algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety holomorphic line bundle; projectivization; Bott towers; torus action; irreducible representation; highest weight; moment map; complex structure; flag variety; Bott-Samelson manifold; connected compact Lie group; virtual character; multiplicities; Demazure's character formulas; Schubert varieties 