Intertwining operators and the restriction of representations of rank-one orthogonal groups

Kobayashi, Toshiyuki; Speh, Birgit

doi:10.1016/j.crma.2013.11.018

Lie algebras/Harmonic analysis

Intertwining operators and the restriction of representations of rank-one orthogonal groups
[Opérateurs dʼentrelacement et restriction des représentations des groupes orthogonaux de rang un]

Présenté par : Michel Duflo

Kobayashi, Toshiyuki ¹ ; Speh, Birgit ²

¹ Kavli IPMU (WPI), Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914, Japan
² Department of Mathematics, Cornell University, Ithaca, 14853-4201, NY, USA

Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 89-94.

Résumé
Abstract

Nous donnons une classification complète des opérateurs dʼentrelacement (opérateurs de brisure de symétrie) entre les représentations des séries principales sphériques de $O (n + 1, 1)$ et de $O (n, 1)$ ainsi que des formules explicites pour les noyaux de Schwartz de ces opérateurs. Par la suite, nous déterminons les opérateurs de brisure de symétrie entre les facteurs irréductibles des séries de composition correspondantes.

We give a complete classification of intertwining operators (breaking symmetry operators) between spherical principal series representations of $O (n + 1, 1)$ and $O (n, 1)$ together with explicit formulae of the distribution kernels. Further we use this to determine the breaking symmetry operators between their irreducible composition factors.

Reçu le : 2013-08-02
Accepté le : 2013-11-26
Publié le : 2014-01-02

DOI : 10.1016/j.crma.2013.11.018

Affiliations des auteurs :

Kobayashi, Toshiyuki ¹ ; Speh, Birgit ²

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Kobayashi, Toshiyuki; Speh, Birgit. Intertwining operators and the restriction of representations of rank-one orthogonal groups. Comptes Rendus. Mathématique, Tome 352 (2014) no. 2, pp. 89-94. doi : 10.1016/j.crma.2013.11.018. https://www.numdam.org/articles/10.1016/j.crma.2013.11.018/

Bibliographie
Cité par

[1] Gross, B.; Prasad, D. On the decomposition of a representations of ${SO}_{n}$ when restricted to ${SO}_{n - 1}$ , Can. J. Math., Volume 44 (1992), pp. 974-1002

[2] Juhl, A. Families of Conformally Covariant Differential Operators, Q-curvature and Holography, Progress in Math., vol. 275, Birkhäuser, 2009 (xiv+488 p)

[3] Kobayashi, T. Discrete decomposability of the restriction of $A_{q} (λ)$ with respect to reductive subgroups III – restriction of Harish-Chandra modules and associated varieties, Invent. Math., Volume 131 (1998), pp. 229-256

[4] Kobayashi, T. F-method for constructing equivariant differential operators, Contemp. Math., vol. 598, Amer. Math. Soc., 2013, pp. 141-148

[5] Kobayashi, T. F-method for symmetry breaking operators, Differ. Geom. Appl. (2013) (available at) | arXiv | DOI

[6] Kobayashi, T.; Oshima, T. Finite multiplicity theorems for induction and restriction, Adv. Math., Volume 248 (2013), pp. 921-944 (available at) | arXiv

[7] Loke, H.Y. Trilinear forms on ${gl}_{2}$ , Pac. J. Math., Volume 197 (2001), pp. 119-144

[8] Sun, B.; Zhu, C.-B. Multiplicity one theorems: the Archimedean case, Ann. Math., Volume 175 (2012), pp. 23-44

[9] Wallach, N. Real Reductive Groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, 1992 (xiv+454 p)

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