GEOMETRY OF REGULAR HESSENBERG VARIETIES

Let g be a complex semisimple Lie algebra. For a regular element x in g and a Hessenberg space H ⊆ g , we consider a regular Hessenberg variety X ( x, H ) in the ag variety associated with g . We take a Hessenberg space so that X ( x, H ) is irreducible, and show that the higher cohomology groups of the structure sheaf of X ( x, H ) vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.