On the cortex of a class of exponential Lie algebras (Q2905195)

Given a locally compact group \(G,\) the set of all unitary irreducible representations of \(G\) which cannot be Hausdorff-separated from the identity representation of the group is called the cortex of \(G.\) If \(G\) is a simply connected, connected exponential solvable Lie group, then its unitary dual is parametrized by the coadjoint orbits of the group \(G\) acting on the dual of its Lie algebra \(\mathfrak{g}\). Using the technique of jump indices, and a layering method, the author was able to compute the cortex of a class of exponential solvable Lie groups by obtaining a precise parametrization of the coadjoint orbits. The author proves that if \(\mathfrak{z}\) is the center of \(\mathfrak{g},\) and if the codimension of a generic coadjoint orbit in the complement of the dual vector space of the center of the Lie algebra is either zero or one, then the cortex of \(\mathfrak{g}^{\ast}\) is in fact the set of common zeros of all \(G\)-invariant polynomials \(P\) on \(\mathfrak{g}^{\ast}\) with \(P\left( 0\right) =0.\) Also, the author deals with some non-nilpotent exponential solvable Lie group. Let \(\mathfrak{g}\) be a real exponential solvable Lie algebra, \(\mathfrak{g}^{\ast}\) its dual, \(\mathfrak{z}\) its center, \(m\) the number of elements in half of the jump indices corresponding to complex non-real roots of the complexification of \(\mathfrak{g.}\) If the generic coadjoint orbits \(\mathcal{O}\) satisfy the equation NEWLINE\[NEWLINE \dim\mathfrak{z}+\dim\mathcal{O}+m=\dim\mathfrak{g} NEWLINE\]NEWLINE then the cortex of \(\mathfrak{g}^{\ast}\) is the vector space of linear functionals vanishing on the dual of the center of the Lie algebra. Finally, the author computes many examples explicitly.