algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety simply connected nilpotent Lie group; Lie algebras; irreducible unitary representations; induced representation; restricted representation; group actions; decompositions; Kirillov correspondence; multiplicity; Pukanszky's parametrization; unipotent action; semi-algebraic sets 