Virtual character modules of semisimple Lie groups and representations of Weyl groups

Kyo NISHIYAMA

doi:10.2969/jmsj/03740719

Kyo NISHIYAMA

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JOURNAL FREE ACCESS

1985 Volume 37 Issue 4 Pages 719-740

DOI https://doi.org/10.2969/jmsj/03740719

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Published: 1985 Received: July 27, 1984 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: CITATION Details: Wrong : 1) D. Barbasch and D. Vogan, Weyl group representations and nilpotent orbits, in Representation theory of reductive groups, edited by C. Trombi, Birkhäuser, 1983.
2) V. Bargman, Irreducible unitary representations of the Lorentz group, Ann. of Math., 48 (1947), 568-640.
3) N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5 et 6, Herman, Paris, 1968.
4) I. M. Gel'fand, M. I. Graev and N. Ya. Vilenkin, Generalized functions, vol. 5, Academic Press, New York and London, 1966.
5) Harish-Chandra, Invariant eigendistributions on semisimple Lie groups, Trans. Amer. Math. Soc., 119 (1965), 457-508.
6) T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups I, Case of SU(p, q), Japan. J. Math., 39 (1970), 1-68.
7) T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups II, General theory for semisimple Lie groups, Japan. J. Math., New Series, 2 (1976), 27-89.
8) T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups III, Methods of construction for semisimple Lie groups, Japan. J. Math., New Series, 2 (1976), 269-341.
9) T. Hirai, Classification and the characters of irreducible representations of SU(p,l), Proc. Japan Acad., 42 (1966), 907-912.
10) A. Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra I, II, J. Algebra, 65 (1980), 269-283, 284-306.
11) D. R. King, The character polynomial of the annihilator of an irreducible Harish Chandra module, Amer. J. Math., 103 (1981), 1195-1240.
12) A. W. Knapp and G. J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math., 116 (1982), 389-501.
13) J. Lepowski, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc., 176 (1973), 1-44.
14) D. E. Littlewood, The theory of group characters and matrix representations of groups, Oxford, 1950.
15) G. Lustig and D. Vogan, Singularities of closures of K-orbits on flag manifolds, Invent. Math., 71 (1983), 265-379.
16) K. Nishiyama, Decompositions of tensor products of infinite and finite dimensional representations of semisimple groups, J. Math. Kyoto Univ., 25 (1985), 1-20.
17) K. Nishiyama, Representations of Weyl group and its subgroups on the virtual character modules, Proc. Japan Acad., 60 (1985), 193-196.
18) G. Warner, Harmonic analysis on semisimple Lie groups I, Springer-Verlag, 1972.
19) G. J. Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math., 106 (1977), 295-308.

Right : [1] D. Barbasch and D. Vogan, Weyl group representations and nilpotent orbits, in Representation theory of reductive groups, edited by C. Trombi, Birkhäuser, 1983.
[2] V. Bargman, Irreducible unitary representations of the Lorentz group, Ann. of Math., 48 (1947), 568-640.
[3] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5 et 6, Herman, Paris, 1968.
[4] I. M. Gel'fand, M. I. Graev and N. Ya. Vilenkin, Generalized functions, vol. 5, Academic Press, New York and London, 1966.
[5] Harish-Chandra, Invariant eigendistributions on semisimple Lie groups, Trans. Amer. Math. Soc., 119 (1965), 457-508.
[6] T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups I, Case of SU (p, q), Japan. J. Math., 39 (1970), 1-68.
[7] T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups II, General theory for semisimple Lie groups, Japan. J. Math., New Series, 2 (1976), 27-89.
[8] T. Hirai, Invariant eigendistributions of Laplace operators on real simple Lie groups III, Methods of construction for semisimple Lie groups, Japan. J. Math., New Series, 2 (1976), 269-341.
[9] T. Hirai, Classification and the characters of irreducible representations of SU (p,l), Proc. Japan Acad., 42 (1966), 907-912.
[10] A. Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra I, II, J. Algebra, 65 (1980), 269-283, 284-306.
[11] D. R. King, The character polynomial of the annihilator of an irreducible Harish Chandra module, Amer. J. Math., 103 (1981), 1195-1240.
[12] A. W. Knapp and G. J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math., 116 (1982), 389-501.
[13] J. Lepowski, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc., 176 (1973), 1-44.
[14] D. E. Littlewood, The theory of group characters and matrix representations of groups, Oxford, 1950.
[15] G. Lustig and D. Vogan, Singularities of closures of K-orbits on flag manifolds, Invent. Math., 71 (1983), 265-379.
[16] K. Nishiyama, Decompositions of tensor products of infinite and finite dimensional representations of semisimple groups, J. Math. Kyoto Univ., 25 (1985), 1-20.
[17] K. Nishiyama, Representations of Weyl group and its subgroups on the virtual character modules, Proc. Japan Acad., 60 (1985), 193-196.
[18] G. Warner, Harmonic analysis on semisimple Lie groups I, Springer-Verlag, 1972.
[19] G. J. Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math., 106 (1977), 295-308.

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