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Analytic families of eigenfunctions on a reductive symmetric space


Authors: E. P. van den Ban and H. Schlichtkrull
Journal: Represent. Theory 5 (2001), 615-712
MSC (2000): Primary 22E30, 22E45
DOI: https://doi.org/10.1090/S1088-4165-01-00146-7
Published electronically: December 12, 2001
MathSciNet review: 1870604
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Abstract: Let $X=G/H$ be a reductive symmetric space, and let $\mathbb D(X)$ denote the algebra of $G$-invariant differential operators on $X$. The asymptotic behavior of certain families $f_\lambda$ of generalized eigenfunctions for $\mathbb D(X)$ is studied. The family parameter $\lambda$ is a complex character on the split component of a parabolic subgroup. It is shown that the family is uniquely determined by the coefficient of a particular exponent in the expansion. This property is used to obtain a method by means of which linear relations among partial Eisenstein integrals can be deduced from similar relations on parabolic subgroups. In the special case of a semisimple Lie group considered as a symmetric space, this result is closely related to a lifting principle introduced by Casselman. The induction of relations will be applied in forthcoming work on the Plancherel and the Paley-Wiener theorem.


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  • 1. J. Arthur, A Paley-Wiener theorem for real reductive groups, Acta Math. 150 (1983), 1-89. MR 85k:22044
  • 2. E. P. van den Ban, Invariant differential operators on a semisimple symmetric space and finite multipilicities in a Plancherel formula, Ark. Mat. 25 (1987), 175-187. MR 89g:22019
  • 3. E. P. van den Ban, Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces, Indag. Math. 49 (1987), 225-249. MR 89c:22025
  • 4. E. P. van den Ban, The principal series for a reductive symmetric space I. $H$-fixed distribution vectors, Ann. Sci. École Norm. Sup. 21 (1988), 359-412. MR 90a:22016
  • 5. E. P. van den Ban, The principal series for a reductive symmetric space II. Eisenstein integrals, J. Funct. Anal. 109 (1992), 331-441. MR 93j:22025
  • 6. E. P. van den Ban and H. Schlichtkrull, Asymptotic expansions and boundary values of eigenfunctions on Riemannian symmetric spaces, J. Reine Angew. Math. 380 (1987), 108-165. MR 89g:43010
  • 7. E. P. van den Ban and H. Schlichtkrull, Fourier transforms on a semisimple symmetric space, Invent. Math. 130 (1997), 517-574. MR 98m:22016
  • 8. E. P. van den Ban and H. Schlichtkrull, Expansions for Eisenstein integrals on semisimple symmetric spaces, Ark. Mat. 35 (1997), 59-86. MR 98e:22003
  • 9. E. P. van den Ban and H. Schlichtkrull, The most continuous part of the Plancherel decomposition for a reductive symmetric space. Ann. of Math. 145 (1997), 267-364. MR 99e:22021
  • 10. E. P. van den Ban and H. Schlichtkrull, A residue calculus for root systems, Compositio Math. 123 (2000), 27-72. CMP 2001:01
  • 11. E. P. van den Ban and H. Schlichtkrull, Fourier inversion on a reductive symmetric space, Acta Math. 182 (1999), 25-85. MR 2000k:43005
  • 12. E. P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space, I-II, in preparation; See arXiv:math.
  • 13. E. P. van den Ban and H. Schlichtkrull, A Paley-Wiener theorem for reductive symmetric spaces, in preparation.
  • 14. O. A. Campoli, Paley-Wiener type theorems for rank-$1$ semisimple Lie groups, Rev. Union Mat. Argentina 29 (1980), 197-221. MR 82i:22013
  • 15. J. Carmona, Terme constant des fonctions tempérées sur un espace symétrique réductif, J. Reine Angew. Math. 491 (1997), 17-63. MR 99b:22016; Erratum 507 (1999), 233. MR 99j:22011
  • 16. R. G. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28., John Wiley, London, 1972. MR 53:10946
  • 17. W. Casselman and D. Miliçic, Asymptotic behaviour of matrix coefficients of admissible representations, Duke Math. J. 49 (1982), 869-930. MR 85a:22024
  • 18. P. Delorme, Intégrales d'Eisenstein pour les espaces symétriques réductifs. Tempérance. Majorations Petite matrice B, J. Funct. Anal. 136 (1996), 422-509. MR 96m:22027
  • 19. P. Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. of Math. 147 (1998), 417-452. MR 99d:22022
  • 20. R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. 93 (1971), 150-165. MR 44:6912
  • 21. R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, 1965. MR 31:4927
  • 22. Harish-Chandra, Spherical functions on a semisimple Lie group, I, II. Amer. J. Math. 80 (1958), 241-310, 553-613. MR 20:925; MR 21:92
  • 23. Harish-Chandra, On the theory of the Eisenstein integral, Lecture Notes in Math., No. 266, Springer-Verlag, 1972, pp. 123-149.
  • 24. Harish-Chandra, Harmonic analysis on real reductive groups, I. The theory of the constant term, J. Funct. Anal. 19 (1975), 104-204. MR 53:3201
  • 25. Harish-Chandra, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. 104 (1976), 117-201. MR 55:12875
  • 26. S. Helgason, A duality for symmetric spaces with applications to group representations, II. Differential equations and eigenspace representations, Adv. Math. 22 (1976), 187-219. MR 55:3169
  • 27. S. Helgason, Groups and geometric analysis, Academic Press, 1984. MR 86c:22017; Corrected reprint, Mathematical Surveys and Monographs, 83, Amer. Math. Soc., Providence, RI, 2000. MR 2001h:22001
  • 28. M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. 107 (1978), 1-39. MR 81f:43013
  • 29. T. Oshima and T. Matsuki, Boundary value problems for system of linear partial differential equations with regular singularities, Adv. Stud. Pure Math. 4 (1984), 433-497. MR 87c:58121
  • 30. T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), 1-81. MR 81k:43014
  • 31. H. Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Birkhäuser, 1984. MR 86g:22021
  • 32. P. C. Trombi and V. S. Varadarajan, Spherical transforms on semisimple Lie groups, Ann. of Math. 94 (1971), 246-303. MR 44:6913
  • 33. N. R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lecture Notes in Math. 1024, Berlin-Heidelberg-New York, 1983, 287-369. MR 85g:22029

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Additional Information

E. P. van den Ban
Affiliation: Mathematisch Institut, Universiteit Utrecht, PO Box 80 010, 3508 TA Utrecht, The Netherlands
Email: ban@math.uu.nl

H. Schlichtkrull
Affiliation: Matematisk Institut, Københavns Universitet, Universitetsparken 5, 2100 København Ø, Denmark
Email: schlicht@math.ku.dk

DOI: https://doi.org/10.1090/S1088-4165-01-00146-7
Received by editor(s): February 20, 2001
Received by editor(s) in revised form: September 6, 2001
Published electronically: December 12, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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