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Harmonic Spinors on Hyperbolic Space

Published online by Cambridge University Press:  20 November 2018

Pierre-Yves Gaillard
Pierre-Yves Gaillard*
Affiliation:
Département de mathématiques Université du Québec CP 8888, suce. A Montréal, Québec H3C 3P8
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Abstract

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The purpose for this short note is to describe the space of harmonic spinors on hyperbolic n-space Hn. This is a natural continuation of the study of harmonic functions on Hn in [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms on Hn were considered. The connection between invariant differential equations on a Riemannian semisimple symmetric space G/K and homological aspects of the representation theory of G, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

Keywords

55R2558G25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 3 , 01 September 1993 , pp. 257 - 262
Copyright
Copyright © Canadian Mathematical Society 1993

References

[Atiyah] Atiyah, M. F., Classical groups and classical differential operators on manifolds. In: Differential operators on manifolds, C.I.M.E., (ed. Cremonese), Rome, (1975).Google Scholar
[Borel & Wallach] Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive Lie groups, Ann. Math. Study 94, Princeton University Press, 1980.Google Scholar
[Chevalley] Chevalley, C., The algebraic theory of spinors, Columbia U. Press, New York, 1954.Google Scholar
[Collingwood] Collingwood, D., Representations of rank one Lie groups, Pitman, 1985.Google Scholar
[Gaillard 1] Gaillard, P. Y., Transformation de Poisson déformes différentielles. Le cas de l'espace hyperbolique, Comment Math. Helv. 61(1986), 581–616.Google Scholar
[Gaillard 2] Gaillard, P. Y., Eigenforms of the Laplacian on real and complex hyperbolic spaces, J. Funct. Anal. 78(1988), 99–115.Google Scholar
[Gaillard 3] Gaillard, P. Y., Invariant syzygies and semisimple groups, Advances in Math. 92(1992), 27–46.Google Scholar
[Guichardet] Guichardet, A., Sur les catégories de (g, K)-modules de longueur finie, manuscript.Google Scholar
[Helgason] Helgason, S., Eigenspaces of the Laplacian, integral representations and irreducibility, J. Funct. Anal. 17(1974), 328–353.Google Scholar
[KKMOOT] Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T. and Tanaka, M., Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. 107(1978), 1–39.Google Scholar
[Minemura] Minemura, K., Harmonic functions on real hyperbolic space, Hiroshima Math. J., (1973), 121- 151.Google Scholar
[Schmid] Schmid, W., Boundary value problems for group invariant differential equations. In: Elie Cartan et les Mathématiques d'aujourd'hui, Astérisque, No hors série, (1985), 311-321.Google Scholar
[Thieleker 1] Thieleker, E., On the quasi-simple irreducible representations of the Lorentz groups, TAMS 179(1973), 465–505.Google Scholar
[Thieleker 2] Thieleker, E., On the integrable and square integrable representations o/Spin(l,2m), TAMS 230 (1977), 1–40.Google Scholar