupG′, each of finite multiplicity. As an application to purely analytic problems, new results on discrete series are also obtained for some pseudo-Riemannian (non-symmetric) spherical homogeneous spaces, which fit nicely into this framework. Some explicit examples of a decomposition formula are also found in the cases whereAq is not necessarily a highest weight module.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A1] Adams, J.: Discrete spectrum of the dual reductive pair (O(p, q), Sp(2m)). Invent. Math.74, 449–475 (1984)Google Scholar
  2. [A2] Adams, J.: Unitary highest weight modules. Adv. Math.63, 113–137 (1987)Google Scholar
  3. [Bo] Borel, A.: Some remarks about Lie groups transitive on spheres and tori. Bull.Am. Math. Soc.55, 580–587 (1949)Google Scholar
  4. [BoW] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Princeton: Princeton University Press 1980Google Scholar
  5. [Br] Brion, M.: Classification des espaces homogenes spheriques. Compos. Math.63-2, 189–208 (1987)Google Scholar
  6. [C] Chang, J.T.: Remarks on localization and standard modules: The duality theorem on a generalized flag variety. Proc. Am. Math. Soc.117, 585–591 (1993)Google Scholar
  7. [EPWW] Enright, T.J., Parthasarathy, R., Wallach, N.R., Wolf, J.A.: Unitary derived functor module with small spectrum. Acta. Math.154, 105–136 (1985)Google Scholar
  8. [FJ] Flensted-Jensen, M.: Analysis on Non-Riemannian Symmetric Spaces. (Conf. Board, vol. 61) Providence, RI: Am. Math. Soc. 1986Google Scholar
  9. [GG] Gelfand, I.M., Graev, M.I.: Geometry of homogeneus spaces, representations of groups in homogeneous spaces, and related questions of integral geometry. Transl., II. Ser., Am. Math. Soc.37, 351–429 (1964)Google Scholar
  10. [GGP] Gelfand, I.M., Graev, M.I., Piatecki-Ŝapiro, I.: Representation theory and automorphic functions, Phiadelphia: Saunders 1969Google Scholar
  11. [HMSW] Hecht, H., Miličic, D., Schmid, W., Wolf, J.A.: Localization and standard modules for real semisimple Lie groups. Invent. Math.90, 297–332 (1987)Google Scholar
  12. [HS] Hecht, H., Schmid, W.: A proof of Blattner's conjecture. Invent. Math.31, 129–154 (1976)Google Scholar
  13. [He] Helgason, S.: Differential geometry, Lie groups and symmetric spaces. (Pure Appl. Math., vol. 80) New York London: Academic Press, 1978Google Scholar
  14. [Ho] Howe, R.: θ-series and invariant theory. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations and L-functions. (Proc. Symp. Pure Math., vol. 33, pp. 275–285) Providence, RI: Am. Math. Soc. 1979Google Scholar
  15. [HT] Howe, R., Tan, E.: Homogeneous functions on light cones: The infinitesimal structures of some degenerate principal series representations. Bull. Am. Math. Soc.28, 1–74 (1993)Google Scholar
  16. [J] Jakobsen, H.P.: Tensor products, reproducing kernels, and power series. J. Funct. Anal.31, 293–305 (1979)Google Scholar
  17. [JV] Jakobsen, H.P., Vergne, M.: Restrictions and expansions of holomorphic representations. J. Funct. Anal.34, 29–53 (1979)Google Scholar
  18. [KV] Kashiwara, M., Vergne, M.: On the Segal-Shale-Weil representations and harmonic polynomials. Invent. Math.44, 1–47 (1978)Google Scholar
  19. [Ko1] Kobayashi, T.: Proper action on a homogeneous space of reductive type. Math. An.,285, 249–263 (1989)Google Scholar
  20. [Ko2] Kobayashi, T.: Unitary representations realized in L2-sections of vector bundles over semisimple symmetric spaces. In: Proceedings at the 27-th. Symp. of Functional Analysis and Real Analysis, pp. 39–54 (in Japanese). Math. Soc. Japan 1989Google Scholar
  21. [Ko3] Kobayashi, T.: Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds\(U(p,q; \mathbb{F})/U(p - m,q; \mathbb{F})\). Mem. Am. Math. Soc.462 (1992)Google Scholar
  22. [Ko4] Kobayashi, T.: Discrete decomposability of the restriction ofA g(λ) with respect to reductive subgroups. II. Classification for classical symmetric pairs. (in preparation)Google Scholar
  23. [Kr1] Krämer, M.: Multilicity free subgroups of compact connected Lie groups. Arch. Math.27, 28–35 (1976)Google Scholar
  24. [Kr2] Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compos. Math.38-2, 129–153 (1979)Google Scholar
  25. [L] Lipsman, R.: Restrictions of principal series to a real form. Pac. J. Math.89, 367–390 (1980)Google Scholar
  26. [M] Martens, S.: The characters of the holomorphic discrete series. Proc. Natl. Acad. Sci. USA72, 3275–3276 (1975)Google Scholar
  27. [MO1] Matsuki, T., Oshima, T.: A description of discrete series for semisimple symmetric spaces. Adv. Stud. Pure Math.4, 331–390 (1984)Google Scholar
  28. [MO2] Matsuki, T., Oshima, T.: Embeddings of discrete series into principal series In: Dulfo, M. et al. (eds.) The orbit method in representation theory. (Prog. Math., vol. 80, pp. 147–175. Boston Buset Stuttgart: Birkhäuser 1990Google Scholar
  29. [OO] Olafsson, G., Ørsted, B.: The holomorphic discrete series of an affine symmetric space. I. J. Funct. Anal.81, 126–159 (1988)Google Scholar
  30. [Os] Oshima, T.: Asymptotic behavior of spherical functions on semisimple symmetric spaces. Adv. Stud. Pure Math.14, 561–601 (1988)Google Scholar
  31. [R] Repka, J.: Tensor products of holomorphic discrete series representations. Can. J. Math.31, 836–844 (1979)Google Scholar
  32. [S1] Schlichtkrull, H.: A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group. Invent. Math.68, 497–516 (1982)Google Scholar
  33. [S2] Schlichtkrull, H.: Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms. J. Funct. Anal.70, 194–219 (1987)Google Scholar
  34. [V1] Vogan, D.: Representations of real reductive Lie groups, Boston Basel Stuttgart: Birkhäuser 1981Google Scholar
  35. [V2] Vogan, D.: Unitary representations of reductive Lie groups. Princeton, NJ: Princeton University Press 1987Google Scholar
  36. [V3] Vogan, D.: Irreducibility of discrete series representations for semisimple symmetric spaces. Adv. Stud. Pure Math.14, 191–221 (1988)Google Scholar
  37. [VZ] Vogan, D., Zuckerman, G.J.: Unitary representations with non-zero cohomology. Compos. Math.53, 51–90 (1984)Google Scholar
  38. [Wa1] Wallach, N.: Real reductive groups. I. (Pure Appl. Math., vol. 132), Boston: Academic Press 1988Google Scholar
  39. [War] Warner, G.: Harmonic analysis on semisimple Lie groups. I. Berlin Heidelberg, New York: Springer 1972Google Scholar
  40. [Wi] Williams, F.: Tensor products of principal series representations. (Lect. Notes Math., vol. 358) Berlin Heidelberg New York: Springer 1973Google Scholar
  41. [Y] Yamashita, H.: Criteria for the finiteness of restriction of U(g) to subalgebras and applications to Harish-Chandra modules. Proc. Japan Acad.68, 316–321 (1992)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Toshiyuki Kobayashi
    • 1
  1. 1.Department of Mathematical SciencesUniversity of TokyoTokyoJapan

Personalised recommendations