algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety conjugacy classes; representation theory of finite groups of Lie type; connected reductive algebraic group; Frobenius morphism; finite group of F-fixed points; BN-pair; Weyl group; Borel subgroup; maximal torus; intersections of parabolic subgroups; virtual representation; generalized character; \(\ell \)-adic cohomology; flag variety; geometric conjugacy; Jordan decomposition; unipotent elements; Deligne-Lusztig characters; irreducible characters; regular characters; Gelfand-Graev representation; cuspidal characters; unipotent characters; intersection cohomology; Springer correspondence; representations of Coxeter groups; Dynkin diagrams; bibliography; exceptional groups; character tables 