Irreducible Characteristic Cycles for Orbit Closures of a Symmetric Subgroup

TL;DR

This paper investigates the geometry and characteristic cycles of certain orbit closures in the flag variety of GL(n), demonstrating their irreducibility and computing their Chern-Mather classes using resolutions and localization techniques.

Contribution

It proves the irreducibility of characteristic cycles for these orbit closures and provides a method to compute their Chern-Mather classes via explicit resolutions and localization.

Findings

Orbit closures have irreducible characteristic cycles.

Resolutions are smooth and strongly reduced.

Chern-Mather classes are computed explicitly using localization.

Abstract

Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a central role in the infinite-dimensional representation theory of the real Lie group $U(p,q)$, and has applications to degeneracy loci and combinatorics. In this paper we use small resolutions to study orbit closures in this family. We prove that the fibers of these resolutions are smooth and strongly reduced, as well as a general result that if a variety has a resolution of singularities with these properties, then its characteristic cycle is irreducible. Hence these orbit closures have irreducible characteristic cycles. A result of Jones then allows us to calculate the torus-equivariant Chern-Mather classes of these orbit closures. We describe torus fixed points and tangent spaces of the resolutions, and use localization to obtain a formula for these classes. We conjecture that the Chern-Mather classes of a $K$-orbit closure are equivariantly positive when expressed in a Schubert basis of equivariant Borel-Moore homology, and use our results to verify the conjecture in an example.