algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety connected reductive algebraic group; cohomology groups; relative Lie algebra cohomology; Eisenstein series; Eisenstein cohomology classes; Euler products; automorphic \(L\)-functions; intertwining operators; mixed motives; Shimura varieties; cusp cohomology; Hecke algebra submodules; Anderson's mixed motives; Hodge-de Rham extension class; Beilinson- Deligne conjectures; topological trace formula 