Skip to main content
SpringerLink
Log in

Dimension of the space of intertwining operators from degenerate principal series representations

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

Let X be a homogeneous space of a real reductive Lie group G. It was proved by T. Kobayashi and T. Oshima that the regular representation \(C^{\infty }(X)\) contains each irreducible representation of G at most finitely many times if a minimal parabolic subgroup P of G has an open orbit in X, or equivalently, if the number of P-orbits on X is finite. In contrast to the minimal parabolic case, for a general parabolic subgroup Q of G, we find a new example that the regular representation \(C^{\infty }(X)\) contains degenerate principal series representations induced from Q with infinite multiplicity even when the number of Q-orbits on X is finite.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenbud, A., Gourevitch, D.: Schwartz functions on Nash manifolds. Int. Math. Res. Not. IMRN 5, 37 (2008). Art. ID rnm 155

    MathSciNet  MATH  Google Scholar 

  2. Aizenbud, A., Gourevitch, D., Minchenko, A.: Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs. Sel. Math. (N.S.) 22(4), 2325–2345 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bien, F.: Orbit, multiplicities, and differential operators. Contemp. Math. Am. Math. Soc. 145, 199–227 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gelfand, I.M., Shilov, G.E.: Generalized Functions. Vol. I: Properties and Operations. Academic, New York (1964). xvii+423 pp

    Google Scholar 

  5. Harish-Chandra, K.: Representations of semisimple Lie groups. II. Trans. Am. Math. Soc. 76, 26–65 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kashiwara, M.: On the maximally overdetermined system of linear differential equations. I. Publ. Res. Inst. Math. Sci. 10, 563–579 (1974/75)

  7. Kashiwara, M.: Systems of Microdifferential Equations. Progr. Math. 34 (1983). Birkhäuser, xv+159 pp

  8. Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (Proceedings of Conference, Katata, 1971; dedicated to the memory of André Martineau). In: Lecture Notes in Mathematics, Vol. 287. Springer, Berlin, pp. 265–529 (1973)

  9. Kimelfeld, B.: Homogeneous domains in flag manifolds. J. Math. Anal. Appl. 121, 506–588 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kobayashi, T.: Introduction to harmonic analysis on real spherical homogeneous spaces. In: Sato, F. (ed.) Proceedings of the 3rd Summer School on Number Theory Homogeneous Spaces and Automorphic Forms in Nagano, pp. 22–41 (1995) (in Japanese)

  11. Kobayashi, T.: Shintani functions, real spherical manifolds, and symmetry breaking operators. Dev. Math. 37, 127–159 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Kobayashi, T.: A program for branching problems in the representation theory of real reductive groups, Progress in Mathematics 312, Representations of Reductive Groups, 277-322, In honor of the 60th Birthday of David A. Vogan, Jr (2015). See also arXiv:1509.08861

  13. Kobayashi, T., Oshima, T.: Finite multiplicity theorems for induction and restriction. Adv. Math. 248, 921–944 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kobayashi, T., Speh, B.: Symmetry breaking for representations of rank one orthogonal groups. Mem. Am. Math. Soc. 238, 118 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Kobayashi, T., Speh, B.: Symmetry breaking for orthogonal groups and a conjecture by B. Gross and D. Prasad. In: proceedings of the Simons Symposium on Geometric Aspects of the Trace Formula, Simons Symposia. Springer, Cham. arXiv: 1702.00263

  16. Kobayashi, T., Yoshino, T.: Compact Clifford–Klein forms of symmetric spaces-revisited. Pure Appl. Math. Q. 1, 603–684 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Matsuki, T.: Orbits on flag manifolds. In: Proceedings of the International Congress of Mathematicians, Kyoto 1990, Vol. II. Springer, pp. 807–813 (1991)

  18. Möllers, J.: Symmetry breaking operators for strongly spherical reductive pairs and the Gross-Prasad conjecture for complex orthogonal groups. arXiv:1705.06109

Download references

Acknowledgements

The author is grateful to Professor Toshiyuki Kobayashi for his much helpful advice and constant encouragement. The author also thanks the referee for his careful comments.

Author information

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, Tokyo, 153-8914, Japan

    Taito Tauchi

Authors
  1. Taito Tauchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taito Tauchi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tauchi, T. Dimension of the space of intertwining operators from degenerate principal series representations. Sel. Math. New Ser. 24, 3649–3662 (2018). https://doi.org/10.1007/s00029-018-0422-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-018-0422-2

Keywords

Mathematics Subject Classification

Search

Navigation