A program for branching problems in the representation theory of real reductive groups

  • Toshiyuki Kobayashi
Part of the Progress in Mathematics book series (PM, volume 312)


We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G′ (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are a priori known to be “nice” in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.

Key words

branching law symmetry breaking operator unitary representation Zuckerman–Vogan’s \(A_{\mathfrak{q}}(\lambda )\) module reductive group spherical variety multiplicity-free representation 

MSC (2010):

Primary 22E46 Secondary 53C35 


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The author thanks J.-L. Clerc, T. Kubo, T. Matsuki, B. Ørsted, T. Oshima, Y. Oshima, M. Pevzner, P. Somberg, V. Souček, B. Speh for their collaboration on the papers which are mentioned in this article. This article is based on the lecture that the author delivered at the conference Representations of reductive groups in honor of David Vogan on his 60th birthday at MIT, 19-23 May 2014. He would like to express his gratitude to the organizers, Roman Bezrukavnikov, Pavel Etingof, George Lusztig, Monica Nevins, and Peter Trapa, for their warm hospitality during the stimulating conference.

This work was partially supported by Grant-in-Aid for Scientific Research (A) (25247006), Japan Society for the Promotion of Science.


  1. 1.
    J. Adams, Unitary highest weight modules, Adv. in Math. 63 (1987), 113–137.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Aizenbud and D. Gourevitch, Multiplicity one theorem for \((GL_{n+1}(\mathbb{R}),GL_{n}(\mathbb{R}))\), Selecta Math. 15 (2009), 271–294.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Berger, Les espaces symétriques non compacts, Ann. Sci. École Norm. Sup. 74 (1957), 85–177.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    I. N. Bernstein and S. I. Gelfand, Meromorphic property of the functions \(P^{\lambda }\), Funktsional Anal, i Prilozhen., 3 (1969), 84–85.Google Scholar
  5. 5.
    F. Bien, Orbit, multiplicities, and differential operators, Contemp. Math. 145, Amer. Math. Soc., 1993, 199–227.Google Scholar
  6. 6.
    M. Brion, Classification des espaces homogènes sphériques, Compos. Math. 63 (1986), 189–208.MathSciNetzbMATHGoogle Scholar
  7. 7.
    R. Brylinski, Geometric quantization of real minimal nilpotent orbits, Differential Geom. Appl. 9 (1998), 5–58.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J.-L. Clerc, T. Kobayashi, B. Ørsted, and M. Pevzner, Generalized Bernstein–Reznikov integrals, Math. Ann., 349 (2011), 395–431.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann., 217 (1975), 271–285.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. of Math. (2), 147 (1998), 417–452.Google Scholar
  11. 11.
    G. van Dijk and M. Pevzner, Ring structures for holomorphic discrete series and Rankin-Cohen brackets, J. Lie Theory, 17, (2007), 283–305.MathSciNetzbMATHGoogle Scholar
  12. 12.
    M. Duflo and J. A. Vargas, Branching laws for square integrable representations, Proc. Japan Acad. Ser. A, Math. Sci. 86 (2010), 49–54.Google Scholar
  13. 13.
    T. Enright and J. Willenbring, Hilbert series, Howe duality and branching for classical groups, Ann. of Math. (2) 159 (2004), 337–375.Google Scholar
  14. 14.
    B. Gross and D. Prasad, On the decomposition of a representations of SO n when restricted to SO n−1, Canad. J. Math. 44 (1992), 974–1002.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    Harish-Chandra, Representations of semisimple Lie groups on a Banach space, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 170–173.Google Scholar
  17. 17.
    M. Harris and H. P. Jakobsen, Singular holomorphic representations and singular modular forms, Math. Ann., 259 (1982), 227–244.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R. Howe, \(\theta\) -series and invariant theory, Proc. Symp. Pure Math. 33 (1979), Amer. Math. Soc., 275–285.Google Scholar
  19. 19.
    R. Howe, Reciprocity laws in the theory of dual pairs, Progr. Math. Birkhäuser, 40, (1983), 159–175.MathSciNetzbMATHGoogle Scholar
  20. 20.
    R. Howe and C. Moore, Asymptotic properties of unitary representations. J. Funct. Anal. 32, (1979), 72–96.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Juhl, Families of conformally covariant differential operators, Q-curvature and holography. Progr. Math., 275. Birkhäuser, 2009.Google Scholar
  22. 22.
    M. Kashiwara and M. Vergne, K-types and singular spectrum, In: Lect. Notes in Math., Vol. 728, 1979, Springer-Verlag, 177–200.Google Scholar
  23. 23.
    B. Kimelfeld, Homogeneous domains in flag manifolds of rank 1, J. Math. Anal. & Appl. 121, (1987), 506–588.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    A. W. Knapp and D. Vogan, Jr., Cohomological Induction and Unitary Representations, Princeton U.P., 1995.CrossRefzbMATHGoogle Scholar
  25. 25.
    T. Kobayashi, The restriction of \(A_{\mathfrak{q}}(\lambda )\) to reductive subgroups, Proc. Japan Acad., 69 (1993), 262–267.MathSciNetCrossRefGoogle Scholar
  26. 26.
    T. Kobayashi, Discrete decomposability of the restriction of \(A_{\mathfrak{q}}(\lambda )\) with respect to reductive subgroups and its applications, Invent. Math., 117, (1994), 181–205.MathSciNetCrossRefGoogle Scholar
  27. 27.
    T. Kobayashi, Introduction to harmonic analysis on real spherical homogeneous spaces, Proceedings of the 3rd Summer School on Number Theory “Homogeneous Spaces and Automorphic Forms” in Nagano (F. Sato, ed.), 1995, 22–41 (in Japanese).Google Scholar
  28. 28.
    T. Kobayashi, Discrete decomposability of the restriction of \(A_{\mathfrak{q}}(\lambda )\) with respect to reductive subgroups II – micro-local analysis and asymptotic K-support, Ann. of Math., 147, (1998), 709–729.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    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), 229–256.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Translations, Series II, Selected Papers on Harmonic Analysis, Groups, and Invariants (K. Nomizu, ed.), 183, (1998), Amer. Math. Soc., 1–31.Google Scholar
  31. 31.
    T. Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal., 152, (1998), 100–135.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    T. Kobayashi, Discretely decomposable restrictions of unitary representations of reductive Lie groups —examples and conjectures, Advanced Study in Pure Math., 26, (2000), 98–126.MathSciNetzbMATHGoogle Scholar
  33. 33.
    T. Kobayashi, Restrictions of unitary representations of real reductive groups, Progr. Math., Vol. 229, pages 139–207, Birkhäuser, 2005.Google Scholar
  34. 34.
    T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci. 41, (2005), 497–549.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    T. Kobayashi, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, 45–109, Progr. Math., Vol. 255, Birkhäuser, Boston, 2008.Google Scholar
  36. 36.
    T. Kobayashi, Branching problems of Zuckerman derived functor modules, In: Representation Theory and Mathematical Physics (in honor of Gregg Zuckerman), Contemporary Mathematics, 557, 23–40. Amer. Math. Soc., Providence, RI, 2011.CrossRefGoogle Scholar
  37. 37.
    T. Kobayashi, Restrictions of generalized Verma modules to symmetric pairs, Transform. Group, 17, (2012), 523–546.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    T. Kobayashi, F-method for constructing equivariant differential operators, Geometric Analysis and Integral Geometry Contemporary Mathematics (in honor of S. Helgason), 598, Amer. Math. Soc., 2013, 141–148.Google Scholar
  39. 39.
    T. Kobayashi, Propagation of multiplicity-freeness property for holomorphic vector bundles, Progr. Math., Vol. 306, Birkhäuser, 2013, 113–140.Google Scholar
  40. 40.
    T. Kobayashi, F-method for symmetry breaking operators, Differential Geom. Appl. 33 (2014), 272–289, Special issue in honor of M. Eastwood.Google Scholar
  41. 41.
    T. Kobayashi, Shintani functions, real spherical manifolds, and symmetry breaking operators, Developments in Mathematics, 37 (2014), 127–159.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    T. Kobayashi, T. Kubo, and M. Pevzner, Vector-valued covariant differential operators for the Möbius transformation, Springer Proceedings in Mathematics & Statistics, 111, 67–86, 2015.Google Scholar
  43. 43.
    T. Kobayashi and G. Mano, The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p,q), Mem. Amer. Math. Soc. (2011), 212, no. 1000, vi+132 pages.Google Scholar
  44. 44.
    T. Kobayashi and T. Matsuki, Classification of finite-multiplicity symmetric pairs, Transform. Groups 19 (2014), 457–493, Special issue in honor of Dynkin for his 90th birthday.Google Scholar
  45. 45.
    T. Kobayashi and B. Ørsted, Analysis on the minimal representation ofO (p,q). Part I, Adv. Math., 180, (2003) 486–512; Part II, ibid, 513–550; Part III, ibid, 551–595.Google Scholar
  46. 46.
    T. Kobayashi, B. Ørsted, and M. Pevzner, Geometric analysis on small unitary representations of \(GL(n, \mathbb{R})\), J. Funct. Anal., 260, (2011) 1682–1720.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    T. Kobayashi, B. Ørsted, P. Somberg, and V. Souček, Branching laws for Verma modules and applications in parabolic geometry, Adv. Math., 285, (2015), 1796–1852 doi: 10.1016/j.aim.2015.08.020, (available also at arXiv:1305.6040).Google Scholar
  48. 48.
    T. Kobayashi and T. Oshima, Finite multiplicity theorems for induction and restriction, Adv. Math., 248, (2013), 921–944.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    T. Kobayashi and Y. Oshima, Classification of discretely decomposable \(A_{\mathfrak{q}}(\lambda )\) with respect to reductive symmetric pairs, Adv. Math., 231 (2012), 2013–2047.MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    T. Kobayashi and Y. Oshima, Classification of symmetric pairs with discretely decomposable restrictions of \((\mathfrak{g},K)\) -modules, Journal für die reine und angewandte Mathematik, 2015, (2015), no.703, 201–223.MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    T. Kobayashi and M. Pevzner, Differential symmetry breaking operators. I . General theory and F-method, to appear in Selecta Math., 44 pages; II . Rankin–Cohen operators for symmetric pairs, to appear in Selecta Math., 64 pages, (available also at arXiv:1301.2111).Google Scholar
  52. 52.
    T. Kobayashi and B. Speh, Symmetry breaking for representations of rank one orthogonal groups, (2015), Memoirs of Amer. Math. Soc. 238, no.1126, 118 pages.Google Scholar
  53. 53.
    M. Krämer, Multiplicity free subgroups of compact connected Lie groups, Arch. Math. (Basel) 27, (1976), 28–36.MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    T. Matsuki, Orbits on flag manifolds, Proceedings of the International Congress of Mathematicians, Kyoto 1990, Vol. II (1991), Springer-Verlag, 807–813.Google Scholar
  55. 55.
    T. Matsuki and T. Oshima, A description of discrete series for semisimple symmetric spaces, Adv. Stud. Pure Math. 4, (1984), 331–390.MathSciNetzbMATHGoogle Scholar
  56. 56.
    I. V. Mikityuk, Integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Math. USSR-Sbornik 57, (1987), 527–546.CrossRefzbMATHGoogle Scholar
  57. 57.
    V. F. Molchanov, Tensor products of unitary representations of the three-dimensional Lorentz group, Math. USSR, Izv. 15, (1980), 113–143.Google Scholar
  58. 58.
    J. Möllers and B. Ørsted, Estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles, to appear in Int. Math. Res. Not. IMRN.Google Scholar
  59. 59.
    B. Ørsted and B. Speh, Branching laws for some unitary representations of \(SL(4, \mathbb{R})\), SIGMA 4, (2008), doi:10.3842/SIGMA.2008.017.Google Scholar
  60. 60.
    Y. Oshima, Discrete branching laws of Zuckerman’s derived functor modules, Ph.D. thesis, the University of Tokyo, 2013.Google Scholar
  61. 61.
    J. Peetre, Une caractérisation abstraite des opérateurs différentiels, Math. Scand., 7, (1959), 211–218.MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    R. A. Rankin, The construction of automorphic forms from the derivatives of a given form, J. Indian Math. Soc., 20, (1956), 103–116.MathSciNetzbMATHGoogle Scholar
  63. 63.
    J. Repka, Tensor products of holomorphic discrete series representations, Can. J. Math., 31, (1979), 836–844.MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    R. Richardson, G. Röhrle, and R. Steinberg, Parabolic subgroup with abelian unipotent radical, Invent. Math., 110 (1992), 649–671.MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    W. Schmid, Die Randwerte holomorphe Funktionen auf hermetisch symmetrischen Raumen, Invent. Math., 9, (1969–70), 61–80.Google Scholar
  66. 66.
    W. Schmid, Boundary value problems for group invariant differential equations, Astérisque (1985), 311–321.Google Scholar
  67. 67.
    J. Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39, (1987), 127–138.MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case, Ann. of Math. 175, (2012), 23–44.Google Scholar
  69. 69.
    D. A. Vogan, Jr., Representations of Real Reductive Lie Groups, Progr. Math., Vol. 15, Birkhäuser, 1981.Google Scholar
  70. 70.
    D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. 120, (1984), 141–187.MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    D. A. Vogan, Jr., Irreducibility of discrete series representations for semisimple symmetric spaces, Adv. Stud. Pure Math. 14, (1988), 191–221.MathSciNetzbMATHGoogle Scholar
  72. 72.
    D. A. Vogan, Jr., Associated varieties and unipotent representations, Progr. Math., Vol. 101, (1991), Birkhäuser, 315–388.Google Scholar
  73. 73.
    D. A. Vogan, Jr. and G. J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), 51–90.MathSciNetzbMATHGoogle Scholar
  74. 74.
    N. R. Wallach, Real reductive groups. I, II, Pure and Applied Mathematics, Vol. 132, Academic Press, Inc., Boston, MA, 1988.Google Scholar
  75. 75.
    H. Wong, Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations, J. Funct. Anal. 129, (1995), 428–454.MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    D. Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. (Math. Sci.) 104, (1994), 57–75.MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    F. Zhu and K. Liang, On a branching law of unitary representations and a conjecture of Kobayashi, C. R. Acad. Sci. Paris, Ser. I, 348, (2010), 959–962.Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Kavli IPMU (WPI) and Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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