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Causal compactification and Hardy spaces


Authors: G. Ólafsson and B. Ørsted
Journal: Trans. Amer. Math. Soc. 351 (1999), 3771-3792
MSC (1991): Primary 43A85, 22E46; Secondary 43A65, 53C35
DOI: https://doi.org/10.1090/S0002-9947-99-02101-7
Published electronically: March 1, 1999
MathSciNet review: 1458309
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Abstract: Let $\mathcal{M}=G/H$ be a irreducible symmetric space of Cayley type. Then $\mathcal{M}$ is diffeomorphic to an open and dense $G$-orbit in the Shilov boundary of $G/K\times G/K$. This compactification of $\mathcal{M}$ is causal and can be used to give answers to questions in harmonic analysis on $\mathcal{M}$. In particular we relate the Hardy space of $\mathcal{M}$ to the classical Hardy space on the bounded symmetric domain $G/K\times G/K$. This gives a new formula for the Cauchy-Szegö kernel for $\mathcal{M}$.


References [Enhancements On Off] (What's this?)

  • 1. M. Chadli, Domaine complexe assocé à un espace symétrique de type Cayley. C.R. Acad. Sci. Paris 321 (1995), 1157-1161. MR 97f:22022
  • 2. Harish-Chandra, Representations of semi-simple Lie groups. IV. Amer. J. Math. 77 (1955), 743-777. MR 17:282c
  • 3. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press. New York/London, 1978. MR 80k:53081
  • 4. S. Helgason, Groups and geometric analysis. Integral geometry, invariant differential operators and spherical functions. Academic Press. New York/London, 1984. MR 86c:22017
  • 5. J. Hilgert, G. Ólafsson, Causal Symmetric Spaces, Geometry and Harmonic Analysis. Perspectives in Mathematics 18, Academic Press, 1996. MR 97m:43006
  • 6. J. Hilgert, G. Ólafsson, B. Ørsted, Hardy spaces on affine symmetric spaces. J. reine und angew. Math. 415 (1991), 189-218. MR 92h:22030
  • 7. S. Kaneyuki, On orbit structure of compactifications of parahermitian symmetric spaces. Japan. J. Math. 13 (1987), 333-370. MR 88m:53094
  • 8. A. Korányi, J. A Wolf, Realization of Hermitian symmetric spaces as generalized half-planes. Ann. of Math. 81 (1965), 265-288. MR 30:4980
  • 9. K. Koufany, Réalisation des espaces symétriques de type Cayley. C. R. Acad. Sci. Paris. 318 (1994), 425-428. MR 95h:32036
  • 10. K. Koufany, B. Ørsted, Espace de Hardy sur le semi-groupe métaplectique. C. R. Acad. Sci. Paris, to appear. MR 96m:22017
  • 11. O. Loos, Symmetric Spaces, I: General Theory. W.A. Benjamin, Inc., New York, 1969. MR 39:365a
  • 12. V. F. Molchanov, Holomorphic discrete series for hyperboloids of Hermitian type, J. Funct. Anal. 147 (1997), 26-50. CMP 97:13
  • 13. G. Ólafsson, Fourier and Poisson transformation associated to a semisimple symmetric space. Invent. math. 90 (1987), 605-629. MR 89d:43011
  • 14. G. Ólafsson, Causal symmetric spaces. Mathematica Gottingensis. 15 (1990).
  • 15. G. Ólafsson, Symmetric spaces of Hermitian type. Diff. Geom. and Appl. 1 (1991), 195-233. MR 94g:22034
  • 16. G. Ólafsson, Spherical Functions and Spherical Laplace Transform on Ordered Symmetric Spaces. Preprint, 1997.
  • 17. G. Ólafsson, B. Ørsted, The holomorphic discrete series for affine symmetric spaces, I. J. Funct. Anal. 81 (1988), 126-159. MR 89m:22021
  • 18. G. Ólafsson, B. Ørsted, B. The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels. Trans. Amer. Math. Soc. 326 (1991), 385-405. MR 91j:22014
  • 19. W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen. Invent. Math. 9 (1969), 61-80. MR 41:3806
  • 20. R. T. Stanton, Analytic extension of the holomorphic discrete series. Amer. J. of Math. 108 (1986), 1411-1424. MR 88b:22013
  • 21. E. T. Whittaker, G. N. Watson, Modern Analysis. Cambridge 1946. MR 97k:01072

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Additional Information

G. Ólafsson
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: olafsson@math.lsu.edu

B. Ørsted
Affiliation: Matematisk Institut, Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark
Email: orsted@imada.ou.dk

DOI: https://doi.org/10.1090/S0002-9947-99-02101-7
Keywords: Causal compactification, Hardy spaces, holomorphic discrete series
Received by editor(s): April 29, 1996
Received by editor(s) in revised form: March 10, 1997
Published electronically: March 1, 1999
Additional Notes: The first named author was supported by NSF grant DMS-9626541, LEQSF grant (1996-99)-RD-A-12 and the Danish Research Council
Article copyright: © Copyright 1999 American Mathematical Society

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