Abstract
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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References
Bergeron, N.: Lefschetz properties for arithmetic real and complex hyperbolic manifolds. Int. Math. Res. Not. 20, 1089–1122 (2003)
Burger, M., Li, J.-S., Sarnak, P.: Ramanujan duals and automorphic spectrum. Bull. Am. Math. Soc. (N.S.) 26(2), 253–257 (1992)
Burger, M., Sarnak, P.: Ramanujan duals. II. Invent. Math. 106(1), 1–11 (1991)
Cowling, M., Dooley, A., Korányi, A., Ricci, F.: An approach to symmetric spaces of rank one via groups of Heisenberg type. J. Geom. Anal. 8(2), 199–237 (1998)
Engliš, M., Hille, S.C., Peetre, J., Rosengren, H., Zhang, G.: A new kind of Hankel type operators connected with the complementary series. Arab. J. Math. Sci. 6, 49–80 (2000)
Faraut, J., Koranyi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64–89 (1990)
Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford University Press, Oxford (1994)
Jacobsen, H., Vergne, M.: Restriction and expansions of holomorphic representations. J. Funct. Anal. 34, 29–53 (1979)
Johnson, K.D.: Composition series and intertwining operators for the spherical principal series. II. Trans. Am. Math. Soc. 215, 269–283 (1976)
Johnson, K.D., Wallach, N.R.: Composition series and intertwining operators for the spherical principal series. I. Trans. Am. Math. Soc. 229, 137–173 (1977)
Juhl, A.: Families of conformally covariant differential operators. In: \(Q\)-Curvature and Holography, Progress in Mathematics, vol. 275. Birkhäuser Verlag, Basel (2009)
Kobayashi, T.: Branching problems of unitary representations. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing), pp. 615–627. Higher Ed. Press (2002)
Kobayashi, T.: Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs. In: Representation Theory and Automorphic Forms, Progr. Math., vol. 255, pp. 45–109. Birkhäuser, Boston (2008)
Kobayashi, T.: F-method for construction equivariant differential operators. Geometric analysis and integral geometry. Contemporary Mathematic, vol. 598, pp. 139–146. American Mathematical Society, Providence, RI (2013)
Kostant, B.: On the existence and irreducibility of certain series of representations. Bull. Am. Math. Soc. 75, 627–642 (1969)
Möllers, J., Oshima, Y.: Restriction of complementary series representations of O(1, N) to symmetric subgroups. arXiv:1209.2312v3 [math.RT]
Mukunda, N.: Unitary representations of the Lorentz groups: reduction of the supplementary series under a noncompact subgroup. J. Math. Phys. 9, 417–431 (1968)
Neretin, Y.: Plancherel formula for Berezin deformation of \({L^2}\) on Riemannian symmetric space. J. Funct. Anal. 189(2), 336–408 (2002)
Rossi, H., Vergne, M.: Analytic continuation of the holomorphic discrete series of a semisimple Lie group. Acta Math. 136, 1–59 (1976)
Ørsted, B., Speh, B.: Branching laws for some unitary representations of SL(4,\({\mathbb{R}}\)). SIGMA 4 (2008)
Speh, B., Venkataramana, T.N.: Discrete components of some complementary series representations. Indian J. Pure Appl. Math. 41(1), 145–151 (2010)
van Dijk, G., Hille, S.C.: Canonical representations related to hyperbolic spaces. J. Funct. Anal. 147, 109–139 (1997)
Vershik, A.M., Graev, M.I.: The structure of complementary series and the spherical representations of the group \( o(n, 1)\) and \(u(n, 1)\). Preprint. arXiv:math.RT/0610215
Wallach, N.: The analytic continuation of the discrete series, I. II. Trans. Am. Math. Soc. 251(1–17), 19–37 (1979)
Zhang, G.: Discrete components in restriction of unitary representations of rank one semisimple lie groups. J. Funct. Anal. 269(12), 3689–3713 (2015)
Zhang, G.: Berezin transform on real bounded symmetric domains. Trans. Am. Math. Soc. 353, 3769–3787 (2001)
Zhang, G.: Branching coefficients of holomorphic representations and Segal-Bargmann transform. J. Funct. Anal. 195, 306–349 (2002)
Zhang, G.: Degenerate principal series representations and their holomorphic extensions. Adv. Math. 223(5), 1495–1520 (2010)
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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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DOI: https://doi.org/10.1007/s00209-016-1614-0