Instability in Invariant Theory

Let v be a point of the representation space V of a reductive algebraic group G.The point v is called unstable if any polynomial function on V, which is invariant under G and vanishes at 0 in V, must vanish at v.There is a related notion of instability.Let λ : G m → G be a one-parameter subgroup of G.The point v is called λ-unstable if limit t→0The Hilbert-Mumford numerical criterion for unstability states that v is unstable if and only if v is λ-unstable for some one parameter subgroup λ of G.This criterion was given in Chapter 2 of 4 for linearly reductive groups.Subsequently, it has been established for arbitrary reductive groups 5,6 due to the efforts of C. S.