Science China Mathematics

, Volume 60, Issue 11, pp 2337–2348 | Cite as

Tensor products of complementary series of rank one Lie groups

  • GenKai ZhangEmail author


We consider the tensor product πα ⊗ πβ of complementary series representations πα and πβ of classical rank one groups SO 0(n; 1), SU(n; 1) and Sp(n; 1). We prove that there is a discrete component πα+β for small parameters α and β (in our parametrization). We prove further that for SO0(n; 1) there are finitely many complementary series of the form πα+β+2j , j = 0, 1,..., k, appearing in the tensor product πα ⊗ πβ of two complementary series πα and πβ where k = k(α, β n) depends on α, β and n.


semisimple Lie groups unitary representations tensor products complementary series intertwining operators 


22E45 43A80 43A85 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by the Swedish Science Council (VR). The author dedicates this article to the memory of Professor Minde Cheng. The author met Cheng first time in the autumn of 1988 in Peking University, when he was then a graduate student in Fudan University with Professor Shaozong Yan, and Professor Lizhong Peng arranged him a visit to Peking University for 7–8 weeks. Yan had a great respect to Cheng; Yan instructed the author that this would be a very good chance to learn some hard classical harmonic analysis in Peking University, and he commissioned the author to hand over his personal letter to Cheng. In Peking University, the author attended the Harmonic Analysis Seminars there, with Professors Minde Cheng, Donggao Deng, Ruiling Long and Lizhong Peng and their graduate students. He was overwhelmed by the academic rigor and intensive research activities there. He did a talk on his work on Hankel operators, and to his surprise they asked him to give a second talk to present all technical details; their emphasis on details struck him. The author has benefited tremendously of the interaction with the harmonic analysis group in Peking University during the years to come, in particular with Peng. He takes this opportunity to thank them all, for their inspiration, encouragement, sharing their knowledge and practical material help. He thanks Jean-Louis Clerc for some stimulating discussions and careful reading of an earlier version of this paper, and Yurii Neretin for explaining some results in [20,21]. Thanks are also due to the referees for expert comments and criticisms.


  1. 1.
    Asmuth C, Repka J. Tensor products for SL 2(K), I: Complementary series and the special representation. Pacific J Math, 1981, 97: 271–282CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ben Saïd S, Koufany K, Zhang G. Invariant trilinear forms on spherical principal series of real-rank one semisimple Lie groups. Internat J Math, 2014, 25: 1450017CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Burger M, Li J-S, Sarnak P. Ramanujan duals and automorphic spectrum. Bull Amer Math Soc (NS), 1992, 26: 253–257CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Clerc J-L. Singular conformally invariant trilinear forms and covariant differential operators on the sphere. ArXiv: 1102.1861, 2011Google Scholar
  5. 5.
    Clerc J-L, Beckmann R. Singular conformally invariant trilinear forms and generalized Rankin-Cohen operators. ArXiv: 1104.3461, 2011Google Scholar
  6. 6.
    Clerc J-L, Kobayashi T, Ørsted B, et al. Generalized Bernstein-Reznikov integrals. Math Ann, 2011, 349: 395–431CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Clerc J-L, Ørsted B. Conformally invariant trilinear forms on the sphere. ArXiv:1001.2851, 2010zbMATHGoogle Scholar
  8. 8.
    Clozel L. Spectral theory of automorphic forms. In: Automorphic Forms and Applications. IAS/Park City Mathematics Series, vol. 12. Providence: Amer Math Soc, 2007, 43–93Google Scholar
  9. 9.
    Connes A, Moscovici H. Rankin-Cohen brackets and the Hopf algebra of transverse geometry. Mosc Math J, 2004, 4: 111–130zbMATHMathSciNetGoogle Scholar
  10. 10.
    Cowling M, Dooley A, Korányi A, et al. An approach to symmetric spaces of rank one via groups of Heisenberg type. J Geom Anal, 1998, 8: 199–237CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dijk G V, Hille S C. Canonical representations related to hyperbolic spaces. J Funct Anal, 1997, 147: 109–139CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Faraut J, Koranyi A. Function spaces and reproducing kernels on bounded symmetric domains. J Funct Anal, 1990, 88: 64–89CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Helgason S. Groups and Geometric Analysis. London-New York: Academic Press, 1984zbMATHGoogle Scholar
  14. 14.
    Johnson K D, Wallach N R. Composition series and intertwining operators for the spherical principal series. I. Trans Amer Math Soc, 1977, 229: 137–173CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Knapp A. Representation Theory of Semisimple Groups. Princeton: Princeton University Press, 1986CrossRefzbMATHGoogle Scholar
  16. 16.
    Kobayashi T, Pevzner M. Rankin-Cohen operators for symmetric pairs. ArXiv:1301.2111, 2013zbMATHGoogle Scholar
  17. 17.
    Molchanov V F. Canonical representations and overgroups. In: Lie groups and symmetric spaces. Advances in the Mathematical Sciences, vol. 54. Providence: Amer Math Soc, 2003, 213–224Google Scholar
  18. 18.
    Möllers J, Oshima Y. Restriction of most degenerate representations of O(1;N) with respect to symmetric pairs. J Math Sci Univ Tokyo, 2015, 22: 279–338zbMATHMathSciNetGoogle Scholar
  19. 19.
    Naimark M A. Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations. III. Tr Mosk Mat Obs, 1961, 10: 181–216MathSciNetGoogle Scholar
  20. 20.
    Neretin Y A. Plancherel formula for Berezin deformation of L 2 on Riemannian symmetric space. J Funct Anal, 2002, 189: 336–408CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Neretin Y A, Ol′shanskii G I. Boundary values of holomorphic functions, singular unitary representations of the groups O(p; q) and their limits as q → ∞. J Math Sci (NY), 1997, 87: 3983–4035CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ovsienko V, Redou P. Generalized transvectants-Rankin-Cohen brackets. Lett Math Phys, 2003, 63: 19–28CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Peng L, Zhang G. Tensor products of holomorphic representations and bilinear differential operators. J Funct Anal, 2004, 210: 171–192CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Pukánszky L. On the Kronecker products of irreducible representations of the 2 × 2 real unimodular group. I. Trans Amer Math Soc, 1961, 100: 116–152CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Repka J. Tensor products of unitary representations of SL 2(R). Amer J Math, 1978, 100: 930–932CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Repka J. Tensor products of holomorphic discrete series representations. Canada J Math, 1979, 31: 836–844CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Speh B, Venkataramana T N. Discrete components of some complementary series representations. Indian J Pure Appl Math, 2010, 41: 145–151CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Speh B, Zhang G. Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups. Math Z, 2016, 283: 629–647CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Vogan D A, Wallach N R. Intertwining operators for real reductive groups. Adv Math, 1990, 82: 203–243CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Zhang G. Discrete components in restriction of unitary representations of rank one semisimple Lie groups. J Funct Anal, 2015, 269: 3689–3713CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Mathematical SciencesChalmers University of TechnologyGöteborgSweden
  2. 2.Mathematical SciencesGöteborg UniversityGöteborgSweden

Personalised recommendations