Related papers: Kempf collapsing and quiver loci

Let $G$ be a reductive group acting on a path algebra $kQ$ as automorphisms. We assume that $G$ admits a graded polynomial representation theory, and the action is polynomial. We describe the quiver $Q_G$ of the smash product algebra $kQ\#…

Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the…

Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases,…

The work of the first author on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an…

A. Vistoli proved a decomposition theorem for the rational equivariant algebraic K-theory of a variety under the action of a finite group $G$. We generalize his result to more general algebraic (co)homology theories having the Mackey…

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…

We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In…

We classify graded Hopf algebras structures over path coalgebras, that is over free pointed coalgebras, using Hopf quivers which are analogous to Cayley graphs. The description involves formulas for the product besides the canonical…

In the setting of a variety $X$ admitting a tilting bundle $T$ we consider the problem of constructing $X$ as a quiver GIT quotient of the endomorphism algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting bundle. We…

Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Mat_d(Q). The algebra D_q is…

Given a simple, simply connected compact Lie group G, let M be a G-space. We describe the quantization of the category of positive energy representations of the loop group of G at a given level and parametrized over the loop space LM. This…

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of…

Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the…

An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work…

This paper is a continuation of our previous work in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of…

The moduli stack of representations of a quiver, or coherent sheaves on a proper curve, carries two structures on its cohomology: a Hall algebra and braided vertex coalgebra. We show that they are compatible, by developing a formulation of…

Let $\mathcal A$ be a hyperplane arrangement in a vector space $V$ and $G \leq GL(V)$ a group fixing $\mathcal A$. In case when $G$ is a complex reflection group and $\mathcal A=\mathcal A(G)$ is its reflection arrangement in $V$, Douglass,…

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group $G_\aff$ [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck…

We give an answer to the abstract Capelli problem: Let $(G, V)$ be a multiplicity-free finite-dimensional representation of a connected reductive complex Lie group $G$ and $G'$ be its derived subgroup. Assume that the categorical quotient…

We define the equivariant degree and local degree of a proper $G$-equivariant map between smooth $G$-manifolds when $G$ is a compact Lie group and prove a local to global result. We show the local degree can be used to compute the…