Regions in the dominant chamber and nilpotent orbits

Equivariant coherent sheaves on the exotic nilpotent cone

Let $G=Sp_{2n}(\mathbb{C})$, and $\mathfrak{N}$ be Kato's exotic nilpotent cone. Following techniques used by Bezrukavnikov in [5] to establish a bijection between $Λ^+$, the dominant weights for a simple algebraic group $H$, and $\textbf{O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $\widetilde{\mathfrak{N}}$ have vanishing higher cohomology, and compute their g…

Birational geometry and deformations of nilpotent orbits

This is a continuation of math.AG/0408274, where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution. The purpose of this paper is, to describe the remainder. We shall first construct a deformation of the nilpotent orbit closure in a canonical manner according to Brieskorn and Slodowy, and next describe all its crepant simultaneous resolutions. This construction enables us to d…

On the equivariant K-theory of the nilpotent cone

Let G be a simple algebraic group over the complex numbers. Let N be the cone of nilpotent elements in the Lie algebra of G. Let K_{G x C^*}(N) denote the Grothendieck group of the category of G x C^*-equivariant coherent sheaves on N. In this note we construct a Kazhdan-Lusztig type canonical basis of K_{G x C^*}(N) over representation ring of C^*. This basis is parametrized by the set of dominant weights for G. On the other hand we conjecture that this basis is close to the basis consisting of irreducible G-equivariant bundles on nilpotent orbits. This would give us a natural construction of…

Equations for some nilpotent varieties

Let $\mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ over $\mathbb C$, induced from a Levi subalgebra whose simple roots are orthogonal short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{ \mathcal{O}}$ in $S \mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broer's result for the subregular nilpotent orbit. Along the way we give a…

Geometric Invariant Theory and Generalized Eigenvalue Problem

Let $H$ be a connected reductive subgroup of a complex connected reductive group $G$. Fix maximal tori and Borel subgroups of $H$ and $G$. Consider the pairs $(V,V')$ of irreducible representations of $H$ and $G$ such that $V$ is a submodule of $V'$. We are interested in the cone $LR(G,H)$ generated by the pairs of dominant weights of such a pair of representations. Our main result gives a minimal set of inequalities describing $LR(G,H)$ as a part of the dominant chamber. In way, we obtain results about the faces of the Dolgachev-Hu's $G$-ample cone and variations of this cone.

彼得森簇的几何不变量理论

Geometric Invariant Theory of Peterson Varieties

本文研究彼得森簇的几何不变量理论，利用理查森分层明确其GIT商的稳定与半稳定轨迹，确定腔分解与跨壁态射，还刻画正则性并证明商光滑的充要条件，揭示彼得森簇奇异几何与GIT商变化的关联。