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

SIGMA 13 (2017), 026, 18 pages      arXiv:1612.01856      https://doi.org/10.3842/SIGMA.2017.026

### Another Approach to Juhl's Conformally Covariant Differential Operators from $S^n$ to $S^{n-1}$

Jean-Louis Clerc
Institut Elie Cartan de Lorraine, Université de Lorraine, France

Received December 07, 2016, in final form April 11, 2017; Published online April 19, 2017

Abstract
A family $({\mathbf D}_\lambda)_{\lambda\in \mathbb C}$ of differential operators on the sphere $S^n$ is constructed. The operators are conformally covariant for the action of the subgroup of conformal transformations of $S^n$ which preserve the smaller sphere $S^{n-1}\subset S^n$. The family of conformally covariant differential operators from $S^n$ to $S^{n-1}$ introduced by A. Juhl is obtained by composing these operators on $S^n$ and taking restrictions to $S^{n-1}$.

Key words: conformally covariant differential operators; Juhl's covariant differential operators.

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References

1. Beckmann R., Clerc J.-L., Singular invariant trilinear forms and covariant (bi-)differential operators under the conformal group, J. Funct. Anal. 262 (2012), 4341-4376.
2. Clerc J.-L., Covariant bi-differential operators on matrix space, Ann. Inst. Fourier (Grenoble), to appear, arXiv:1601.07016.
3. Eelbode D., Souček V., Conformally invariant powers of the Dirac operator in Clifford analysis, Math. Methods Appl. Sci. 33 (2010), 1558-1570.
4. Fischmann M., Juhl A., Somberg P., Conformal symmetry breaking differential operators on differential forms, arXiv:1605.04517.
5. Gel'fand I.M., Shilov G.E., Generalized functions. Vol. I: Properties and operations, Academic Press, New York - London, 1964.
6. Juhl A., Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, Vol. 275, Birkhäuser Verlag, Basel, 2009.
7. Knapp A.W., Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, Vol. 36, Princeton University Press, Princeton, NJ, 1986.
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., Pevzner M., Differential symmetry breaking operators: I. General theory and F-method, Selecta Math. (N.S.) 22 (2016), 801-845, arXiv:1301.2111.
10. Kobayashi T., Pevzner M., Differential symmetry breaking operators: II. Rankin-Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22 (2016), 847-911, arXiv:1301.2111.
11. 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.
12. Olver P.J., Classical invariant theory, London Mathematical Society Student Texts, Vol. 44, Cambridge University Press, Cambridge, 1999.