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Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra


Author: Thierry Levasseur
Journal: Trans. Amer. Math. Soc. 353 (2001), 4189-4202
MSC (1991): Primary 14L30, 16S32, 17B20, 22E46
DOI: https://doi.org/10.1090/S0002-9947-01-02851-3
Published electronically: June 1, 2001
MathSciNet review: 1837227
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Abstract: Let $\mathfrak{g}$ be a semisimple complex Lie algebra with adjoint group $G$ and $\mathcal{D}(\mathfrak{g})$ be the algebra of differential operators with polynomial coefficients on $\mathfrak{g}$. If $\mathfrak{g}_0$ is a real form of $\mathfrak{g}$, we give the decomposition of the semisimple $ \mathcal{D}(\mathfrak{g})^G$-module of invariant distributions on $\mathfrak{g}_0$ supported on the nilpotent cone.


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  • 1. J. Arthur, The $L^2$-Lefschetz numbers of Hecke operators, Invent. Math., 97 (1989), 257-290. MR 91i:22024
  • 2. D. Barbasch and D. Vogan, The local structure of characters, J. Funct. Anal., 37 (1980), 27-55. MR 82e:22024
  • 3. -, Primitive ideals and orbital integrals in complex classical groups, Math. Ann., 259 (1982), 153-199. MR 83m:22026
  • 4. -, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra, 80 (1983), 350-382. MR 84h:22038
  • 5. A. Bouaziz, Intégrales orbitales sur les algèbres de Lie réductives, Invent. Math., 115 (1994), 163-207. MR 95a:22017
  • 6. R. W. Carter, Finite Groups of Lie Type, Conjugacy Classes and Complex Characters, John Wiley & Sons, Chichester, 1985. MR 94k:20020
  • 7. D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand, New York, 1993.MR 94j:17001
  • 8. V. Ginsburg, Intégrales sur les orbites nilpotentes et représentations des groupes de Weyl, C. R. Acad. Sci. Paris, 296 (1983), 249-252. MR 85b:22019
  • 9. R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math., 75 (1984), 327-358. MR 87i:22041
  • 10. T. Levasseur, Equivariant $\mathrm{D}$-modules attached to nilpotent orbits in a semisimple Lie algebra, to appear in Transformation Groups, (1998).
  • 11. T. Levasseur and J. T. Stafford, Invariant differential operators and an homomorphism of Harish-Chandra, J. Amer. Math. Soc., 8 (1995), 365-372. MR 2001b:16025
  • 12. -, The kernel of an homomorphism of Harish-Chandra, Ann. Sci. Éc. Normale Sup., 29 (1996), 385-397. MR 97b:22019
  • 13. -, Semi-simplicity of invariant holonomic systems on a reductive Lie algebra, Amer. J. Math., 119 (1997), 1095-1117. MR 99g:17020
  • 14. R. Rao, Orbital integrals on reductive groups, Ann. Math., 96 (1972), 505-510.
  • 15. W. Rossmann, Nilpotent Orbital Integrals in a Real Semisimple Lie Algebra and Representations of Weyl Groups, in Actes du colloque en l'honneur de Jacques Dixmier, Progress in Math. 92, Birkhäuser, Boston, 1990, 333-397. MR 92c:22022
  • 16. -, Invariant Eigendistributions on a Semisimple Lie Algebra and Homology Classes on the Conormal Variety. II. Representations of Weyl Groups, J. Funct. Anal., 96 (1991), 155-193. MR 92g:22034
  • 17. V. S. Varadarajan, Harmonic Analysis on Real Reductive Groups, Part I, Lecture Notes in Mathematics 576, Springer-Verlag, Berlin/New York, 1977. MR 57:12789
  • 18. N. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc., 6 (1993), 779-816. MR 94a:17014

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Additional Information

Thierry Levasseur
Affiliation: Département de Mathématiques, Université de Brest, 29285 Brest, France
Email: Thierry.Levasseur@univ-brest.fr

DOI: https://doi.org/10.1090/S0002-9947-01-02851-3
Keywords: Semisimple Lie algebra, invariant distribution, nilpotent orbit, Weyl group representation
Received by editor(s): November 17, 1998
Published electronically: June 1, 2001
Additional Notes: Research partially supported by EC TMR network “Algebraic Lie Representations”, Grant No. ERB FMRX-CT97-0100
Article copyright: © Copyright 2001 American Mathematical Society

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