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A twistor correspondence and Penrose transform for odd-dimensional hyperbolic space

Authors: Toby N. Bailey and Edward G. Dunne
Journal: Proc. Amer. Math. Soc. 126 (1998), 1245-1252
MSC (1991): Primary 22E46, 32L25; Secondary 53C35
MathSciNet review: 1443811
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Abstract: For odd-dimensional hyperbolic space $\mathcal{H}$, we construct transforms between the cohomology of certain line bundles on $\mathcal{T}$ (a twistor space for $\mathcal{H}$) and eigenspaces of the Laplacian $\Delta$ and of the Dirac operator $D$ on $\mathcal{H}$. The transforms are isomorphisms. As a corollary we obtain that every eigenfunction of $\Delta$ or $D$ on $\mathcal{H}$ extends as a holomorphic eigenfunction of the corresponding holomorphic operator on a certain region of the complexification of $\mathcal{H}$. We also obtain vanishing theorems for the cohomology of a class of line bundles on $\mathcal{T}$.

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  • [AS] M. Atiyah & W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1-62. MR 57:3310
  • [BES] T.N. Bailey, M.G. Eastwood & M.A. Singer, The Penrose transform and involutive structures, preprint.
  • [BKZ] L. Barchini, A.W. Knapp & R. Zierau, Intertwining operators into Dolbeault cohomology, Jour. Funct. Anal. 107 (1992), 302-341. MR 93e:22026
  • [BE] R.J. Baston & M.G. Eastwood, The Penrose Transform: its Interaction with Representation Theory, Oxford University Press, 1989. MR 92j:32112
  • [D] E.G. Dunne, Involutive structures on smooth manifolds, electronic preprint, 1995. Available at:[?~?] dunne/ElectronicPapers.html.
  • [H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977. MR 57:3116
  • [H1] S. Helgason, Eigenspaces of the Laplacian: integral representations and irreducibility, Jour. Funct. Anal. 17 (1974), 328-353. MR 51:3353
  • [H2] S. Helgason, Groups and Geometric Analysis, Academic Press, 1984. MR 86c:22017
  • [J] P.E. Jones, Minitwistors, D.Phil. thesis, Oxford University, 1984.
  • [JT] P.E. Jones & K.P. Tod, Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum Grav. 2 (1985), 565-577. MR 87b:53115
  • [S] H. Schlichtkrull, Eigenspaces of the Laplacian on hyperbolic spaces: composition series and integral transforms, Jour. Funct. Anal. 70 (1987), 194-219. MR 88f:22040
  • [T] F. Treves, Hypo-analytic structures, Princeton University Press, 1992. MR 94e:35014
  • [Ts] C-C. Tsai, The Penrose transform for Einstein-Weyl and related spaces, Ph.D. thesis, University of Edinburgh, 1996.
  • [W] H-W. Wong, Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations, Jour. Funct. Anal. 129 (1995), 428-454. MR 96c:22024

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Additional Information

Toby N. Bailey

Edward G. Dunne
Address at time of publication: American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940

Keywords: Penrose transform, twistor theory, involutive cohomology, hyperbolic space, eigenspaces of the Laplacian, Dirac operator
Received by editor(s): October 3, 1996
Additional Notes: The authors are grateful for support from the EPSRC. The second author would also like to thank the Department of Mathematics at the University of Edinburgh for their hospitality.
Communicated by: Roe Goodman
Article copyright: © Copyright 1998 American Mathematical Society

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