Generalized Laplacians for generalized Poisson-Cauchy transforms on classical domains
Eisuke Imamura, Kiyosato Okamoto, Michiroh Tsukamoto, and Atsushi Yamamori
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 82, Number 9
(2006), 167-172.
Abstract
We develop a group-theoretic method of generalizing the Laplace-Beltrami operators on the classical domains. In [18], we defined the generalized Poisson-Cauchy transforms on the classical domains. We show that the generalized Poisson-Cauchy transforms give us eigenfunctions of the generalized Laplacians defined in this paper.
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Permanent link to this document: http://projecteuclid.org/euclid.pja/1165244966
Mathematical Reviews number (MathSciNet): MR2293504
Digital Object Identifier: doi:10.3792/pjaa.82.167
Zentralblatt MATH identifier: 1154.43006
References
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Mathematical Reviews (MathSciNet): MR2091638
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Mathematical Reviews (MathSciNet): MR104743
Zentralblatt MATH: 0091.10802
Harish-Chandra, Representations of semisimple Lie groups. V, VI, Amer. J. Math. 78 (1956), 1--41, 564--628.
Mathematical Reviews (MathSciNet): MR82055
Zentralblatt MATH: 0070.11602
Digital Object Identifier: doi:10.2307/2372481
JSTOR: links.jstor.org
M. Hashizume, A. Kowata, K. Minemura and K. Okamoto, An integral representation of an eigenfunction of the Laplacian on the Euclidean space, Hiroshima Math. J. 2 (1972), 535--545.
Mathematical Reviews (MathSciNet): MR430161
Zentralblatt MATH: 0269.35012
Project Euclid: euclid.hmj/1206137631
M. Hashizume, K. Minemura and K. Okamoto, Harmonic functions on hermitian hyperbolic spaces, Hiroshima Math. J. 3 (1973), 81--108.
Mathematical Reviews (MathSciNet): MR361628
Zentralblatt MATH: 0273.31011
Project Euclid: euclid.hmj/1206137443
S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962.
Mathematical Reviews (MathSciNet): MR145455
S. Helgason, A duality for symmetric spaces with applications to group representations, Advances in Math. 5 (1970), 1--154.
Mathematical Reviews (MathSciNet): MR263988
Digital Object Identifier: doi:10.1016/0001-8708(70)90037-X
S. Helgason, Eigenspaces of the Laplacian; integral representations and irreducibility, J. Functional Analysis 17 (1974), 328--353.
Mathematical Reviews (MathSciNet): MR367111
Digital Object Identifier: doi:10.1016/0022-1236(74)90045-7
S. Helgason, Groups and geometric analysis, Academic Press, Orlando, FL, 1984.
Mathematical Reviews (MathSciNet): MR754767
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Mathematical Reviews (MathSciNet): MR622203
Zentralblatt MATH: 0389.35015
Project Euclid: euclid.hmj/1206135664
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translated from the Russian by Leo Ebner and Adam Koranyi, Amer. Math. Soc., Providence, R.I., 1963.
Mathematical Reviews (MathSciNet): MR171936
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Mathematical Reviews (MathSciNet): MR364550
Project Euclid: euclid.hmj/1206137071
E. Imamura, K. Okamoto, M. Tsukamoto and A. Yamamori, Eigenvalues of Generalized Laplacians for Generalized Poisson-Cauchy transforms on classical domains. (to appear).
Mathematical Reviews (MathSciNet): MR2293504
Project Euclid: euclid.pja/1165244966
M. Kashiwara, A. Kowata, K. Minemura and K. Okamoto, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. (2) 107 (1978), no. 1, 1--39.
Mathematical Reviews (MathSciNet): MR485861
Digital Object Identifier: doi:10.2307/1971253
A. Kowata and K. Okamoto, Harmonic functions and the Borel-Weil theorem, Hiroshima Math. J. 4 (1974), 89--97.
Mathematical Reviews (MathSciNet): MR457796
Project Euclid: euclid.hmj/1206137155
M. S. Narasimhan and K. Okamoto, An analogue of the Borel-Weil-Bott theorem for hermitian symmetric pairs of non-compact type, Ann. of Math. (2) 91 (1970), 486--511.
Mathematical Reviews (MathSciNet): MR274657
Digital Object Identifier: doi:10.2307/1970635
JSTOR: links.jstor.org
K. Okamoto, Harmonic analysis on homogeneous vector bundles, in Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), 255--271. Lecture Notes in Math., 266, Springer, Berlin.
Mathematical Reviews (MathSciNet): MR486323
Zentralblatt MATH: 0234.43009
K. Okamoto, M. Tsukamoto and K. Yokota, Generalized Poisson and Cauchy kernel functions on classical domains, Japan. J. Math. (N.S.) 26 (2000), no. 1, 51--103.
Mathematical Reviews (MathSciNet): MR1771435
K. Okamoto, M. Tsukamoto and K. Yokota, Vector bundle-valued Poisson and Cauchy kernel functions on classical domains, Acta Appl. Math. 63 (2000), no. 1-3, 323--332.
Mathematical Reviews (MathSciNet): MR1834228
Digital Object Identifier: doi:10.1023/A:1010741717802
Zentralblatt MATH: 0973.22005
T. Oshima, Poisson transformations on affine symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 9, 323--327.
Mathematical Reviews (MathSciNet): MR555057
Project Euclid: euclid.pja/1195517108
T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), no. 1, 1--81.
Mathematical Reviews (MathSciNet): MR564184
Digital Object Identifier: doi:10.1007/BF01389818
Zentralblatt MATH: 0434.58020
D. Šmí d, The Poisson transform for higher order differential operators, Rend. Circ. Mat. Palermo (2) Suppl. No. 75 (2005), 317--323.
Mathematical Reviews (MathSciNet): MR2152370
G. Warner, Harmonic analysis on semi-simple Lie groups. I, II, Springer, New York, 1972.
Mathematical Reviews (MathSciNet): MR498999
Zentralblatt MATH: 0265.22020
A. A. Artemov, The Poisson transform for a hyperboloid of one sheet, translation in Sb. Math. 195 (2004), 643--667.
Mathematical Reviews (MathSciNet): MR2091638
M. I. Graev, Irreducible unitary representations of the group of third order matrices conserving an indefinite Hermite form, Dokl. Akad. Nauk SSSR (N.S.) 113 (1957), 966--969.
Mathematical Reviews (MathSciNet): MR104743
Harish-Chandra, Representations of semisimple Lie groups V , VI, Amer. J. Math. 78 (1956), 1--41, 564--628.
Mathematical Reviews (MathSciNet): MR82055
Digital Object Identifier: doi:10.2307/2372481
JSTOR: links.jstor.org
Zentralblatt MATH: 0070.11602
M. Hashizume, A. Kowata, K. Minemura and K. Okamoto, An integral representaion of an eigenfunction of the Laplacian on the Euclidean space, Hiroshima Math. J. 2 (1972), 535--545.
Mathematical Reviews (MathSciNet): MR430161
Project Euclid: euclid.hmj/1206137631
M. Hashizume, K. Minemura and K. Okamoto, Harmonic functions on Hermitian hyperbolic spaces, Hiroshima Math. J. 3 (1973), 81--108.
Mathematical Reviews (MathSciNet): MR361628
Project Euclid: euclid.hmj/1206137443
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press Inc. (1962).
Mathematical Reviews (MathSciNet): MR145455
Zentralblatt MATH: 0111.18101
S. Helgason, A duality for symmetric spaces with applications to group representations, Advan. Math. 5 (1970), 1--154.
Mathematical Reviews (MathSciNet): MR263988
Digital Object Identifier: doi:10.1016/0001-8708(70)90037-X
S. Helgason, Eigenspaces of the Laplacian; integral representations and irreducibility, J. Functional Analysis 17 (1974), 328--353.
Mathematical Reviews (MathSciNet): MR367111
Digital Object Identifier: doi:10.1016/0022-1236(74)90045-7
Zentralblatt MATH: 0303.43021
S. Helgason, Groups and Geometric Analysis, Academic Press Inc. (1984).
Mathematical Reviews (MathSciNet): MR754767
Zentralblatt MATH: 0543.58001
K. Hiraoka, S. Matsumoto and K. Okamoto, Eigenfunctions of the Laplacian on a real hyperboloid of one Sheet, Hiroshima Math. J. 7 (1977), 855--864.
Mathematical Reviews (MathSciNet): MR622203
Project Euclid: euclid.hmj/1206135664
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, A.M.S. (Translations of mathematical monographs) 6 (1963).
Mathematical Reviews (MathSciNet): MR171936
T. Inoue, K. Okamoto and M. Tanaka, An integral representaion of an eigenfunction of invariant differential operators on a symmetric space, Hiroshima Math. J. 4 (1974), 413--419.
Mathematical Reviews (MathSciNet): MR364550
Project Euclid: euclid.hmj/1206137071
E. Imamura, K. Okamoto, M. Tsukamoto and A. Yamamori, Eigenvalues of Generalized Laplacians for Generalized Poisson-Cauchy transforms on classical domains, to appear.
Mathematical Reviews (MathSciNet): MR2293504
Project Euclid: euclid.pja/1165244966
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.
Mathematical Reviews (MathSciNet): MR485861
Digital Object Identifier: doi:10.2307/1971253
A. Kowata, and K. Okamoto, Harmonic functions and the Borel-Weil theorem, Hiroshima Math. J. 4 (1974), 89--97.
Mathematical Reviews (MathSciNet): MR457796
Project Euclid: euclid.hmj/1206137155
M. S. Narasimhan and K. Okamoto, An analogue of the Borel-Weil-Bott theorem for hermitian symmetric pairs of non-compact type, Ann. of Math. 91 (1970), 486--511.
Mathematical Reviews (MathSciNet): MR274657
Digital Object Identifier: doi:10.2307/1970635
JSTOR: links.jstor.org
K. Okamoto, Harmonic analysis on homogeneous vector bundles, Lecture Notes in Mathematics, Springer-Verlag 266 (1971), 255--271.
Mathematical Reviews (MathSciNet): MR486323
Zentralblatt MATH: 0234.43009
K. Okamoto, M. Tsukamoto and K. Yokota, Generalized Poisson and Cauchy kernel functions on classical domains, Japan. J. Math. 26 No.1 (2000), 51-103.
Mathematical Reviews (MathSciNet): MR1771435
K. Okamoto, M. Tsukamoto and K. Yokota, Vector bundle valued Poisson and Cauchy kernel functions on classical domains, Acta. Appl. Math.63 (2000), 323--332.
Mathematical Reviews (MathSciNet): MR1834228
Digital Object Identifier: doi:10.1023/A:1010741717802
Zentralblatt MATH: 0973.22005
T. Oshima, Poisson transformations on affine symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 323--327.
Mathematical Reviews (MathSciNet): MR555057
Project Euclid: euclid.pja/1195517108
T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), 1--81.
Mathematical Reviews (MathSciNet): MR564184
Digital Object Identifier: doi:10.1007/BF01389818
Zentralblatt MATH: 0434.58020
D. Smid, The Poisson transform for higher order differential operators, Rend. Circ. Mat. Palermo Suppl. 75 (2005), 317--323.
Mathematical Reviews (MathSciNet): MR2152370
G. Warner, Harmonic analysis of semi-simple Lie groups I , II, Springer-Verlag, Berlin-Heidelberg-New York (1973).
Mathematical Reviews (MathSciNet): MR498999
Zentralblatt MATH: 0265.22020
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