Proceedings of the Japan Academy, Series A, Mathematical Sciences

Symmetry breaking operators for the restriction of representations of indefinite orthogonal groups $O(p,q)$

Toshiyuki Kobayashi and Alex Leontiev

Full-text: Open access

Abstract

For the pair $(G, G') =(O(p+1, q+1), O(p,q+1))$, we construct and give a complete classification of intertwining operators (symmetry breaking operators) between most degenerate spherical principal series representations of $G$ and those of the subgroup $G'$, extending the work initiated by Kobayashi and Speh [Mem. Amer. Math. Soc. 2015] in the real rank one case where $q=0$. Functional identities and residue formulæ of the regular symmetry breaking operators are also provided explicitly. The results contribute to Program C of branching problems suggested by the first author [Progr. Math. 2015].

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 93, Number 8 (2017), 86-91.

Dates
First available in Project Euclid: 3 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.pja/1506996023

Digital Object Identifier
doi:10.3792/pjaa.93.86

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 53C35: Symmetric spaces [See also 32M15, 57T15]

Keywords
Representation theory reductive group branching law broken symmetry conformal geometry symmetry breaking operator

Citation

Kobayashi, Toshiyuki; Leontiev, Alex. Symmetry breaking operators for the restriction of representations of indefinite orthogonal groups $O(p,q)$. Proc. Japan Acad. Ser. A Math. Sci. 93 (2017), no. 8, 86--91. doi:10.3792/pjaa.93.86. https://projecteuclid.org/euclid.pja/1506996023


Export citation

References

  • J. Bernstein and A. Reznikov, Estimates of automorphic functions, Mosc. Math. J. 4 (2004), no. 1, 19–37.
  • J.-L. Clerc, T. Kobayashi, B. Ørsted, and M. Pevzner, Generalized Bernstein-Reznikov integrals, Math. Ann. 349 (2011), no. 2, 395–431.
  • R. E. Howe and E.-C. Tan, Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 1, 1–74.
  • A. Juhl, Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, 275, Birkhäuser Verlag, Basel, 2009.
  • T. Kobayashi, Discrete decomposability of the restriction of $A_{q}(\lambda)$ with respect to reductive subgroups. II. Micro-local analysis and asymptotic $K$-support, Ann. of Math. (2) 147 (1998), no. 3, 709–729.
  • T. Kobayashi, Discrete decomposability of the restriction of $A_{q}(\lambda)$ with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), no. 2, 229–256.
  • T. Kobayashi, Shintani functions, real spherical manifolds, and symmetry breaking operators, in Developments and retrospectives in Lie theory, Dev. Math., 37, Springer, Cham, 2014, pp. 127–159.
  • T. Kobayashi, A program for branching problems in the representation theory of real reductive groups, in Representations of reductive groups: in honor of the 60th birthday of D. Vogan (MIT, 2014), 277–322, Progr. Math., 312, Birkhäuser/Springer, Cham, 2015.
  • T. Kobayashi, T. Kubo and M. Pevzner, Conformal symmetry breaking operators for differential forms on spheres, Lecture Notes in Mathematics, 2170, Springer, Singapore, 2016.
  • T. Kobayashi, T. Kubo and M. Pevzner, Conformal symmetry breaking operators for anti-de Sitter spaces, arXiv:1610.09475. (to appear in Trends Math.).
  • T. Kobayashi and T. Matsuki, Classification of finite-multiplicity symmetric pairs, Transform. Groups 19 (2014), no. 2, 457–493. (In special issue in honour of Professor Dynkin for his 90th birthday).
  • T. Kobayashi and B. Ørsted, Analysis on the minimal representation of $\mathrm{O}(p,q)$. I. Realization via conformal geometry, Adv. Math. 180 (2003), no. 2, 486–512.
  • T. Kobayashi, B. Ørsted, P. Somberg and V. Souček, Branching laws for Verma modules and applications in parabolic geometry. I, Adv. Math. 285 (2015), 1796–1852.
  • T. Kobayashi and T. Oshima, Finite multiplicity theorems for induction and restriction, Adv. Math. 248 (2013), 921–944.
  • T. Kobayashi and M. Pevzner, Differential symmetry breaking operators: I. General theory and F-method, Selecta Math. (N.S.) 22 (2016), no. 2, 801–845.
  • T. Kobayashi and B. Speh, Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc. 238 (2015), no. 1126, v+110 pp.
  • N. R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, 132-II, Academic Press, Inc., Boston, MA, 1992.