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Local systems on nilpotent orbits and weighted Dynkin diagrams


Authors: Pramod N. Achar and Eric N. Sommers
Journal: Represent. Theory 6 (2002), 190-201
MSC (2000): Primary 17B10, 32L20
DOI: https://doi.org/10.1090/S1088-4165-02-00174-7
Published electronically: September 5, 2002
MathSciNet review: 1927953
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Abstract: We study the Lusztig-Vogan bijection for the case of a local system. We compute the bijection explicitly in type A for a local system and then show that the dominant weights obtained for different local systems on the same orbit are related in a manner made precise in the paper. We also give a conjecture (putatively valid for all groups) detailing how the weighted Dynkin diagram for a nilpotent orbit in the dual Lie algebra should arise under the bijection.


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  • 1. P. N. Achar, Equivariant coherent sheaves on the nilpotent cone for complex reductive Lie groups, Ph.D. thesis, Massachusetts Institute of Technology, 2001.
  • 2. R. Bezrukavnikov, On tensor categories attached to cells in affine Weyl groups, arXiv:math.RT/0010089.
  • 3. -, Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, arXiv:math.RT/0201073.
  • 4. -, Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone, arXiv:math.RT/0102039.
  • 5. T. Chmutova and V. Ostrik, Calculating distinguished involutions in the affine Weyl groups, arXiv:math.RT/0106011.
  • 6. D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 94j:17001
  • 7. M. Demazure, A very simple proof of Bott's Theorem, Invent. Math. 33 (1976), 271-272. MR 54:2670
  • 8. V. Hinich, On the singularities of nilpotent orbits, Israel J. Math 73 (1991), no. 3, 297-308. MR 92m:14005
  • 9. G. Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, 107, Princeton University Press, Princeton, N.J., 1984. MR 86j:20038
  • 10. -, Cells in affine Weyl groups. IV, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 36 (1989), 297-328. MR 90k:20068
  • 11. -, Notes on unipotent classes, Asian J. Math. 1 (1997), 194-207. MR 98k:20078
  • 12. W. McGovern, Rings of regular functions on nilpotent orbits and their covers, Invent. Math. 97 (1989), 209-217. MR 90g:22022
  • 13. V. Ostrik, On the equivariant $K$-theory of the nilpotent cone, Represent. Theory 4 (2000), 296-305. MR 2001g:19001
  • 14. D. Panyushev, Rationality of singularities and the Gorenstein property of nilpotent orbits, Funct. Anal. Appl. 25 (1991), no. 3, 225-226. MR 92i:14047
  • 15. E. Sommers, Lusztig's canonical quotient and generalized duality, J. of Algebra (2001), no. 243, 790-812. MR 2002f:20066
  • 16. N. Xi, The based ring of two-sided cells of affine Weyl groups of type $\widetilde A\sb {n-1}$, Memoirs of the Amer. Math. Soc. 157 (2002), no. 749.

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Additional Information

Pramod N. Achar
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: pramod@math.uchicago.edu

Eric N. Sommers
Affiliation: Department of Mathematics, University of Massachusetts—Amherst, Amherst, Massachusetts 01003
Address at time of publication: School of Mathematics, IAS, Princeton, New Jersey 08540

DOI: https://doi.org/10.1090/S1088-4165-02-00174-7
Received by editor(s): December 14, 2001
Received by editor(s) in revised form: July 26, 2002
Published electronically: September 5, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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