$K$-Orbit closures and Hessenberg varieties

TL;DR

This paper investigates the geometric and cohomological properties of Hessenberg varieties linked to semisimple operators with two eigenvalues, revealing their irreducibility, dimension, and intersection characteristics, with connections to Catalan numbers.

Contribution

It establishes conditions for irreducibility, computes dimensions, and analyzes intersection properties of Hessenberg varieties associated with specific semisimple operators.

Findings

Number of such Hessenberg varieties equals a Catalan number

Proved irreducibility under certain conditions

Computed cohomology classes and intersection multiplicities

Abstract

This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under which these semisimple Hessenberg varieties are irreducible. We determine the dimension of each irreducible Hessenberg variety under consideration and show that the number of such varieties is a Catalan number. We then apply a theorem of Brion to compute a polynomial representative for the cohomology class of each such variety. Additionally, we calculate the intersections of a standard (Schubert) hyperplane section of the flag variety with each of our Hessenberg varieties and prove this intersection possess a cohomological multiplicity-free property.