Decomposing Hessenberg varieties over classical groups

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Hessenberg varieties are a family of subvarieties of the flag variety, including the Springer fibers, the Peterson variety, and the entire flag variety itself. The seminal example arises from a problem in numerical analysis and consists for a fixed linear operator M of the full flags V₁ V₂ >... Vₙ in GLₙ with M Vᵢ contained in Vᵢ₊₁ for all i. In this paper I show that all Hessenberg varieties in type Aₙ and semisimple and regular nilpotent Hessenberg varieties in types Bₙ,Cₙ, and Dₙ can be paved by affine spaces. Moreover, this paving is the intersection of a particular Bruhat decomposition with the Hessenberg variety. In type Aₙ, an equivalent description of the cells of the paving in terms of certain fillings of a Young diagram can be used to compute the Betti numbers of Hessenberg varieties. As an example, I show that the Poincare polynomial of the Peterson variety in Aₙ is ∑ᵢ ₌₀ⁿ⁻¹ n-1i x²ⁱ.