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.

本文研究了三个相互关联的品种的几何形状和组合：斯普林格纤维，斯坦伯格品种和抛物线型海森伯格品种。我们证明了每个抛物型的Hessenberg变种都是Steinberg变种在旗形变种到适当的部分旗形变种的投影下的回缩，并且给出了此结果的三个应用。第一个应用程序以半标准餐具的形式为所有Lie型A的Steinberg品种构造了一个明确的铺路石。结果，我们得到了斯坦伯格和下村定理的基本证明，即众所周知的科斯特卡数计算了斯坦伯格变种的最大维不可约成分。第二个应用程序通过证明其Betti数等于Schubert变体特定联盟的贝蒂数，证明了某些抛物线型Hesenberg变种的公开猜想。第三个应用证明，抛物线型Hessenberg变种的不可约成分与Steinberg变种的不可约成分是双射的。所有这三个应用程序扩展了我们对这三个品种的几何理解，这是本文的核心，尽管有40多年的工作经验，但即使对于Springer品种，也无法完全了解。