Representations With a Reduced Null Cone

Let G be a complex reductive group and V a G-module. Let π: V \to 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 SL_2, SL_3 or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.

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