Local systems on nilpotent orbits and weighted Dynkin diagrams
HTML articles powered by AMS MathViewer
 by Pramod N. Achar and Eric N. Sommers PDF
 Represent. Theory 6 (2002), 190201 Request permission
Abstract:
We study the LusztigVogan 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:quasiexc —, Quasiexceptional sets and equivariant coherent sheaves on the nilpotent cone, arXiv:math.RT/0102039.
chmostrik:distinv 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/S1088416500000893
 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 twosided cells of affine Weyl groups of type $\widetilde A_ {n1}$, Memoirs of the Amer. Math. Soc. 157 (2002), no. 749.
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), 190201
 MSC (2000): Primary 17B10, 32L20
 DOI: https://doi.org/10.1090/S1088416502001747
 MathSciNet review: 1927953