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

SIGMA 7 (2011), 003, 7 pages      arXiv:1007.5069      https://doi.org/10.3842/SIGMA.2011.003

Intertwinors on Functions over the Product of Spheres

Doojin Hong
Department of Mathematics, University of North Dakota, Grand Forks ND 58202, USA

Received August 23, 2010, in final form December 30, 2010; Published online January 06, 2011

Abstract
We give explicit formulas for the intertwinors on the scalar functions over the product of spheres with the natural pseudo-Riemannian product metric using the spectrum generating technique. As a consequence, this provides another proof of the even order conformally invariant differential operator formulas obtained earlier by T. Branson and the present author.

Key words: intertwinors; conformally invariant operators.

pdf (298 Kb)   tex (10 Kb)

References

  1. Branson T., Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), 199-291.
  2. Branson T., Harmonic analysis in vector bundles associated to the rotation and spin groups, J. Funct. Anal. 106 (1992), 314-328.
  3. Branson T., Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995), 3671-3742.
  4. Branson T., Nonlinear phenomena in the spectral theory of geometric linear differential operators, in Quantization, Nonlinear Partial Differential Equations, and Operator Algebra (Cambridge, MA, 1994), Proc. Sympos. Pure Math., Vol. 59, Amer. Math. Soc., Providence, RI, 1996, 27-65.
  5. Branson T., Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997), 334-383.
  6. Branson T.P., Hong D., Spectrum generating on twistor bundle, Arch. Math. (Brno) 42 (2006), suppl., 169-183, math.DG/0606524.
  7. Branson T., Hong D., Translation to bundle operators, SIGMA 3 (2007), 102, 14 pages, math.DG/0606552.
  8. Branson T., Ólafsson G., Ørsted B., Spectrum generating operators, and intertwining operators for representations induced from a maximal parabolic subgroups, J. Funct. Anal. 135 (1996), 163-205.
  9. Hong D., Eigenvalues of Dirac and Rarita-Schwinger operators, in Clifford Algebras (Cookeville, TN, 2002), Prog. Math. Phys., Vol. 34, Birkhäuser Boston, Boston, MA, 2004, 201-210.
  10. Hong D., Spectra of higher spin operators, Ph.D. Thesis, University of Iowa, 2004.
  11. Ikeda A., Taniguchi Y., Spectra and eigenforms of the Laplacian on Sn and Pn(C), Osaka J. Math. 15 (1978), 515-546.
  12. Kobayashi T., Ørsted B., Analysis on the minimal representation of O(p,q). I. Realization via conformal geometry, Adv. Math. 180 (2003), 486-512, math.RT/0111083.

Previous article   Next article   Contents of Volume 7 (2011)