Abstract
Recent results on the harmonic analysis of spinor fields on the complex hyperbolic space H n(C) are reviewed. We discuss the action of the invariant differential operators on the Poisson transforms, the theory of spherical functions and the spherical transform. The inversion formula, the Paley–Wiener theorem, and the Plancherel theorem for the spherical transform are obtained by reduction to Jacobi analysis on L 2(R).
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References
Arthur, J.: A Paley-Wiener theorem for real reductive groups, Acta Math. 150 (1983), 1-89.
Branson, T. P., Olafsson, G. and Schlichtkrull, H.: A bundle-valued Radon transform with applications to invariant wave equations, Quart. J. Math. Oxford (2) 45 (1994), 429-461.
Campoli, O.: Paley-Wiener type theorems for rank one semisimple Lie groups, Rev. Un. Mat. Argentina 29 (1980), 197-221.
Camporesi, R.: The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces, J. Lie Theory 7 (1997), 29-60.
Camporesi, R.: The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces, Pacific J. Math. 179 (1997), 263-300.
Camporesi, R.: Harmonic analysis for spinor fields in complex hyperbolic spaces, Adv. Math. 154 (2000), 367-442.
Camporesi, R. and Pedon, E.: Harmonic analysis for spinor fields on real hyperbolic spaces, Colloq. Math. 87 (2001), 245-286.
Camporesi, R. and Pedon, E.: The continuous spectrum of the Dirac operator on noncompact Riemannian symmetric spaces of rank one, Proc. Amer. Math. Soc. 130 (2002), 507-516.
Van Dijk, G. and Pasquale, A.: Harmonic analysis on vector bundles over Sp(1, n)/Sp(1) × Sp(n), Enseign. Math. 45 (1999), 219-252.
Goette, S. and Semmelmann, U.: The point spectrum of the Dirac operator on noncompact symmetric spaces, Proc. Amer. Math. Soc. 130 (2002), 915-923.
Goodman, R. and Wallach, N. R.: Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and Its Applications, Vol. 68, Cambridge Univ. Press, Cambridge, 1998.
Knapp, A.W.: Representation Theory of Semisimple Groups. An Overview Based on Examples, Princeton Univ. Press, Princeton, NJ, 1986. </del>
Koornwinder, T. H.: Jacobi functions and analysis on noncompact semisimple Lie groups, In: R. Askey et al. (eds), Special Functions, Group Theoretical Aspects and Applications, Reidel, Dordrecht, 1984, pp. 1-85.
Pedon, E.: Analyse harmonique des formes differentielles sur l'espace hyperbolique reel, Thesis, Université Henri Poincaré, Nancy, 1997.
Pedon, E.: Harmonic analysis for differential forms on complex hyperbolic spaces, J. Geom. Phys. 32 (1999), 102-130.
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Camporesi, R. Spherical Harmonic Analysis for Spinors on H n(C). Acta Applicandae Mathematicae 73, 15–38 (2002). https://doi.org/10.1023/A:1019758200630
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DOI: https://doi.org/10.1023/A:1019758200630