Abstract
So far we have been working with symmetry breaking for a pair of the orthogonal groups (O(n + 1, 1), O(n, 1)). On the other hand, the Gross–Prasad conjectures (Chapters 11 and 13) are formulated for special orthogonal groups rather than orthogonal groups. In this chapter, we explain how to translate the results for (G, G′) = (O(n + 1, 1), O(n, 1)) to those for the pair of special orthogonal groups \((\overline {G},\overline {G'})=(SO(n+1,1), SO(n,1))\).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
A. Borel, N.R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups: Second edition. Math. Surveys Monogr., vol. 67 (Amer. Math. Soc., Providence, RI, 2000), xviii+260 pp.; First ed., Ann. of Math. Stud., vol. 94 (Princeton Univ. Press, Princeton, NJ, 1980), xvii+388 pp.
D.H. Collingwood, Representations of Rank One Lie Groups. Res. Notes in Math., vol. 137 (Pitman (Advanced Publishing Program), Boston, MA, 1985), vii+244 pp.
T. Hirai, On irreducible representations of the Lorentz group of n-th order. Proc. Jpn. Acad. 38, 258–262 (1962). https://projecteuclid.org/euclid.pja/1195523378
A.W. Knapp, G.J. Zuckerman, Classification of irreducible tempered representations of semisimple groups. Ann. Math. (2) 116, 389–455 (1982); II, ibid, 457–501
T. Kobayashi, T. Kubo, M. Pevzner, Conformal Symmetry Breaking Operators for Differential Forms on Spheres. Lecture Notes in Math., vol. 2170 (Springer, 2016), iv+192 pp. ISBN: 978-981-10-2657-7. http://dx.doi.org/10.1007/978-981-10-2657-7
T. Kobayashi, T. Oshima, Finite multiplicity theorems for induction and restriction. Adv. Math. 248, 921–944 (2013). http://dx.doi.org/10.1016/j.aim.2013.07.015
T. Kobayashi, B. Speh, Symmetry Breaking for Representations of Rank One Orthogonal Groups. Mem. Amer. Math. Soc., vol. 238 (Amer. Math. Soc., Providence, RI, 2015), v+112 pp. ISBN: 978-1-4704-1922-6. http://dx.doi.org/10.1090/memo/1126
T. Kobayashi, B. Speh, Symmetry breaking for orthogonal groups and a conjecture by B. Gross and D. Prasad, in Geometric Aspects of the Trace Formula, Simons Symposia, ed. by W. Müller et al. (Springer, 2018), pp. 245–266. https://doi.org/10.1007/978-3-319-94833-1_8. Available also at arXiv:1702.00263. http://arxiv.org/abs/1702.00263
R.P. Langlands, On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups. Math. Surveys Monogr., vol. 31 (Amer. Math. Soc., Providence, RI, 1989), pp. 101–170
B. Sun, C.-B. Zhu, Multiplicity one theorems: the Archimedean case. Ann. Math. (2) 175, 23–44 (2012). http://dx.doi.org/10.4007/annals.2012.175.1.2
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Kobayashi, T., Speh, B. (2018). Appendix II: Restriction to \(\overline G=SO(n+1,1)\) . In: Symmetry Breaking for Representations of Rank One Orthogonal Groups II. Lecture Notes in Mathematics, vol 2234. Springer, Singapore. https://doi.org/10.1007/978-981-13-2901-2_15
Download citation
DOI: https://doi.org/10.1007/978-981-13-2901-2_15
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2900-5
Online ISBN: 978-981-13-2901-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)