algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Weil representations; cubic hypersurfaces; irreducible subgroups; cubic invariants; automorphism groups; subgroups of \(\text{PSL}_ n(\mathbb{C})\); cyclic extensions of groups; classical groups; smallest Suzuki group; complex algebraic groups 