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 subset of 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.