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