AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Symmetry breaking for representations of rank one orthogonal groups
About this Title
Toshiyuki Kobayashi, Kavli IPMU (WPI), Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914 Japan and Birgit Speh, Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 238, Number 1126
ISBNs: 978-1-4704-1922-6 (print); 978-1-4704-2615-6 (online)
DOI: https://doi.org/10.1090/memo/1126
Published electronically: May 12, 2015
Keywords: Branching law,
reductive Lie group,
symmetry breaking,
Lorentz group,
conformal geometry,
Verma module,
complementary series.
MSC: Primary 22E46; Secondary 33C45, 53C35
Table of Contents
Chapters
- 1. Introduction
- 2. Symmetry breaking for the spherical principal series representations
- 3. Symmetry breaking operators
- 4. More about principal series representations
- 5. Double coset decomposition $P’ \backslash G/P$
- 6. Differential equations satisfied by the distribution kernels of symmetry breaking operators
- 7. $K$-finite vectors and regular symmetry breaking operators $\widetilde {\mathbb {A}} _{\lambda , \nu }$
- 8. Meromorphic continuation of regular symmetry breaking operators ${K}_{{\lambda },{\nu }}^{\mathbb {A}}$
- 9. Singular symmetry breaking operator $\widetilde {\mathbb {B}}_{\lambda ,\nu }$
- 10. Differential symmetry breaking operators
- 11. Classification of symmetry breaking operators
- 12. Residue formulae and functional identities
- 13. Image of symmetry breaking operators
- 14. Application to analysis on anti-de Sitter space
- 15. Application to branching laws of complementary series
- 16. Appendix
Abstract
We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of $G=O(n+1,1)$ and $G’=O(n,1)$. We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explicitly. Symmetry breaking operators at exceptional discrete parameters are thoroughly studied.
We obtain closed formulae for the functional equations which the composition of the symmetry breaking operators with the Knapp–Stein intertwining operators of $G$ and $G’$ satisfy, and use them to determine the symmetry breaking operators between irreducible composition factors of the spherical principal series representations of $G$ and $G’$. Some applications are included.
- Jean-Louis Clerc, Toshiyuki Kobayashi, Bent Ørsted, and Michael Pevzner, Generalized Bernstein-Reznikov integrals, Math. Ann. 349 (2011), no. 2, 395–431. MR 2753827, DOI 10.1007/s00208-010-0516-4
- Michael G. Eastwood and C. Robin Graham, Invariants of conformal densities, Duke Math. J. 63 (1991), no. 3, 633–671. MR 1121149, DOI 10.1215/S0012-7094-91-06327-1
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0065685
- J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl. (9) 58 (1979), no. 4, 369–444 (French). MR 566654
- I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5: Integral geometry and representation theory, Academic Press, New York-London, 1966. Translated from the Russian by Eugene Saletan. MR 0207913
- I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
- Benedict H. Gross and Dipendra Prasad, On the decomposition of a representation of $\textrm {SO}_n$ when restricted to $\textrm {SO}_{n-1}$, Canad. J. Math. 44 (1992), no. 5, 974–1002. MR 1186476, DOI 10.4153/CJM-1992-060-8
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
- Michael Harris and Hans Plesner Jakobsen, Singular holomorphic representations and singular modular forms, Math. Ann. 259 (1982), no. 2, 227–244. MR 656663, DOI 10.1007/BF01457310
- Sigurdur Helgason, Geometric analysis on symmetric spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 2008. MR 2463854
- Joachim Hilgert, Toshiyuki Kobayashi, and Jan Möllers, Minimal representations via Bessel operators, J. Math. Soc. Japan 66 (2014), no. 2, 349–414. MR 3201818, DOI 10.2969/jmsj/06620349
- Kenneth D. Johnson and Nolan R. Wallach, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc. 229 (1977), 137–173. MR 447483, DOI 10.1090/S0002-9947-1977-0447483-0
- Andreas Juhl, Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, vol. 275, Birkhäuser Verlag, Basel, 2009. MR 2521913
- Masaki Kashiwara, Takahiro Kawai, and Tatsuo Kimura, Foundations of algebraic analysis, Princeton Mathematical Series, vol. 37, Princeton University Press, Princeton, NJ, 1986. Translated from the Japanese by Goro Kato. MR 855641
- A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
- Toshiyuki Kobayashi, Discrete decomposability of the restriction of $A_{\mathfrak {q}}(\lambda )$ with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), no. 2, 229–256. MR 1608642, DOI 10.1007/s002220050203
- Toshiyuki Kobayashi, $F$-method for constructing equivariant differential operators, Geometric analysis and integral geometry, Contemp. Math., vol. 598, Amer. Math. Soc., Providence, RI, 2013, pp. 139–146. MR 3156443, DOI 10.1090/conm/598/11998
- Toshiyuki Kobayashi, F-method for symmetry breaking operators, Differential Geom. Appl. 33 (2014), no. suppl., 272–289. MR 3159963, DOI 10.1016/j.difgeo.2013.10.003
- Toshiyuki Kobayashi and Gen Mano, The Schrödinger model for the minimal representation of the indefinite orthogonal group $\textrm {O}(p,q)$, Mem. Amer. Math. Soc. 213 (2011), no. 1000, vi+132. MR 2858535, DOI 10.1090/S0065-9266-2011-00592-7
- T. Kobayashi and T. Matsuki, Classification of finite-multiplicity symmetric pairs, Transform. Groups 19 (2014), no. 2, 457–493. MR 3200432, DOI 10.1007/s00031-014-9265-x
- Toshiyuki Kobayashi and Bent Ørsted, Analysis on the minimal representation of $\mathrm O(p,q)$. I. Realization via conformal geometry, Adv. Math. 180 (2003), no. 2, 486–512. MR 2020550, DOI 10.1016/S0001-8708(03)00012-4
- Toshiyuki Kobayashi and Bent Ørsted, Analysis on the minimal representation of $\mathrm O(p,q)$. II. Branching laws, Adv. Math. 180 (2003), no. 2, 513–550. MR 2020551, DOI 10.1016/S0001-8708(03)00013-6
- Toshiyuki Kobayashi and Bent Ørsted, Analysis on the minimal representation of $\mathrm O(p,q)$. III. Ultrahyperbolic equations on ${\Bbb R}^{p-1,q-1}$, Adv. Math. 180 (2003), no. 2, 551–595. MR 2020552, DOI 10.1016/S0001-8708(03)00014-8
- T. Kobayashi, B. Ørsted, P. Somberg, V. Souček, Branching laws for Verma modules and applications in parabolic geometry, Part I, preprint, http://arxiv.org/abs/1305.6040 arXiv:1305.6040.
- Toshiyuki Kobayashi and Toshio Oshima, Finite multiplicity theorems for induction and restriction, Adv. Math. 248 (2013), 921–944. MR 3107532, DOI 10.1016/j.aim.2013.07.015
- T. Kobayashi, M. Pevzner, Rankin–Cohen operators for symmetric pairs, preprint, 53pp. http://arxiv.org/abs/1301.2111arXiv:1301.2111.
- Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
- Manfred Krämer, Multiplicity free subgroups of compact connected Lie groups, Arch. Math. (Basel) 27 (1976), no. 1, 28–36. MR 399373, DOI 10.1007/BF01224637
- Hung Yean Loke, Trilinear forms of $\mathfrak {gl}_2$, Pacific J. Math. 197 (2001), no. 1, 119–144. MR 1810211, DOI 10.2140/pjm.2001.197.119
- Toshio Ōshima, Poisson transformations on affine symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 9, 323–327. MR 555057
- Y. Sakellaridis, A. Venkatesh Periods and harmonic analysis on spherical varieties, http://arxiv.org/abs/1203.0039 arXiv:1203.0039.
- Henrik Schlichtkrull, Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms, J. Funct. Anal. 70 (1987), no. 1, 194–219. MR 870761, DOI 10.1016/0022-1236(87)90130-3
- Birgit Speh and T. N. Venkataramana, Discrete components of some complementary series, Forum Math. 23 (2011), no. 6, 1159–1187. MR 2855046, DOI 10.1515/FORM.2011.042
- Robert S. Strichartz, Harmonic analysis on hyperboloids, J. Functional Analysis 12 (1973), 341–383. MR 0352884, DOI 10.1016/0022-1236(73)90001-3
- Binyong Sun and Chen-Bo Zhu, Multiplicity one theorems: the Archimedean case, Ann. of Math. (2) 175 (2012), no. 1, 23–44. MR 2874638, DOI 10.4007/annals.2012.175.1.2
- A. M. Vershik and M. I. Graev, The structure of complementary series and special representations of the groups $\textrm {O}(n,1)$ and $\textrm {U}(n,1)$, Uspekhi Mat. Nauk 61 (2006), no. 5(371), 3–88 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 5, 799–884. MR 2328257, DOI 10.1070/RM2006v061n05ABEH004356
- David A. Vogan Jr., Unitary representations of reductive Lie groups, Annals of Mathematics Studies, vol. 118, Princeton University Press, Princeton, NJ, 1987. MR 908078
- David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51–90. MR 762307
- Nolan R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1992. MR 1170566
- Joseph A. Wolf, Spaces of constant curvature, 6th ed., AMS Chelsea Publishing, Providence, RI, 2011. MR 2742530