Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 084, 50 pages      arXiv:1702.02326      https://doi.org/10.3842/SIGMA.2019.084

Knapp-Stein Type Intertwining Operators for Symmetric Pairs II. - The Translation Principle and Intertwining Operators for Spinors

Jan Frahm and Bent Ørsted
Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark

Received May 17, 2019, in final form October 29, 2019; Published online November 02, 2019

Abstract
For a symmetric pair $(G,H)$ of reductive groups we extend to a large class of generalized principal series representations our previous construction of meromorphic families of symmetry breaking operators. These operators intertwine between a possibly vector-valued principal series of $G$ and one for $H$ and are given explicitly in terms of their integral kernels. As an application we give a complete classification of symmetry breaking operators from spinors on a Euclidean space to spinors on a hyperplane, intertwining for a double cover of the conformal group of the hyperplane.

Key words: Knapp-Stein intertwiners; intertwining operators; symmetry breaking operators; symmetric pairs; principal series; translation principle.

pdf (670 kb)   tex (49 kb)  

References

1. Branson T., Ólafsson G., Ørsted B., Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup, J. Funct. Anal. 135 (1996), 163-205.
2. Clerc J.L., Ørsted B., Conformal covariance for the powers of the Dirac operator, J. Lie Theory, to appear, arXiv:1409.4983.
3. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions. Vol. II, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.
4. Fischmann M., Juhl A., Somberg P., Conformal symmetry breaking differential operators on differential forms, Mem. Amer. Math. Soc., to appear, arXiv:1605.04517.
5. Juhl A., Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, Vol. 275, Birkhäuser Verlag, Basel, 2009.
6. Knapp A.W., Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, Vol. 36, Princeton University Press, Princeton, NJ, 1986.
7. Kobayashi T., A program for branching problems in the representation theory of real reductive groups, in Representations of reductive groups, Progr. Math., Vol. 312, Birkhäuser/Springer, Cham, 2015, 277-322, arXiv:1509.08861.
8. Kobayashi T., Kubo T., Pevzner M., Conformal symmetry breaking operators for differential forms on spheres, Lecture Notes in Math., Vol. 2170, Springer, Singapore, 2016.
9. Kobayashi T., Ørsted B., Somberg P., Souček V., Branching laws for Verma modules and applications in parabolic geometry. I, Adv. Math. 285 (2015), 1796-1852, arXiv:1305.6040.
10. Kobayashi T., Speh B., Symmetry breaking for representations of rank one orthogonal groups, Mem. Amer. Math. Soc. 238 (2015), v+110 pages, arXiv:1310.3213.
11. Kobayashi T., Speh B., Symmetry breaking for representations of rank one orthogonal groups II, Lecture Notes in Math., Vol. 2234, Springer, Singapore, 2018, arXiv:1801.00158.
12. Möllers J., Ørsted B., Estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles, Int. Math. Res. Not. 2017 (2017), 3209-3236, arXiv:1308.0298.
13. Möllers J., Ørsted B., The compact picture of symmetry breaking operators for rank one orthogonal and unitary groups, Pacific J. Math., to appear, arXiv:1404.1171.
14. Möllers J., Ørsted B., Oshima Y., Knapp-Stein type intertwining operators for symmetric pairs, Adv. Math. 294 (2016), 256-306, arXiv:1309.3904.
15. Möllers J., Ørsted B., Zhang G., On boundary value problems for some conformally invariant differential operators, Comm. Partial Differential Equations 41 (2016), 609-643, arXiv:1503.01695.
16. Wallach N.R., Cyclic vectors and irreducibility for principal series representations, Trans. Amer. Math. Soc. 158 (1971), 107-113.