Analytic families of eigenfunctions on a reductive symmetric space
HTML articles powered by AMS MathViewer
- by E. P. van den Ban and H. Schlichtkrull
- Represent. Theory 5 (2001), 615-712
- DOI: https://doi.org/10.1090/S1088-4165-01-00146-7
- Published electronically: December 12, 2001
- PDF | Request permission
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.References
- James Arthur, On a family of distributions obtained from Eisenstein series. I. Application of the Paley-Wiener theorem, Amer. J. Math. 104 (1982), no. 6, 1243–1288. MR 681737, DOI 10.2307/2374061
- Erik P. van den Ban, Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula, Ark. Mat. 25 (1987), no. 2, 175–187. MR 923405, DOI 10.1007/BF02384442
- E. P. van den Ban, Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 3, 225–249. MR 914083, DOI 10.1016/1385-7258(87)90013-8
- E. P. van den Ban, The principal series for a reductive symmetric space. I. $H$-fixed distribution vectors, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 359–412. MR 974410, DOI 10.24033/asens.1562
- E. P. van den Ban, The principal series for a reductive symmetric space. II. Eisenstein integrals, J. Funct. Anal. 109 (1992), no. 2, 331–441. MR 1186325, DOI 10.1016/0022-1236(92)90021-A
- 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 916202, DOI 10.1515/crll.1987.380.108
- Erik van den Ban and Henrik Schlichtkrull, Fourier transform on a semisimple symmetric space, Invent. Math. 130 (1997), no. 3, 517–574. MR 1483993, DOI 10.1007/s002220050193
- Erik P. van den Ban and Henrik Schlichtkrull, Expansions for Eisenstein integrals on semisimple symmetric spaces, Ark. Mat. 35 (1997), no. 1, 59–86. MR 1443036, DOI 10.1007/BF02559593
- E. P. van den Ban and H. Schlichtkrull, The most continuous part of the Plancherel decomposition for a reductive symmetric space, Ann. of Math. (2) 145 (1997), no. 2, 267–364. MR 1441878, DOI 10.2307/2951816 BSres E. P. van den Ban and H. Schlichtkrull, A residue calculus for root systems, Compositio Math. 123 (2000), 27-72.
- Erik P. van den Ban and Henrik Schlichtkrull, Fourier inversion on a reductive symmetric space, Acta Math. 182 (1999), no. 1, 25–85. MR 1687176, DOI 10.1007/BF02392823 BSpl E. P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space, I-II, in preparation; See arXiv:math. BSpw E. P. van den Ban and H. Schlichtkrull, A Paley-Wiener theorem for reductive symmetric spaces, in preparation.
- Oscar A. Campoli, Paley-Wiener type theorems for rank-$1$ semisimple Lie groups, Rev. Un. Mat. Argentina 29 (1979/80), no. 3, 197–221. MR 602486
- Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99, DOI 10.1515/crll.1999.507.233
- Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
- William Casselman and Dragan Miličić, Asymptotic behavior of matrix coefficients of admissible representations, Duke Math. J. 49 (1982), no. 4, 869–930. MR 683007, DOI 10.1215/S0012-7094-82-04943-2
- Patrick Delorme, Intégrales d’Eisenstein pour les espaces symétriques réductifs: tempérance, majorations. Petite matrice $B$, J. Funct. Anal. 136 (1996), no. 2, 422–509 (French, with English summary). MR 1380660, DOI 10.1006/jfan.1996.0035
- Patrick Delorme, Formule de Plancherel pour les espaces symétriques réductifs, Ann. of Math. (2) 147 (1998), no. 2, 417–452 (French). MR 1626757, DOI 10.2307/121014
- R. Gangolli, On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups, Ann. of Math. (2) 93 (1971), 150–165. MR 289724, DOI 10.2307/1970758
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- S. S. Pillai, On normal numbers, Proc. Indian Acad. Sci., Sect. A. 10 (1939), 13–15. MR 0000020, DOI 10.1007/BF03170534 HCEis Harish-Chandra, On the theory of the Eisenstein integral, Lecture Notes in Math., No. 266, Springer-Verlag, 1972, pp. 123-149.
- Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 0399356, DOI 10.1016/0022-1236(75)90034-8
- Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117–201. MR 439994, DOI 10.2307/1971058
- Sigurdur Helgason, A duality for symmetric spaces with applications to group representations. II. Differential equations and eigenspace representations, Advances in Math. 22 (1976), no. 2, 187–219. MR 430162, DOI 10.1016/0001-8708(76)90153-5
- R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86, DOI 10.1090/surv/083
- M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Ōshima, and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. (2) 107 (1978), no. 1, 1–39. MR 485861, DOI 10.2307/1971253
- Toshio Ōshima, Boundary value problems for systems of linear partial differential equations with regular singularities, Group representations and systems of differential equations (Tokyo, 1982) Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 391–432. MR 810637, DOI 10.2969/aspm/00410391
- Toshio Ōshima and Jir\B{o} Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), no. 1, 1–81. MR 564184, DOI 10.1007/BF01389818
- R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86, DOI 10.1007/978-1-4612-5298-6
- P. C. Trombi and V. S. Varadarajan, Spherical transforms of semisimple Lie groups, Ann. of Math. (2) 94 (1971), 246–303. MR 289725, DOI 10.2307/1970861
- Nolan R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 287–369. MR 727854, DOI 10.1007/BFb0071436
Bibliographic Information
- E. P. van den Ban
- Affiliation: Mathematisch Institut, Universiteit Utrecht, PO Box 80 010, 3508 TA Utrecht, The Netherlands
- MR Author ID: 30285
- Email: ban@math.uu.nl
- H. Schlichtkrull
- Affiliation: Matematisk Institut, Københavns Universitet, Universitetsparken 5, 2100 København Ø, Denmark
- MR Author ID: 156155
- ORCID: 0000-0002-4681-3563
- Email: schlicht@math.ku.dk
- Received by editor(s): February 20, 2001
- Received by editor(s) in revised form: September 6, 2001
- Published electronically: December 12, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Represent. Theory 5 (2001), 615-712
- MSC (2000): Primary 22E30, 22E45
- DOI: https://doi.org/10.1090/S1088-4165-01-00146-7
- MathSciNet review: 1870604