# Hessenberg varieties of parabolic type

## TL;DR

This paper explores the geometry and combinatorics of parabolic Hessenberg varieties, establishing their relations with Steinberg and Springer varieties, and provides new proofs and conjecture resolutions in Lie type A.

## Contribution

It proves that parabolic Hessenberg varieties are pullbacks of Steinberg varieties and applies this to construct explicit pavings, prove conjectures, and relate irreducible components.

## Key findings

- Explicit paving of Steinberg varieties in Lie type A using semistandard tableaux
- Elementary proof that Kostka numbers count maximal-dimensional irreducible components
- Betti numbers of certain parabolic Hessenberg varieties match those of specific Schubert unions

## Abstract

This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type $A$ in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-known Kostka numbers count the maximal-dimensional irreducible components of Steinberg varieties. The second application proves an open conjecture for certain parabolic Hessenberg varieties in Lie type A by showing that their Betti numbers equal those of a specific union of Schubert varieties. The third application proves that the irreducible components of parabolic Hessenberg varieties are in bijection with the irreducible components of the Steinberg variety. All three of these applications extend our geometric understanding of the three varieties at the heart of this paper, a full understanding of which is unknown even for Springer varieties, despite over forty years' worth of work.