algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety character formula; irreducible representations; semisimple compact Lie group; Borel-Weil-Bott theorem; holomorphic sections; line bundle; Atiyah-Bott-Lefschetz formula; homogeneous space; coadjoint action; Loop groups 