Invariants and moduli revisited: the case of a single root

What is the correct combinatorial object to encode a linear representation?  Many shadows of this problem have been studied:moment polytopes, Duistermaat-Heckman measures, Okounkov bodies.  We suggest that already in very simple cases these miss a crucial feature.  The ring theory, as opposed to just the linear algebra, of the group action on the coordinate ring, depends on some non-trivial lattice geometry and an associated filtration.  Some striking similarities to, and key differences from, the theory of toric varieties ensue.  Finite and non-finite generation phenomena emerge naturally.  We discuss motivations from, and applications to, questions in the effective geometry of moduli of curves.