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Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups

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We consider the spherical complementary series of rank one Lie groups \(H_n={ SO }_0(n, 1; {\mathbb {F}})\) for \({\mathbb {F}}={\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}\). We prove that there exist finitely many discrete components in its restriction under the subgroup \(H_{n-1}={ SO }_0(n-1, 1; {\mathbb {F}})\). This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of \(G_n=SU(n, 1)\), \(SU(n, 1)\times SU(n, 1)\) and SU(2n, 2) and by the branching of holomorphic representations under the corresponding subgroup \(G_{n-1}\).

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Correspondence to B. Speh.

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Research by B. Speh partially supported by NSF Grant DMS-0901024 and research by G. Zhang partially supported by the Swedish Science Council (VR).

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Speh, B., Zhang, G. Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups. Math. Z. 283, 629–647 (2016). https://doi.org/10.1007/s00209-016-1614-0

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