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On Algebraic Groups and Discontinuous Groups

Published online by Cambridge University Press:  22 January 2016

Takashi Ono
Takashi Ono*
Affiliation:
University of Pennsylvania
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Let G be a connected semi-simple algebraic group defined over Q and let Γ be a discrete subgroup of GR (the subgroup of G consisting of points rational over R) such that Γ\GR is compact. The main purpose of the present paper is to prove that for a certain type of group G there exists an invariant algebraic differential from ω on G of highest degree defined over Q such that

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 279 - 322
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Allendoerfer, C. B. and Weil, A., The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943), 10–129.Google Scholar
[2] Alexadroff, P. and Hopf, H., Topologie I, Berlin, 1935.CrossRefGoogle Scholar
[3] Borei, A., Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115–207.Google Scholar
[4] Borel, A., Some finiteness properties of adele groups over number fields, I.H.E.S. No. 16, 1963, 101–126.Google Scholar
[5] Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces, I, Amer. J. Math. 80 (1958), 459–538, II, ibid. 81 (1959), 315–382.Google Scholar
[6] Harish-Chandra, , Differential operators on a semi-simple Lie algebra. Amer. J. Math. 79 (1957), 87–120.CrossRefGoogle Scholar
[7] Chern, S. S., A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45 (1944), 747–752.CrossRefGoogle Scholar
[8] Chevalley, C., Théorie des groupes de Lie II, Paris, 1951.Google Scholar
[9] Chevalley, C., Sur certains groupes simples, Tôhoku Math. J. 7 (1955), 14–66.Google Scholar
[10] Séminaire, C. Chevalley, Classification des groupes de Lie algébriques, E.N.S., 1958.Google Scholar
[11] Helgason, S., Differential geometry and symmetric spaces, Academic Press, New York, 1962.Google Scholar
[12] Klingen, H., Über die Werte der Dedekindschen Zetafunktion, Math. Annalen 145 (1962), 265–272.Google Scholar
[13] Lang, S., Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563.CrossRefGoogle Scholar
[14] Séminaire, S. Lie, E.N.S., 1955.Google Scholar
[15] Leopoldi, H. W., Eine Verallgemeinerung der Bernoullischen Zahlen, Abh. Math, Sem. Hamburg, 22 (1958), 131–140.Google Scholar
[16] Ono, T., Sur les groupes de Chevalley, J. Math. Soc. Japan 10 (1958), 307–313.Google Scholar
[17] Ono, T., On the field of definition of Borel subgroups of semi-simple algebraic groups, J. Math. Soc. Japan 15 (1963), 392–395.Google Scholar
[38] Ono, T., On the relative theory of Tamagawa numbers, Ann. of Math. 82 (1965),88–111.Google Scholar
[19] Ree, R., On some simple groups defined by C. Chevalley, Trans. Amer. Math. Soc. 84 (1957), 392–400.Google Scholar
[20] Samelson, H., Topology of Lie groups, Bull. Amer. Math. Soc. 58 (1952), 2–37.CrossRefGoogle Scholar
[21] Satake, I., The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan 9 (1957), 464–492.CrossRefGoogle Scholar
[22] Siegel, C. L., Symplectic geometry, Amer. J. Math. 65 (1943), 1–86.Google Scholar
[23] Siegel, C. L., Über die analytische Theorie der quadratischen Formen III. Ann. of Math. 38 (1937), 212–291.CrossRefGoogle Scholar
[24] Steinberg, R., Variations on a theme of Chevalley, Pacific J. of Math. 9 (1959), 875–891.Google Scholar
[25] Weil, A., Adeles and algebraic groups, The Institute for Advanced Study, Princeton N.J., 1961.Google Scholar