  We consider ad-nilpotent ideals of a Borel subalgebra of a simple Lie
algebra. The goal of this paper is two-fold. First, we study the ad-nilpotent
ideals lying inside the Heisenberg ideal. The Heisenberg ideal is the nilpotent
radical of the centralizer of the highest root vector. Second, we study the
ideals having the property that the corresponding domain of the Shi arrangement
consists of a single alcove. Such ideals (and the corresponding elements of the
affine Weyl group) are called minimax.
