×

Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups. (English) Zbl 1341.22012

The authors investigate representations \(\pi_\mu\) of rank one Lie groups \(H = \mathrm{SO}_0(n,1;\mathbb F)\) for \(\mathbb F = \mathbb R, \mathbb C, \mathbb H\) and their restrictions to the subgroups \(H_1 = \mathrm{SO}_0(n - 1,1;\mathbb F)\). In general \(\mu \in \mathbb C\), but in this paper only complementary series for which \(\mu \in \mathbb R\) are considered. Such representations are unitarizable. It is proved that for certain parameters \(\mu\) the restricted representation of \(H\) has a direct summand which is isomorphic to a complementary series representation of the smaller group \(H_1\); similar results are also proved for the exceptional Lie groups \(H = F_{4(-20)}\) and \(H_1 = \mathrm{Spin}(8,1)\). The proofs use the imbedding of the complementary series into the analytic continuation of the holomorphic discrete series of the Lie groups \(G = \mathrm{SU}(n,1), \mathrm{SU}(n,1) \times \mathrm{SU}(n,1)\) and \( \mathrm{SU}(2n,2)\) and consider the branching of holomorphic representations.

MSC:

22E46 Semisimple Lie groups and their representations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bergeron, N.: Lefschetz properties for arithmetic real and complex hyperbolic manifolds. Int. Math. Res. Not. 20, 1089-1122 (2003) · Zbl 1036.11021 · doi:10.1155/S1073792803212253
[2] Burger, M., Li, J.-S., Sarnak, P.: Ramanujan duals and automorphic spectrum. Bull. Am. Math. Soc. (N.S.) 26(2), 253-257 (1992) · Zbl 0762.22009
[3] Burger, M., Sarnak, P.: Ramanujan duals. II. Invent. Math. 106(1), 1-11 (1991) · Zbl 0774.11021 · doi:10.1007/BF01243900
[4] 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) · Zbl 0966.53039 · doi:10.1007/BF02921641
[5] 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) · Zbl 0990.47023
[6] Faraut, J., Koranyi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64-89 (1990) · Zbl 0718.32026 · doi:10.1016/0022-1236(90)90119-6
[7] Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford University Press, Oxford (1994) · Zbl 0841.43002
[8] Jacobsen, H., Vergne, M.: Restriction and expansions of holomorphic representations. J. Funct. Anal. 34, 29-53 (1979) · Zbl 0433.22011 · doi:10.1016/0022-1236(79)90023-5
[9] Johnson, K.D.: Composition series and intertwining operators for the spherical principal series. II. Trans. Am. Math. Soc. 215, 269-283 (1976) · Zbl 0295.22016 · doi:10.1090/S0002-9947-1976-0385012-X
[10] 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) · Zbl 0349.43010 · doi:10.1090/S0002-9947-1977-0447483-0
[11] Juhl, A.: Families of conformally covariant differential operators. \[In: Q\] Q-Curvature and Holography, Progress in Mathematics, vol. 275. Birkhäuser Verlag, Basel (2009) · Zbl 1177.53001
[12] 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) · Zbl 1008.43009
[13] 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) · Zbl 1304.22013
[14] 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) · Zbl 1290.22008
[15] Kostant, B.: On the existence and irreducibility of certain series of representations. Bull. Am. Math. Soc. 75, 627-642 (1969) · Zbl 0229.22026 · doi:10.1090/S0002-9904-1969-12235-4
[16] Möllers, J., Oshima, Y.: Restriction of complementary series representations of O(1, N) to symmetric subgroups. arXiv:1209.2312v3 [math.RT] · Zbl 1333.22009
[17] Mukunda, N.: Unitary representations of the Lorentz groups: reduction of the supplementary series under a noncompact subgroup. J. Math. Phys. 9, 417-431 (1968) · Zbl 0159.29301 · doi:10.1063/1.1664595
[18] Neretin, Y.: Plancherel formula for Berezin deformation of \[{L^2}\] L2 on Riemannian symmetric space. J. Funct. Anal. 189(2), 336-408 (2002) · Zbl 1012.43005 · doi:10.1006/jfan.2000.3691
[19] Rossi, H., Vergne, M.: Analytic continuation of the holomorphic discrete series of a semisimple Lie group. Acta Math. 136, 1-59 (1976) · Zbl 0356.32020 · doi:10.1007/BF02392042
[20] Ørsted, B., Speh, B.: Branching laws for some unitary representations of SL \[(4,{\mathbb{R}}\] R). SIGMA 4 (2008) · Zbl 1135.22014
[21] Speh, B., Venkataramana, T.N.: Discrete components of some complementary series representations. Indian J. Pure Appl. Math. 41(1), 145-151 (2010) · Zbl 1197.22008 · doi:10.1007/s13226-010-0020-2
[22] van Dijk, G., Hille, S.C.: Canonical representations related to hyperbolic spaces. J. Funct. Anal. 147, 109-139 (1997) · Zbl 0882.22017 · doi:10.1006/jfan.1996.3057
[23] Vershik, A.M., Graev, M.I.: The structure of complementary series and the spherical representations of the group \[o(n, 1)\] o(n,1) and \[u(n, 1)\] u(n,1). Preprint. arXiv:math.RT/0610215 · Zbl 1148.22017
[24] Wallach, N.: The analytic continuation of the discrete series, I. II. Trans. Am. Math. Soc. 251(1-17), 19-37 (1979) · Zbl 0419.22018 · doi:10.1090/S0002-9947-79-99965-3
[25] Zhang, G.: Discrete components in restriction of unitary representations of rank one semisimple lie groups. J. Funct. Anal. 269(12), 3689-3713 (2015) · Zbl 1345.22009 · doi:10.1016/j.jfa.2015.09.021
[26] Zhang, G.: Berezin transform on real bounded symmetric domains. Trans. Am. Math. Soc. 353, 3769-3787 (2001) · Zbl 0965.22015 · doi:10.1090/S0002-9947-01-02832-X
[27] Zhang, G.: Branching coefficients of holomorphic representations and Segal-Bargmann transform. J. Funct. Anal. 195, 306-349 (2002) · Zbl 1019.22006 · doi:10.1006/jfan.2002.3957
[28] Zhang, G.: Degenerate principal series representations and their holomorphic extensions. Adv. Math. 223(5), 1495-1520 (2010) · Zbl 1242.22022 · doi:10.1016/j.aim.2009.09.014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.