Appears in collection : Representation theory and analysis on homogeneous spaces / Théorie des représentations et Analyse sur les espaces homogènes
In this talk, 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 $\mathfrak{g}$. For this class of Lie algebras we give an explicit algorithm for finding its corresponding cortex.
Next we give an example of 7-dimensional (real) nilpotent Lie algebra for which the cortex is a proper semi-algebraic set in the set of the common zeros of the $G$-invariant polynomials on $\mathfrak{g}^{\star}$ namely the $I$-cortex, where $G$ is the corresponding connected and simply connected Lie group and $\mathfrak{g}^{\star}$ is its dual.
Finally we study the cortex of a class of nilpotent Lie groups given by a semi-direct product of abelian groups $G:=\mathbb{R}^m \rtimes_\pi V$ where $\pi$ is the continuous representation of $\mathbb{R}^n$ on the $m$ -dimensional (real) vector space $V$ defined by
$$
\pi\left(t_1, \ldots, t_n\right)=\exp \left(\sum_{i=1}^n t_i A_i\right)
$$
with $\left\{A_1, \ldots, A_n\right}$ is a set of pairwise commuting nilpotent matrices in $\mathbb{R}^{m \times m}$.
