Note on the cortex of two-step nilpotent Lie algebras

In this paper, we construct an example of a family of $4d$-dimensional two-step nilpotent Lie algebras $(\frak g_d)_{d\geq 2}$ so that the cortex of the dual of each $\frak g_d$ is a projective algebraic set. More precisely, we show that the cortex of each dual $\frak g_d^*$ of $\frak g_d$ is the zero set of a homogeneous polynomial of degree $d$. This example is a generalization of one given in "Irreducible representations of locally compact groups that cannot be Hausdorff separated from the identity representation" by "{\sc M.E.B. Bekka, and E. Kaniuth}".