×
Oshima, Toshio; Sekiguchi, Jiro

Eigenspaces of invariant differential operators on an affine symmetric space. (English) Zbl 0434.58020

Invent. Math. 57, 1-81 (1980).
Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page
scan of Zbl 0434.58020

Cited in 6 Reviews
Cited in 70 Documents

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
22E46 Semisimple Lie groups and their representations
43A90 Harmonic analysis and spherical functions
43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces
58J70 Invariance and symmetry properties for PDEs on manifolds

Keywords:

invariant differential operators; Poisson integral transformation; generalizations of zonal spherical functions; Eisenstein series; affine symmetric spaces; space of hyperfunctions on compact boundaries of symmetric spaces

Citations:

Zbl 0377.43012
Cite Review PDF
Full Text: DOI EuDML

References:

[1] Berger, M.: Les espaces symétriques non compacts. Ann. Sci. Ecole Norm. Sup.74, 85-177 (1957) · Zbl 0093.35602
[2] Bernstein, I.N., Gel’fand, S.I.: Meromorphic property of the functionP ? (in Russian). Funkt. Anal. i Ego Prilozheniya3, 84-85 (1969) English translation: Funct. Anal. Appl.,3, 68-69 (1969)
[3] Bony, J.M.: Propagation des singularités différentiables pour des opérateurs à coefficient analytiques. Astérisque,34-35, 43-91 (1976)
[4] Borel, A., de Siebenthal, J.: Les sous-groupes fermés de rang maximum des groupes de Lie clos. Comm. Math. Helv.,23, 200-221 (1949) · Zbl 0034.30701 · doi:10.1007/BF02565599
[5] Bourbaki, N.: Groupes et Algèbres de Lie, Chapters IV?VI. Paris: Hermann 1968 · Zbl 0186.33001
[6] Bruhat, F.: Sur les représentations induites des groupes de Lie. Bull. Soc. Math. France,84, 97-205 (1956) · Zbl 0074.10303
[7] Flensted-Jensen, M.: Spherical functions on a real semisimple Lie group. A method of reduction to the complex case. J. Functional Analysis,30, 106-146 (1978) · Zbl 0419.22019 · doi:10.1016/0022-1236(78)90058-7
[8] Furstenberg, H.: A Poisson formula for semisimple Lie groups. Ann. of Math.,77, 335-386 (1963) · Zbl 0192.12704 · doi:10.2307/1970220
[9] Gindikin, S.G., Karpelevi?, F.I.: Plancherel measure for Riemannian symmetric spaces of nonpositive curvature. Doklady Akad. Nauk. SSSR,145, 252-255 (1962). English translation: Soviet Math. Dokl.,3-II, 962-965 (1962)
[10] Goto, M.: Faithful representation of Lie groups I. Math. Japonica,1, 107-119 (1948)
[11] Harish-Chandra: Representations of semi-simple Lie groups I. Trans. Amer. Math. Soc.,75, 185-243 (1953) · Zbl 0051.34002 · doi:10.1090/S0002-9947-1953-0056610-2
[12] Harish-Chandra: Spherical functions on a semi-simple Lie group I. Amer. J. Math.,80, 241-310 (1958) · Zbl 0093.12801 · doi:10.2307/2372786
[13] Helgason, S.: Differential Geometry and Symmetric Spaces. New York: Academic Press 1962 · Zbl 0111.18101
[14] Helgason, S.: A duality for symmetric spaces with applications to group representations. Advances in Math.,5, 1-154 (1970) · Zbl 0209.25403 · doi:10.1016/0001-8708(70)90037-X
[15] Helgason, S.: A duality for symmetric spaces with applications to group representations II, ibid,22, 187-219 (1976) · Zbl 0351.53037 · doi:10.1016/0001-8708(76)90153-5
[16] Kashiwara, M., Kawai, T.: Micro-hyperbolic pseudo-differential operators I. J. Math. Soc. Japan,27, 359-404 (1975) · Zbl 0305.35066 · doi:10.2969/jmsj/02730359
[17] Kashiwara, M., Kowata, A., Minemura, K., Okamoto, K., Oshima, T., Tanaka, M.: Eigenfunctions of invariant differential operators on a symmetric space. Ann. of Math.,107, 1-39 (1978) · Zbl 0377.43012 · doi:10.2307/1971253
[18] Kashiwara, M., Oshima, T.: Systems of differential equations with regular singularities and their boundary value problems. Ann. of Math.,106, 145-200 (1977) · Zbl 0358.35073 · doi:10.2307/1971163
[19] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II, New York: Wiley-Interscience 1969 · Zbl 0175.48504
[20] Koh, S.S.: On affine symmetric spaces. Trans. Amer. Math. Soc.,119 291-309 (1965) · Zbl 0139.39502 · doi:10.1090/S0002-9947-1965-0184170-2
[21] Komatsu, H.: Projective and injective limits of weakly compact sequences of locally convex spaces. J. Math. Soc. Japan,19, 366-383 (1967) · Zbl 0168.10603 · doi:10.2969/jmsj/01930366
[22] Langlands, R.P.: On the functional equations satisfied by Eisenstein series. Lecture Notes in Math., No. 544, Berlin-Heidelberg-New York: Springer-Verlag 1976 · Zbl 0332.10018
[23] Lewis, J.: Eigenfunctions on symmetric spaces with distribution-valued boundary forms, J. Functional Analysis29, 287-307 (1978) · Zbl 0398.43010 · doi:10.1016/0022-1236(78)90032-0
[24] Lojasiewicz, S.: Sur le problème de la division. Studia Mathematica, TomXVIII, 87-136 (1959) · Zbl 0115.10203
[25] Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan,31, 331-357 (1979) · Zbl 0396.53025 · doi:10.2969/jmsj/03120331
[26] Matsuki, T., Oshima, T.: Orbits on affine symmetric spaces under the action of the isotropy subgroups, to appear in J. Math. Soc. Japan · Zbl 0451.53039
[27] Minemura, K., Oshima, T.: Boundary value problems with regular singularities and Helgason-Okamoto conjecture. Publ. RIMS, Kyoto Univ.,12 (Suppl.) 257-265 (1977) · Zbl 0419.35017 · doi:10.2977/prims/1195196608
[28] Murakami, S.: Sur la classification des algèbres de Lie réelles et simples. Osaka J. Math.,2, 291-307 (1965) · Zbl 0163.28101
[29] Nomizu, K.: Invariant affine connections on homogeneous spaces. Amer. J. Math.76, 33-65 (1954) · Zbl 0059.15805 · doi:10.2307/2372398
[30] Oshima, T.: A realization of Riemannian symmetric spaces. J. Math. Soc. Japan,30, 117-132 (1978) · Zbl 0364.43010 · doi:10.2969/jmsj/03010117
[31] Oshima, T., Sekiguchi, J.: Boundary value problem on symmetric homogeneous spaces. Proc. Japan Acad.,53 (Ser. A), 81-83 (1977) · Zbl 0452.43013 · doi:10.3792/pjaa.53.81
[32] Satake, I.: On representations and compactifications of symmetric Riemannian spaces. Ann. of Math.,71, 77-110 (1960) · Zbl 0094.34603 · doi:10.2307/1969880
[33] Sato, F.: Eisenstein series for indefinite quadratic forms, preprint
[34] Sato, M.: Theory of hyperfunctions, I, II. J. Fac. Sci. Univ. of Tokyo, Sect. I,8, 139-193, 357-437 (1959) · Zbl 0087.31402
[35] Sato, M., Kawai, T., Kashiwara, M.: Micro-functions and pseudo-differential equations. Proc. Conf. at Katata, 1971. Lecture Notes in Math. No. 287, pp. 265-529. Berlin-Heidelberg-New York: Springer 1973
[36] Schiffmann, G.: Intégrales d’entrelacement et fonctions de Whittaker. Bull. Soc. Math. France,99, 3-72 (1971) · Zbl 0223.22017
[37] Schwartz, L.: Théorie des distributions, I, II. Paris: Hermann 1950-1951 · Zbl 0037.07301
[38] Sekiguchi, J.: Boundary value problem on hyperboloids to appear in Nagoya Math. J.
[39] Sekiguchi, J.: Invariant system of differential equations on a Siegel’s upper half plane, preprint
[40] Selberg, A.: Discontinuous groups and harmonic analysis. Proc. Int. Congress of Math., Stockholm, pp. 177-189, 1962
[41] Wallach, N.: Harmonic Analysis on Homogeneous Spaces. New York: Marcel Dekker, Inc., 1973 · Zbl 0265.22022
[42] Warner, G.: Harmonic Analysis on Semi-simple Lie groups, I, II. Berlin-Heidelberg-New York: Springer-Verlag 1972 · Zbl 0265.22020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.