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On the equivariant $K$-theory of the nilpotent cone

Author: Viktor Ostrik
Journal: Represent. Theory 4 (2000), 296-305
MSC (2000): Primary 20G05; Secondary 14L30
Published electronically: July 31, 2000
MathSciNet review: 1773863
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In this note we construct a ``Kazhdan-Lusztig type'' basis in equivariant $K$-theory of the nilpotent cone of a simple algebraic group $G$. This basis conjecturally is very close to the basis of this $K$-group consisting of irreducible bundles on nilpotent orbits. As a consequence we get a natural (conjectural) construction of Lusztig's bijection between dominant weights and pairs {nilpotent orbit $\mathcal O$, irreducible $G$-bundle on $\mathcal O$}.

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  • 1. H. H. Andersen, Tensor products of quantized tilting modules, Comm. Math. Phys. 149 (1992), 149-159. MR 94b:17015
  • 2. H. H. Andersen, J. C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), 487-525. MR 86g:20057
  • 3. H. H. Andersen, J. C Jantzen, W. Soergel, Representations of quantum groups at $p$th root of unity and of semi-simple groups in characteristic $p$: independence of $p$, Asterisque, 220 (1994). MR 95j:20036
  • 4. A. Broer, Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113 (1993), 1-20. MR 94g:14027
  • 5. A. Broer, Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety, in ``Lie Theory and Geometry. In Honor of Bertram Kostant", (J.-L. Brylinski et al., Eds.), P.M., Vol. 123, 1-19, Birkhäuser, Boston, 1994. MR 96g:14042
  • 6. A. Broer, Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (5) (1998), 929-971. MR 99k:14077
  • 7. N. Chriss, V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston, 1997. MR 98i:22021
  • 8. V. Ginzburg, Perverse sheaves on a loop group and Langlands' duality, preprint alg-geom/9511007.
  • 9. V. Ginzburg, S. Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J., 69 No. 1 (1993), 179-198. MR 94c:17026
  • 10. W. Graham, Functions on the universal cover of principal nilpotent orbit, Invent. Math. 108 (1992), 15-27. MR 93h:22026
  • 11. V. Hinich, On the singularities of nilpotent orbits, Israel J. Math. 73 (1991), 297-308. MR 92m:14005
  • 12. J. Humphreys, Comparing modular representations of semisimple groups and their Lie algebras, Modular Interfaces (Riverside, CA, 1995), 69-80, AMS/IP Stud. Adv. Math., 4, Amer. Math. Soc., Providence, RI, 1997. MR 98h:17006
  • 13. G. Lusztig, Nonlocal finiteness of a $W$-graph, Representation Theory 1 (1997), 25-30. MR 98c:20078
  • 14. G. Lusztig, Cells in affine Weyl groups, Algebraic Groups and Related Topics, Adv. Studies in Pure Math. vol. 6, North Holland and Kinokuniya, Amsterdam and Tokyo, 1985, 255-287; II, J. Algebra 109 (1987), 536-548; III, J. Fac. Sci. Univ. Tokyo (IA) 34 (1987), 223-243; IV, J. Fac. Sci. Univ. Tokyo (IA) 36 (1989), 297-328. MR 87h:20074; MR 88m:20103a; MR 88m:20103b; MR 90k:20068
  • 15. G. Lusztig, Bases in equivariant K-theory, Representation Theory 2 (1998), 298-369. MR 99i:19005
  • 16. W. McGovern, Rings of regular functions on nilpotent orbits and their covers, Invent. Math. 97 (1989) 209-217. MR 90g:22022
  • 17. W. McGovern, A branching law for $Spin(7, {\mathbb C})\to G_2$ and its applications to unipotent representations, J. Alg. 130 (1990), 165-175. MR 91d:22010
  • 18. V. Ostrik, Cohomology of subregular tilting modules for small quantum groups, preprint q-alg/9902094.
  • 19. D. Panyushev, Rationality of singularities and the Gorenstein property for nilpotent orbits, Funct. Anal. Appl. 25 (1991), 225-226. MR 92i:14047
  • 20. W. Soergel, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Representation Theory 1 (1997), 37-68. MR 99d:17023

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Additional Information

Viktor Ostrik
Affiliation: Independent Moscow University, 11 Bolshoj Vlasjevskij per., Moscow 121002 Russia

Received by editor(s): November 16, 1999
Received by editor(s) in revised form: April 19, 2000
Published electronically: July 31, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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