We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the Sn-representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case, and generalizes results obtained by Shareshian-Wachs and Teff. Our proof uses previous work of Stanley, Gasharov, Shareshian-Wachs, and Brosnan-Chow, as well as results of the second author on the geometry and combinatorics of Hessenberg varieties. As part of our arguments, we obtain inductive formulas for the Poincaré polynomials of regular abelian Hessenberg varieties. THE SETUP AND BACKGROUNDLet n be a positive integer. We denote by [n] the set of positive integers {1, 2, . . . , n}. We work in type A throughout, so GL(n, C) is the group of invertible n × n complex matrices and gl(n, C) is the Lie algebra of GL(n, C) consisting of all n × n complex matrices.2.1. Hessenberg varieties. Hessenberg varieties in Lie type A are subvarieties of the (full) flag variety Fℓags(C n ), which is the collection of sequences of nested linear subspaces of C n :Fℓags(C n ) := {V • = ({0} ⊂ V 1 ⊂ V 2 ⊂ · · · V n−1 ⊂ V n = C n ) | dim C (V i ) = i for all i = 1, . . . , n}.

In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent and arbitrary elements of gl n (C) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases the Hessenberg variety has no odd dimensional cohomology.

Abstract. Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs describing a representation of the symmetric group on the cohomology of regular semisimple Hessenberg varieties for GLn(C). A key component of their argument is that the Betti numbers of regular Hessenberg varieties for GLn(C) are palindromic. In this paper, we extend this result to all reductive algebraic groups, proving that the Betti numbers of regular Hessenberg varieties are palindromic.

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In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. These varieties arise in representation theory, algebraic geometry, and combinatorics. We give a connectedness criterion for semisimple Hessenberg varieties that generalizes a criterion given by Anderson and Tymoczko. It also generalizes results of Iveson in type A which prove that all Hessenberg varieties satisfying this criterion are connected. We then show that nilpotent Hessenberg varieties are rationally connected.

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

This manuscript is a contributed chapter in the forthcoming CRC Press volume, titled the Handbook of Combinatorial Algebraic Geometry: Subvarieties of the Flag Variety. The book, as a whole, is aimed at a diverse audience of researchers and graduate students seeking an expository introduction to the area. In our chapter, we give an overview of some of the past research on the cohomology rings of regular nilpotent Hessenberg varieties, with no claim to being exhaustive. For the purposes of this manuscript, we focus mainly on the case of Lie type A, with some brief remarks on the general Lie types. We end the chapter with a selection of topics currently active in this area.

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