On the Invariant Theory of G₀ -Actions from a Geometric Perspective

In this paper we give a strict classification of G₀ -representations. This is done through the notion of a c (t) -pair. Namely if Spec (A) is a G₀ -variety with action β, then a c (t) -pair is a pair of elements (g, h) such that g (t₀ x) = g (x) +c (t₀) h (x). This allows us to describe exactly when an affine, G₀ -stable, sub-variety D (h) is a trivial bundle over D (h) //G₀. If Spec (A) is a G₀ -variety, we define the large pedestal ideal P₆ (A) and the pedestal ideal P (A). If β: G₀ GL (V) is a G₀ -representation, then we classify such a representation on whether: a) the large pedestal ideal P₆ (S₊ (V^) ) is equal to zero. b) the large pedestal ideal is non-zero, but the pedestal ideal is equal to zero. or c) the pedestal ideal is non-zero.