Springer's Weyl group representation via localization
release_6i5r35mizbdjrpjep5jgraexbm
by
Jim Carrell, Kiumars Kaveh
(2017)
Abstract
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. Wellknown 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 parabolicsurjective. 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 parabolicsurjective 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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article
Stage
submitted
Date 20170212
Version
v3
Language
en
^{?}
1505.06404v3
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