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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Local systems on nilpotent orbits and weighted Dynkin diagrams
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by Pramod N. Achar and Eric N. Sommers PDF
Represent. Theory 6 (2002), 190-201 Request permission

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.
References
    achar:thesis P. N. Achar, Equivariant coherent sheaves on the nilpotent cone for complex reductive Lie groups, Ph.D. thesis, Massachusetts Institute of Technology, 2001. bezrukavnikov:tensor R. Bezrukavnikov, On tensor categories attached to cells in affine Weyl groups, arXiv:math.RT/0010089. bezrukavnikov:3 —, Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, arXiv:math.RT/0201073. bezrukavnikov:quasi-exc —, Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone, arXiv:math.RT/0102039. chm-ostrik:dist-inv T. Chmutova and V. Ostrik, Calculating distinguished involutions in the affine Weyl groups, arXiv:math.RT/0106011.
  • David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
  • Michel Demazure, A very simple proof of Bott’s theorem, Invent. Math. 33 (1976), no. 3, 271–272. MR 414569, DOI 10.1007/BF01404206
  • V. Hinich, On the singularities of nilpotent orbits, Israel J. Math. 73 (1991), no. 3, 297–308. MR 1135219, DOI 10.1007/BF02773843
  • George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
  • George Lusztig, Cells in affine Weyl groups. IV, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 2, 297–328. MR 1015001
  • G. Lusztig, Notes on unipotent classes, Asian J. Math. 1 (1997), no. 1, 194–207. MR 1480994, DOI 10.4310/AJM.1997.v1.n1.a7
  • William M. McGovern, Rings of regular functions on nilpotent orbits and their covers, Invent. Math. 97 (1989), no. 1, 209–217. MR 999319, DOI 10.1007/BF01850661
  • Viktor Ostrik, On the equivariant $K$-theory of the nilpotent cone, Represent. Theory 4 (2000), 296–305. MR 1773863, DOI 10.1090/S1088-4165-00-00089-3
  • D. I. Panyushev, Rationality of singularities and the Gorenstein property of nilpotent orbits, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 76–78 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 225–226 (1992). MR 1139878, DOI 10.1007/BF01085494
  • Eric Sommers, Lusztig’s canonical quotient and generalized duality, J. Algebra 243 (2001), no. 2, 790–812. MR 1850659, DOI 10.1006/jabr.2001.8868
  • xi:xi N. Xi, The based ring of two-sided cells of affine Weyl groups of type $\widetilde A_ {n-1}$, 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
  • MR Author ID: 701892
  • 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
  • Received by editor(s): December 14, 2001
  • Received by editor(s) in revised form: July 26, 2002
  • Published electronically: September 5, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Represent. Theory 6 (2002), 190-201
  • MSC (2000): Primary 17B10, 32L20
  • DOI: https://doi.org/10.1090/S1088-4165-02-00174-7
  • MathSciNet review: 1927953