algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety complex symmetric domains; p-adic fields; Lie groups; maximal compact subgroups; non-archimedean groundfields; symmetric spaces; analytic varieties; reductive group; non-archimedean local field; discrete co- compact subgroup; rigid analytic variety; Bruhat-Tits buildings; reductive linear algebraic groups; toroidal embeddings; analytic spaces; projective space; split orthogonal groups 