algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety infinite dimensional representation; reductive group; geometry of flag manifold; fixed point theorem; Weyl's character formula; cohomology group; equivariant line bundle; real semisimple group; invariant eigendistribution; flag variety; Harish-Chandra modules; K-equivariant sheaves 