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An Algebro-Geometric Realization of Equivariant Cohomology of Some Springer Fibers



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Journal of Algebra 368 2012 70 74 Contents lists available at SciVerse ScienceDirect Journal of Algebra www elsevier com locate jalgebra An algebro geometric realization of equivariant cohomology of some Springer bers Shrawan Kumar a 1 Claudio Procesi b a b Department of Mathematics University of North Carolina Chapel Hill NC 27599 3250 USA Dipartimento di Matematica Sapienza Universit di Roma Piazzale Aldo Moro 5 00185 Roma Italy a r t i c l e i n f o a b s t r a c t Article history Received 30 March 2012 Available online xxxx Communicated by Dihua Jiang We give an explicit a ne algebraic variety whose coordinate ring is isomorphic as a W algebra with the equivariant cohomology of some Springer bers 2012 Elsevier Inc All rights reserved Keywords Equivariant cohomology Springer bers 1 Introduction Let G be a connected simply connected semisimple complex algebraic group with a Borel subgroup B and a maximal torus T B Let P B be a standard parabolic subgroup of G Let L T be the Levi subgroup of P and let S be the connected center of L i e S is the identity component of the center of L Then S T We denote the Lie algebras of G T B P L S by the corresponding Gothic characters g t b p l s respectively Let W be the Weyl group of G and W L W the Weyl group of L Let l be a principal nilpotent element of l Let X G B be the full ag variety of G and let X X be the Springer ber corresponding to the nilpotent element i e X is the subvariety of X xed under the left multiplication by Exp endowed with the reduced subscheme structure Observe that S keeps the variety X stable under the left multiplication of S on X De nition 1 1 Let Z l be the reduced closed subvariety of t t de ned by Z l x wx w W x s 1 Corresponding author E mail addresses shrawan email unc edu S Kumar procesi mat uniroma1 it C Procesi Supported by NSF grants 0021 8693 see front matter 2012 Elsevier Inc All rights reserved http dx doi org 10 1016 j jalgebra 2012 06 019 S Kumar C Procesi Journal of Algebra 368 2012



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