Megumi Harada

SURJECTIVITY FOR HAMILTONIAN G-SPACES IN K-THEORY

Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma. We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in admits an equivariant extension in .

A positive Monk formula in the S1-equivariant cohomology of type A Peterson varieties

Peterson varieties are a special class of Hessenberg varieties that have been extensively studied, for example, by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley–Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type A n−1 , with respect to a natural S 1 -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac–Moody flag varieties G/P. Our main results are as follows. First, we identify a computationally convenient basis of H * S1 (Y), which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley–Monk formula for the product of a cohomology-degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both H * S1 (Y) and H*(Y) are generated in degree 2. Finally, by using our Chevalley–Monk formula we give explicit descriptions (via generators and relations) of both the S 1 -equivariant cohomology ring H * S1 (Y) and the ordinary cohomology ring H*(Y) of the type A n−1 Peterson variety. Our methods are both directly from and inspired by those of the GKM (Goresky–Kottwitz–MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.

Hyperpolygon spaces and their cores

Given an n-tuple of positive real numbers (α 1 ,.., an), Konno (2000) defines the hyperpolygon space X(a), a hyperkahler analogue of the Kahler variety M(a) parametrizing polygons in R 3 with edge lengths (α 1 ,..., an). The polygon space M(α) can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, X(a) is the hyperkahler quiver variety defined by Nakajima. A quiver variety admits a natural C*-action, and the union of the precompact orbits is called the core. We study the components of the core of X(α), interpreting each one as a moduli space of pairs of polygons in R 3 with certain properties. Konno gives a presentation of the cohomology ring of X(a); we extend this result by computing the C*-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes

We describe, under certain conditions, the Newton-Okounkov body of a Bott-Samelson variety as a lattice polytope defined by an explicit list of inequalities. The valuation that we use to define the Newton-Okounkov body is different from that used previously in the literature. The polytope that arises is a special case of the Grossberg-Karshon twisted cubes studied by Grossberg and Karshon in connection to character formulae for irreducible

ORBIFOLD COHOMOLOGY OF HYPERTORIC VARIETIES

G

The equivariant cohomology rings of Peterson varieties

-representations and also studied previously by the authors in relation to certain toric varieties associated to Bott-Samelson varieties.

The Equivariant Cohomology Rings of Peterson Varieties in All Lie Types

Hypertoric varieties are hyperkahler analogues of toric varieties, and are constructed as abelian hyperkahler quotients T*ℂn//// T of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrangements. By developing hyperkahler analogues of symplectic techniques developed by Goldin, Holm, and Knutson, we give an explicit combinatorial description of the Chen–Ruan orbifold cohomology of an orbifold hypertoric variety in terms of the combinatorial data of a rational cooriented weighted hyperplane arrangement . We detail several explicit examples, including some computations of orbifold Betti numbers (and Euler characteristics).

Poset Pinball, the Dimension Pair Algorithm, and Type A Regular Nilpotent Hessenberg Varieties

The main result of this note is an efficient presentation of the

Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

S^1

Connectivity properties of moment maps on based loop groups

-equivariant cohomology ring of Peterson varieties (in type