Tensor products of complementary series of rank one Lie groups
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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.
Keywordssemisimple Lie groups unitary representations tensor products complementary series intertwining operators
MSC(2010)22E45 43A80 43A85
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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.
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