Springer's Weyl group representation via localization release_6i5r35mizbdjrpjep5jgraexbm

by Jim Carrell, Kiumars Kaveh

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Let G denote a reductive algebraic group over C and x a nilpotent element of its Lie algebra g. The Springer variety B_x is the closed subvariety of the flag variety B of G parameterizing the Borel subalgebras of g containing x. It has the remarkable property that the Weyl group W of G admits a representation on the cohomology of B_x even though W rarely acts on B_x itself. Well-known constructions of this action due to Springer et al use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when x is what we call parabolic-surjective. The idea is to use localization to construct an action of W on the equivariant cohomology algebra H_S^*(B_x), where S is a certain algebraic subtorus of G. This action descends to H^*(B_x) via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type A and, more generally, all nilpotents for which it is known that W acts on H_S^*(B_x) for some torus S. Our result is deduced from a general theorem describing when a group action on the cohomology of the fixed point set of a torus action on a space lifts to the full cohomology algebra of the space.
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Type  article
Stage   submitted
Date   2017-02-12
Version   v3
Language   en ?
arXiv  1505.06404v3
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