algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety homogeneous cover; ring of regular functions; simply connected semisimple complex Lie group; Lie algebra; nilpotent adjoint \(G\)-orbit; Poisson structure; semisimple Lie algebra; Heisenberg Lie algebra; minimal nilpotent orbit; flag varieties; group of holomorphic automorphisms 