Infinitely small orbits in two-step nilpotent Lie algebras

In this paper, we consider the open question: is the cortex of the dual of a nilpotent Lie algebra an algebraic set? We give a partial answer by considering the class of two-step nilpotent Lie algebra g. For this class of Lie algebras we give an explicit algorithm for finding its corresponding cortex. By the way, we prove that either the cortex coincides with the zero set of invariant homogeneous polynomials and in this case is z⊥ where z denotes the center of g, or it is a proper projective algebraic subset of z⊥. Finally we materialize our algorithm on examples.