algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety vanishing theorems; Frobenius splitting; Schubert varieties; Demazure's character formula; semi-simple algebraic group; flag variety; Weyl group; reductive groups; desingularization; dual Joseph modules; excellent filtration; good filtrations; dual Weyl modules; tensor product; restriction to parabolic subgroups; line bundle; Schubert filtrations 