Let G be a complex reductive group and V a G- module. Let π: V → V/G be the quotient morphism and set N(V) = π−1(π(0)). We consider the following question. Is the null cone N (V) reduced, i.e., is the ideal of N (V) generated by G- invariant polynomials? We have complete results when G is SL2, SL3 or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.
