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Invariant spherical distributions and the Fourier inversion formula on GL(n, C)/GL(n, R)
Shigeru SANO
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Shigeru SANO
Department of Mathematics The Institute of Vocational Training
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1984 Volume 36 Issue 2 Pages 191-219
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- Published: 1984 Received: February 23, 1983 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : Invariant spherical distributions and the Fourier inversion formula on GL(n, C) /GL(n, R)
Right : Invariant spherical distributions and the Fourier inversion formula on GL(n, C)/GL(n, R)
Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl., 58 (1978), 369-444.
2) I. M. Gel'fand, M. I. Graev and N. Ya. Vilenkin, Generalized functions, Vol. 5, Academic Press, 1966.
3) Harish-Chandra, Discrete series for semisimple Lie groups I, II, Acta Math., 113 (1965), 241-318; Acta Math., 116 (1966), 1-111.
4) Harish-Chandra, Harmonic analysis on real reductive groups I, J. Functional Analysis. 19 (1975), 104-204.
5) Harish-Chandra, Harmonic analysis on real reductive groups II, Invent. Math., 36 (1976), 1-55.
6) Harish-Chandra, Harmonic analysis on real reductive groups III, Ann. of Math., 104 (1976), 117-201.
7) S. Helgason, Analysis on Lie groups and homogeneous spaces, Regional Conference Series in Mathematics, Amer. Math. Soc., 14, 1972.
8) T. Hirai, The characters of some induced representations of semisimple Lie groups, J. Math. Kyoto Univ., 8 (1968), 313-363.
9) T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups I, Japan. J. Math., 39 (1970), 1-68.
10) T. Hirai, The Plancherel formula for SU (p, q), J. Math. Soc. Japan, 22 (1970), 134-179.
11) T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups II, Japan. J. Math. New Series, 2 (1976), 27-89.
12) T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups III, Japan. J. Math. New Series, 2 (1976), 269-341.
13) T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups IV, Japan. J. Math. New Series, 3 (1977), 1-48.
14) T. Hirai, The characters of the discrete series for semisimple Lie groups, J. Math. Kyoto Univ., 21 (1981), 418-500.
15) R. Lipsman, On the characters and equivalence of continuous series representations, J. Math. Soc. Japan., 23 (1971), 452-480.
16) H. Midorikawa, On the explicit formu1a of characters in the discrete series, Japan. J. Math. New Series, 3(1977), 313-368.
17) U. F. Molcanov, Analogue of the Plancherel formula for hyperboloids, Soviet Math. Dokl., 9(1968), 1382-1385.
18) S. Moriguchi, K. Udagawa and S. Hitotumatu, Mathematical formulas, II, Iwanami, 1956.
19) T. Oshima and T. Matsuki, Orbits on affine symmetric spaces under the actions of the isotropy subgroups, J. Math. Soc. Japan, 32 (1980), 399-414.
20) T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math., 57 (1980), 1-81.
21) B. D. Romm, Analogue to the Plancherel formula for the real unimodular group of the n-th order, Amer. Math. Soc. Transl. (2), 58 (1966), 155-215.
22) S. Sano and J. Sekiguchi, The Plancherel formula for SL(2, C)/SL(2, R), Sci. Papers College Gen. Ed. Univ. Tokyo, 30 (1980), 93-105.
23) S. Sano, The inversion formula for Sp(p, q), Bull. Inst. Vocational Training, 11-A (1982), 35-52.
24) S. Sano, The Plancherel formula for Sp(n, R), Lecture in Math. Kyoto Univ., 14 (1982), 223-259.
25) T. Shintani, On the decomposition of regular representation of the Lorentz group on a hyperboloid of one sheet, Proc. Japan Acad., 43 (1967), 1-5.
26) R. S. Strichartz, Harmonic analysis on hyperboloids, J. Functional Analysis, 12 (1973), 341-383.
27) M. Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan, 11 (1959), 374-434.
28) M. Sugiura, Unitary representations and harmonic analysis, Kodansha, Tokyo, 1975.
29) R. Takahashi, Sur les functions spheriques et la formule de Plancherel dans le groupe hyperbolique, Japan. J. Math., 31 (1961), 55-90.
30) G. Warner, Harmonic analysis on semi-simple Lie groups I, II, Springer-Verlag, New York, 1972.
Right : [1] J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl., 58 (1978), 369-444.
[2] I. M. Gel'fand, M. I. Graev and N. Ya. Vilenkin, Generalized functions, Vol. 5, Academic Press, 1966.
[3] Harish-Chandra, Discrete series for semisimple Lie groups I, II, Acta Math., 113 (1965), 241-318; Acta Math., 116 (1966), 1-111.
[4] Harish-Chandra, Harmonic analysis on real reductive groups I, J. Functional Analysis. 19 (1975), 104-204.
[5] Harish-Chandra, Harmonic analysis on real reductive groups II, Invent. Math., 36 (1976), 1-55.
[6] Harish-Chandra, Harmonic analysis on real reductive groups III, Ann. of Math., 104 (1976), 117-201.
[7] S. Helgason, Analysis on Lie groups and homogeneous spaces, Regional Conference Series in Mathematics, Amer. Math. Soc., 14, 1972.
[8] T. Hirai, The characters of some induced representations of semisimple Lie groups, J. Math. Kyoto Univ., 8 (1968), 313-363.
[9] T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups I, Japan. J. Math., 39 (1970), 1-68.
[10] T. Hirai, The Plancherel formula for SU(p,q), J. Math. Soc. Japan, 22 (1970), 134-179.
[11] T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups II, Japan. J. Math. New Series, 2 (1976), 27-89.
[12] T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups III, Japan. J. Math. New Series, 2 (1976), 269-341.
[13] T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups IV, Japan. J. Math. New Series, 3 (1977), 1-48.
[14] T. Hirai, The characters of the discrete series for semisimple Lie groups, J. Math. Kyoto Univ., 21 (1981), 418-500.
[15] R. Lipsman, On the characters and equivalence of continuous series representations, J. Math. Soc. Japan., 23 (1971), 452-480.
[16] H. Midorikawa, On the explicit formula of characters in the discrete series, Japan. J. Math. New Series, 3 (1977), 313-368.
[17] U. F. Molcanov, Analogue of the Plancherel formula for hyperboloids, Soviet Math. Dokl., 9 (1968), 1382-1385.
[18] S. Moriguchi, K. Udagawa and S. Hitotumatu, Mathematical formulas, II, Iwanami, 1956.
[19] T. Oshima and T. Matsuki, Orbits on affine symmetric spaces under the actions of the isotropy subgroups, J. Math. Soc. Japan, 32 (1980), 399-414.
[20] T. Oshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math., 57 (1980), 1-81.
[21] B. D. Romm, Analogue to the Plancherel formula for the real unimodular group of the n-th order, Amer. Math. Soc. Transl. (2), 58 (1966), 155-215.
[22] S. Sano and J. Sekiguchi, The Plancherel formula for SL(2,C)/SL(2,R), Sci. Papers College Gen. Ed. Univ. Tokyo, 30 (1980), 93-105.
[23] S. Sano, The inversion formula for Sp(p,q), Bull. Inst. Vocational Training, 11-A (1982), 35-52.
[24] S. Sano, The Plancherel formula for Sp(n,R), Lecture in Math. Kyoto Univ., 14 (1982), 223-259.
[25] T. Shintani, On the decomposition of regular representation of the Lorentz group on a hyperboloid of one sheet, Proc. Japan Acad., 43 (1967), 1-5.
[26] R. S. Strichartz, Harmonic analysis on hyperboloids, J. Functional Analysis, 12 (1973), 341-383.
[27] M. Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan, 11 (1959), 374-434.
[28] M. Sugiura, Unitary representations and harmonic analysis, Kodansha, Tokyo, 1975.
[29] R. Takahashi, Sur les functions spheriques et la formule de Plancherel dans le groupe hyperbolique, Japan. J. Math., 31 (1961), 55-90.
[30] G. Warner, Harmonic analysis on semi-simple Lie groups I, II, Springer-Verlag, New York, 1972.
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