Polynomials for GL_p x GL_q orbit closures in the flag variety

TL;DR

This paper introduces a family of polynomials representing cohomology classes of K=GL_p x GL_q orbit closures in the flag variety, linking algebraic, geometric, and singularity aspects.

Contribution

It defines new polynomials for orbit closures, studies K-orbit determinantal ideals, and connects local singularity measures with Kazhdan-Lusztig-Vogan polynomials.

Findings

Polynomials specialize to cohomology class representatives.

K-orbit determinantal ideals support geometric interpretations.

Established an analogy between H-polynomials and Kazhdan-Lusztig-Vogan polynomials.

Abstract

The subgroup K=GL_p x GL_q of GL_{p+q} acts on the (complex) flag variety GL_{p+q}/B with finitely many orbits. We introduce a family of polynomials that specializes to representatives for cohomology classes of the orbit closures in the Borel model. We define and study K-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the H-polynomials and the Kazhdan-Lusztig-Vogan polynomials.