algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety homogeneous vector bundles; holomorphic functions; Lie groups; complex analysis; Lie group actions; complex spaces; Lie homomorphism; Lie algebra; automorphism group; compact homogeneous complex manifolds; flag manifolds; normalizer theorem; Tits fibration; induced representations; characters; linear algebraic group; complexification; Fourier expansion; Fréchet vector spaces; geometric invariant theory 