1. CMB Online first
 Carrell, Jim; Kaveh, Kiumars

Springer's Weyl Group Representation via Localization
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.
Keywords:Springer variety, Weyl group action, equivariant cohomology Categories:14M15, 14F43, 55N91 

2. CMB 2016 (vol 59 pp. 483)
 Crooks, Peter; Holden, Tyler

Generalized Equivariant Cohomology and Stratifications
For $T$ a compact torus and $E_T^*$ a generalized $T$equivariant
cohomology theory, we provide a systematic framework for computing
$E_T^*$ in the context of equivariantly stratified smooth complex
projective varieties. This allows us to explicitly compute $E_T^*(X)$
as an $E_T^*(\text{pt})$module when $X$ is a direct limit of
smooth complex projective $T_{\mathbb{C}}$varieties with finitely
many $T$fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$,
$K_T^*$, and $MU_T^*$. We perform this computation on the affine
Grassmannian of a complex semisimple group.
Keywords:equivariant cohomology theory, stratification, affine Grassmannian Categories:55N91, 19L47 

3. CMB 2014 (vol 58 pp. 80)
 Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya

The Equivariant Cohomology Rings of Peterson Varieties in All Lie
Types
Let $G$ be a complex semisimple linear algebraic group and let
$Pet$ be the associated Peterson variety in the flag
variety $G/B$.
The main theorem of this note gives an efficient presentation
of the equivariant cohomology ring $H^*_S(Pet)$ of the
Peterson variety as a quotient of a polynomial ring by an ideal
$J$ generated by quadratic polynomials, in the spirit of the
Borel presentation of the cohomology of the flag variety. Here
the group $S \cong \mathbb{C}^*$ is a certain subgroup of a maximal
torus $T$ of $G$.
Our description of the ideal $J$ uses the Cartan matrix and is
uniform across Lie types. In our arguments we use the Monk formula
and Giambelli formula for the equivariant cohomology rings of
Peterson varieties for all Lie types, as obtained in the work
of Drellich. Our result generalizes a previous theorem of FukukawaHaradaMasuda,
which was only for Lie type $A$.
Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula Categories:55N91, 14N15 

4. CMB 2007 (vol 50 pp. 365)
 Godinho, Leonor

Equivariant Cohomology of $S^{1}$Actions on $4$Manifolds
Let $M$ be a symplectic $4$dimensional manifold equipped with a
Hamiltonian circle action with isolated fixed points. We describe a
method for computing its integral equivariant cohomology in terms of
fixed point data. We give some examples of these computations.
Categories:53D20, 55N91, 57S15 
