Hessenberg Varieties for the Minimal Nilpotent Orbit

For a connected, simply-connected complex simple algebraic group G, we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we compute the Poincare polynomials and irreducible components of these varieties in Lie type A. Furthermore, we show these Hessenberg varieties to be GKM with respect to the action of a maximal torus T.G. The corresponding GKM graphs are then explicitly determined. Finally, we present the ordinary and T-equivariant cohomology rings of our varieties as quotients of those of the flag variety.