Harmonic analysis for differential forms on complex hyperbolic spaces

Journal of Geometry and Physics

Volumn 32, Issue 2, 1999, Pages 102-130

  Pedon, Emmanuel ^a  

^a UNIVERSITÉ DE LORRAINE   (France)

Author keywords

Complex hyperbolic space; Differential forms; Fourier transform; Heat kernel; Hodge de rham laplacian spectrum; Plancherel theorem; Spherical functions

Indexed keywords

EID: 0033451109     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(99)00026-1     Document Type: Article

References (40)

1. M.W. Baldoni-Silva, Branching theorems for semisimple Lie groups of real rank one, Rend. Sem. Mat. Univ. Padova 61 (1979) 229-250.
2. M.W. Baldoni-Silva, The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one, Trans. Am. Math. Soc. 261 (1980) 303-369.
3. 2-cohomology of negatively curved Riemannian symmetric spaces, Ann. Acad. Scient. Fenn. Ser. A.I. Math. 10 (1985) 95-105.
4. A. Borel, N.R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive Lie groups, Ann. Math. Study 94, Princeton University Press, Princeton, NJ, 1980.
5. F. Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956) 97-205.
6. U. Bunke, M. Olbrich, Selberg zeta and theta functions, A Differential Operator Approach, Mathematical Research 83, Akademie Verlag, Berlin, 1995.
7. O. Campoli, Paley-Wiener type theorems for rank one semisimple Lie groups, Rev. Un. Mat. Argentina 29 (1980) 197-221.
8. R. Camporesi, The Helgason fourier transform for homogeneous vector bundles over Riemannian symmetric spaces, Pacific J. Math. 179 (1997) 263-300.
9. R. Camporesi, The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces, J. Lie Theory 7 (1997) 29-60.
10. R. Camporesi, A. Higuchi, The plancherel measure for p-forms in real hyperbolic spaces, J. Geom. Phys. 15 (1994) 57-94.
11. A. Deitmar, Invariant operators on higher K-types, J. Reine Angew. Math. 412 (1990) 97-107.
12. 2 cohomology and index theorem for the bergman metric, Ann. Math. 118 (1983) 593-618.
13. P.-Y. Gaillard, Transformation de poisson de formes différentielles. Le cas de l'espace hyperbolique, Comment. Math. Helv. 61 (1986) 581-616.
14. P.-Y. Gaillard, Eigenforms of the laplacian on real and complex hyperbolic spaces, J. Funct. Anal. 78 (1988) 99-115.
15. 2-hodge theory, J. Diff. Geom. 33 (1991) 263-292.
16. S. Helgason, Eigenspaces of the laplacian, integral representations and irreducibility, J. Funct. Anal. 17 (1974) 328-353.
17. S. Helgason, Groups and Geometric Analysis; Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York, 1984.
18. S. Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs 39, Am. Math. Soc., Providence, Rhode Island, 1994.
19. J.-S. Huang, T. Oshima, N.R. Wallach, Dimensions of spaces of generalized spherical functions, Am. J. Math. 118 (1996) 637-652.
20. n(ℂ), Osaka J. Math. 15 (1978) 515-546.
21. P. Julg, G. Kasparov, Operator K-theory for the group SU(n, 1), J. Reine Angew. Math. 463 (1995) 99-152.
22. M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima, M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. Math. 107 (1978) 1-39.
23. A.W. Knapp, Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series 36, Princeton University Press, Princeton, NJ, 1986.
24. A.W. Knapp, Lie groups beyond an introduction, Progress in Mathematics 140, Birkhäuser, Basel, 1996.
25. A.W. Knapp, E.M. Stein, Intertwining operators for semisimple groups, Ann. Math. 93 (1971) 489-578.
26. T.H. Koornwinder, A note on the multiplicity free reduction of certain orthogonal and unitary groups, Indag. Math. 44 (1982) 215-218.
27. J. Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom. 35 (1992) 471-510.
28. R.J. Miatello, On the plancherel measure for linear Lie groups of rank one, Manuscripta Math. 29 (1979) 249-276.
29. K. Minemura, Invariant differential operators and spherical sections on a homogeneous vector bundle, Tokyo J. Math. 15 (1992) 231-245.
30. M. Olbrich, Die Poisson-transformation Für Homogene Vektorbündel, Ph.D. Thesis, Humboldt Universität, Berlin, 1994.
31. E. Pedon, Analyse Harmonique des Formes Différentielles Sur L'espace Hyperbolique Réel, Ph.D. Thesis, Université Henri-Poincaré, Nancy, 1997.
32. E. Pedon, Analyse harmonique des formes différentielles sur l'espace hyperbolique réel I. Transformation de poisson et fonctions sphériques, C. R. Acad. Sci. Paris Sér. I 326 (1998) 671-676.
33. E. Pedon, Analyse harmonique des formes différentielles sur l'espace hyperbolique réel II. Transformation de fourier sphérique et applications, C. R. Acad. Sci. Paris Sér. I 326 (1998) 781-786.
34. M. Rumin, Un complexe de formes différentielles sur les variétés de contact, C. R. Acad. Sci. Paris Sér. I 310 (1990) 401-404.
35. M. Rumin, Formes différentielles sur les variétés de contact, J. Diff. Geom. 39 (1994) 281-330.
36. N. Shimeno, Eigenspaces of invariant differential operators on a homogeneous line bundle on a Riemannian symmetric space, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990) 210-234.
37. N. Shimeno, The plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of hermitean type, J. Funct. Anal. 121 (1994) 330-388.
38. H. van der Ven, Vector valued poisson transforms on Riemannian symmetric spaces of rank one, J. Funct. Anal. 119 (1994) 358-400.
39. R.O. Wells, Differential analysis on complex manifolds, Graduate Texts in Mathematics, 2nd Ed., vol. 65, Springer, Berlin, 1980.
40. A. Yang, Poisson transforms on vector bundles, Trans. Am. Math. Soc. 350 (1998) 857-887.