Abstract view
Springer's Weyl Group Representation via Localization


Published:20170503
Printed: Sep 2017
Jim Carrell,
Department of Mathematics, University of British Columbia, Vancouver, B.C.
Kiumars Kaveh,
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA
Abstract
Let $G$ denote a reductive algebraic group over
$\mathbb{C}$
and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer
variety $\mathcal{B}_x$
is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing
the
Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable
property that
the Weyl group $W$ of $G$ admits a representation on the cohomology
of $\mathcal{B}_x$
even though $W$ rarely acts on $\mathcal{B}_x$ 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 $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic
subtorus of $G$.
This action descends to $H^*(\mathcal{B}_x)$ 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
$H_S^*(\mathcal{B}_x)$ 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.
MSC Classifications: 
14M15, 14F43, 55N91 show english descriptions
Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] Other algebrogeometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Equivariant homology and cohomology [See also 19L47]
14M15  Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14F43  Other algebrogeometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 55N91  Equivariant homology and cohomology [See also 19L47]
