Advertisement

Symmetry Breaking for Orthogonal Groups and a Conjecture by B. Gross and D. Prasad

  • Toshiyuki Kobayashi
  • Birgit Speh
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

We consider irreducible unitary representations Ai of G = SO(n + 1, 1) with the same infinitesimal character as the trivial representation and representations Bj of H = SO(n, 1) with the same properties and discuss H-equivariant homomorphisms \(\operatorname {Hom}_H(A_i,B_j)\). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.

Keywords

Special orthogonal group Tempered representations Branching laws Gross-Prasad conjectures 

Notes

Acknowledgements

Many of the results were obtained while the authors were supported by the Research in Pairs program at the Mathematisches Forschungsinstitut in Oberwolfach (MFO), Germany.

The research by T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research (A) (25247006), Japan Society for the Promotion of Science.

The research by B. Speh was partially supported by NSF grant DMS-1500644. Part of this research was conducted during a visit of the second author at the Graduate School of Mathematics of the University of Tokyo, Komaba. She would like to thank it for its support and hospitality during her stay.

The authors thank an anonymous referee for careful reading of the manuscript and comments.

References

  1. 1.
    J. Adams, D. Barbasch, D. Vogan, The Langlands Classification and Irreducible Characters for Real Reductive Groups, Progr. Math. 104, Birkhäuser, Boston–Basel–Berlin, 1992.Google Scholar
  2. 2.
    A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. 2nd edition. Mathematical Surveys and Monographs, 67, Amer. Math. Soc., Providence, 2000.Google Scholar
  3. 3.
    W.-T. Gan, B. Gross, D. Prasad: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Astérisque 346, 1–109, 2012.MathSciNetzbMATHGoogle Scholar
  4. 4.
    B. Gross and M. Reeder, From Laplace to Langlands via representations of orthogonal groups, Bull. Amer. Math. Soc. (N.S.), 43, (2006), 163–205.MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Gross and D. Prasad, On the decomposition of a representation of \(\operatorname {SO}_n\) when restricted to \(\operatorname {SO}_{n-1}\), Canad. J. Math. 44, (1992), no. 5, 974–1002.MathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Gross and N. Wallach, Restriction of small discrete series representations to symmetric subgroups, Proc. Sympos. Pure Math. 68, (2000), Amer. Math. Soc., 255–272.Google Scholar
  7. 7.
    A. W. Knapp and D. Vogan, Cohomological induction and unitary representations. Princeton Mathematical Series, 45. Princeton University Press, Princeton, NJ, 1995. xx+948 pp. ISBN: 0-691-03756-6.Google Scholar
  8. 8.
    T. Kobayashi, Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds U(p, q; F)∕U(p − m, q; F), Mem. Amer. Math. Soc. 462, Amer. Math. Soc., 1992. 106 pp. ISBN:9810210906.
  9. 9.
    T. Kobayashi, Discrete decomposability of the restriction of \(A_{\mathfrak {q}}(\lambda )\) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131, (1998), no. 2, 229–256.
  10. 10.
    T. Kobayashi, T. Kubo, M. Pevzner, Conformal Symmetry Breaking Differential Operators for Differential Forms on Spheres, Lecture Notes in Math., vol.2170, ix +  192 pages, 2016. ISBN: 978-981-10-2657-7.
  11. 11.
    T. Kobayashi and T. Oshima, Finite multiplicity theorems for induction and restriction, Adv. Math. 248, (2013), 921–944.MathSciNetCrossRefGoogle Scholar
  12. 12.
    T. Kobayashi and B. Speh, Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc., vol. http://dx.doi.org/238, No. 1126, (2015), v +  112 pages, ISBN: 978-1-4704-1922-6.Google Scholar
  13. 13.
    T. Kobayashi and B. Speh, Symmetry breaking for representations of rank one orthogonal groups, Part II, Lecture Notes in Math., 2234, Springer, 2018, ISBN: 978-981-13-2900-5; available also at arXiv:1801.00158.Google Scholar
  14. 14.
    R. Langlands, On the classification of irreducible representations of real reductive groups, Math. Surveys and Monographs 31, Amer. Math. Soc., Providence, 1988.Google Scholar
  15. 15.
    B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case, Ann. of Math. (2), 175, (2012), no. 1, http://dx.doi.org/23–44.Google Scholar
  16. 16.
    D. Vogan, The local Langlands conjecture, Contemp. Math., 145, (1993), 305–379, Amer. Math. Soc.Google Scholar
  17. 17.
    D. Vogan and G. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53, (1984), no. 1, 51–90.MathSciNetzbMATHGoogle Scholar
  18. 18.
    N. R. Wallach, Real Reductive Groups. II, Pure and Applied Mathematics, 132, Academic Press, Inc., Boston, MA, 1992. ISBN 978-0127329611.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of Mathematical Sciences and Kavli IPMUThe University of TokyoTokyoJapan
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

Personalised recommendations